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(metastable)f(states)h(lik)n(e)f(sup)r(erco)r(oled)g(liquids,)456 3186 y(sup)r(ersaturated)26 b(v)-5 b(ap)r(ors)26 b(and)h(solutions,)g (or)f(ferromagnets)g(with)h(magnetization)g(opp)r(osite)456 3286 y(to)k(the)h(\014eld.)50 b(The)32 b(phenomenon)g(of)f (metastabilit)n(y)h(o)r(ccurs)f(in)h(thermo)r(dynamic)f(systems)456 3385 y(close)c(to)g(a)h(\014rst)g(order)e(phase)h(transition.)38 b(A)28 b(system)f(is)h(initially)g(prepared)f(in)h(an)g(equilib-)456 3485 y(rium)j(pure)h(phase,)g(and)g(then)h(the)f(thermo)r(dynamic)f (parameters)f(are)h(c)n(hanged)g(to)h(v)-5 b(alues)456 3584 y(for)33 b(whic)n(h)h(there)g(is)g(a)g(di\013eren)n(t)g (equilibrium)g(pure)g(phase.)56 b(Under)34 b(suitable)g(conditions,)456 3684 y(the)k(system)g(do)r(es)f(not)h(undergo)f(a)g(phase)h (transition,)i(but)e(it)h(remains)e(in)h(an)g(apparen)n(t)456 3784 y(equilibrium,)g(the)f Fu(metastable)f(state)p Fw(,)j(whic)n(h)d (is)h(v)n(ery)e(similar)h(to)g(the)h(initial)f(one.)64 b(This)456 3883 y(state)33 b(p)r(ersists)g(un)n(til)h(some)f(\(ev)n(en) h(sligh)n(t\))f(disturbance)h(leads)f(the)h(system)f(to)h(the)g(stable) 456 3983 y(equilibrium.)59 b(F)-7 b(or)35 b(instance)g(w)n(e)g(can)f (prepare)g(a)h(glass)f(of)h(w)n(ater)f(in)i(an)e(en)n(vironmen)n(t)h (at)456 4083 y(temp)r(erature)29 b(sligh)n(tly)g(b)r(elo)n(w)g(zero)g (degrees)f(cen)n(tigrade.)41 b(The)30 b(glass)e(lo)r(oks)h(apparen)n (tly)f(in)456 4182 y(an)d(equilibrium)h(liquid)f(phase,)h(and)f(this)h (state)f(ma)n(y)g(last)g(for)h(a)f(long)f(time;)j(but)f(ev)n(en)n (tually)456 4282 y(an)h(irrev)n(ersible)e(pro)r(cess)i(tak)n(es)f (place)i(and)f(the)h(glass)e(of)i(w)n(ater)e(suddenly)i(freezes.)555 4381 y(A)38 b(\014rst)e(theoretical)g(explanation)g(of)h(metastabilit)n (y)g(go)r(es)f(bac)n(k)g(to)h(the)g(classical)f(v)-5 b(an)456 4481 y(der)40 b(W)-7 b(aals)40 b(Maxw)n(ell)f(theory)h(of)g (liquid-v)-5 b(ap)r(or)40 b(phase)g(transition.)75 b(In)41 b(this)f(theory)g(the)456 4581 y(notion)29 b(of)h(metastable)g(states)f (emerges)g(in)h(a)f(natural)g(w)n(a)n(y)-7 b(,)30 b(as)f(describing)g (homogeneous,)456 4680 y(almost)h(equilibrium)i(phases)f(whic)n(h)g(ho) n(w)n(ev)n(er)f(ha)n(v)n(e)g(higher)h(free)g(energy)f(with)i(resp)r (ect)g(to)456 4780 y(the)c(corresp)r(onding)d(stable)j(equilibria.)555 4880 y(As)f(a)f(matter)g(of)g(fact,)h(the)g(app)r(earance)d(of)j (metastable)f(states)f(is)i(a)f(common)f(feature)i(of)456 4979 y(all)c(the)h(microscopic)e Fu(mean)i(\014eld)g(theories)f Fw(of)g(phase)h(transition.)34 b(In)24 b(fact,)h(due)f(to)g(the)g Fu(mean)p 456 5053 499 4 v 461 5145 a Ft(2000)j(AMS)e(Subje)l(ct)g (Classi\014c)l(ation.)33 b Fy(35Q99,)24 b(47J05,)h(82C24.)456 5242 y Ft(Key)f(wor)l(ds)k(and)e(phr)l(ases.)33 b Fy(In)n(terfaces,)25 b(critical)e(droplet,)h(phase)g(transition,)g(unstable)g(manifold.)1933 5315 y Fs(1)p eop %%Page: 2 2 2 1 bop 456 251 a Fs(2)756 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fu(\014eld)36 b(appro)n(ximation)p Fw(,)h(the)f(range)f(of)h(the)h(in)n(teraction)e(b)r(et)n(w)n(een)h (the)h(particles)e(coincides)456 550 y(with)27 b(the)g(macroscopic)e (size)h(of)h(the)g(system,)f(and)h(this)g(fact)g(implies)g(the)g(non)f (con)n(v)n(exit)n(y)g(of)456 649 y(the)i(free)f(energy)-7 b(,)27 b(th)n(us)g(allo)n(wing)f(metastable)i(branc)n(hes)e(in)i(the)g (phase)f(diagram.)555 749 y(Ho)n(w)n(ev)n(er,)37 b(the)f(mean)g (\014eld)h(appro)n(ximation)d(is)i(unrealistic,)i(and)e(in)g(fact)h(it) f(pro)r(duces)456 849 y(nonph)n(ysical)25 b(features)i(lik)n(e)f (thermo)r(dynamic)h(instabilities)g(and)g(the)g(\(already)f(men)n (tioned\))456 948 y(non)h(con)n(v)n(exit)n(y)f(of)i(the)g(free)f (energy)-7 b(.)555 1048 y(A)27 b(more)f(realistic)f(mean)h(\014eld)h (appro)n(ximation,)e(called)h(the)h Fu(lo)r(cal)f(mean)g(\014eld)h (limit)p Fw(,)g(has)456 1147 y(b)r(een)i(in)n(tro)r(duced)f(in)h(the)g (60's)f(b)n(y)g(Kac,)h(Uhlen)n(b)r(ec)n(k,)g(and)f(Hemmer,)h([15)o(].) 41 b(They)29 b(consider)456 1247 y(in)n(teractions,)41 b(usually)e(called)g Fu(Kac)f(p)r(oten)n(tials)p Fw(,)43 b(that)c(dep)r(end)h(on)f(a)h(scaling)e(parameter)456 1347 y Fr(\015)27 b(>)c Fw(0,)j(studying)g(the)g(limit)h Fr(\015)h Fq(#)22 b Fw(0)k(where)f(the)i(range)d(of)i(the)h(in)n (teraction)e(b)r(ecomes)h(in\014nite.)456 1446 y(This)i(program)e(has)i (b)r(een)g(carried)f(out)i(b)n(y)e(Kac,)h(Uhlen)n(b)r(ec)n(k)g(and)g (Hemmer,)h([15)o(],)g(in)f(some)456 1546 y(particular)33 b(mo)r(dels,)j(and)e(then)h(b)n(y)f(Leb)r(o)n(witz)g(and)g(P)n(enrose)e (in)j(a)f(more)f(general)g(class)h(of)456 1646 y(systems,)28 b([16)o(].)41 b(These)29 b(results)f(pro)n(v)n(e)f(that)i(the)h(phase)e (diagram)f(con)n(v)n(erges,)g(for)h(an)n(y)g(tem-)456 1745 y(p)r(erature,)23 b(to)h(the)g(v)-5 b(an)23 b(der)h(W)-7 b(aals)22 b(phase)i(diagram,)e(comprehensiv)n(e)h(of)g(the)h(Maxw)n (ell)f(equal)456 1845 y(area)j(rule.)555 1944 y(Because)37 b(of)h(the)g(dynamical)f(nature)g(of)h(the)g(phenomenon,)i(a)d(fully)i (satisfactory)d(de-)456 2044 y(scription)31 b(of)h(metastabilit)n(y)f (can)h(b)r(e)g(found)g(only)g(in)g(the)g(framew)n(ork)e(of)i(non)f (equilibrium)456 2144 y(statistical)i(mec)n(hanics.)55 b(In)34 b(this)h(setting)e(the)i(\014rst)f(rigorous)d(approac)n(h)h(to) i(metastabilit)n(y)456 2243 y(go)r(es)25 b(bac)n(k)h(to)g(Leb)r(o)n (witz)g(and)h(P)n(enrose,)d([17],)i(who)h(giv)n(e)e(a)h(general)f (metho)r(d)i(for)f(describing)456 2343 y(metastable)j(states.)44 b(Moreo)n(v)n(er,)27 b(they)j(apply)g(this)g(metho)r(d)h(to)f(systems)f (with)h(Kac)f(p)r(oten-)456 2443 y(tials,)d(giving)g(a)g(rigorous)f (justi\014cation)h(\(for)h(what)f(concerns)g(the)h(static)f(prop)r (erties\))g(of)h(the)456 2542 y(v)-5 b(an)27 b(der)g(W)-7 b(aals)27 b(description)g(of)h(metastable)f(states.)555 2642 y(In)35 b(more)f(recen)n(t)g(y)n(ears,)h([8],)h(Ising)f(spin)g (systems)f(with)h(Glaub)r(er)g(dynamics)f(and)g(Kac)456 2742 y(p)r(oten)n(tials)h(ha)n(v)n(e)f(b)r(een)h(in)n(tro)r(duced)g(in) h(order)e(to)h(analyze)f(non)h(equilibrium)h(phenomena)456 2841 y(lik)n(e)f(phase)h(separation)e(and)i(in)n(terface)f(dynamics.)62 b(In)36 b(this)g(mo)r(del)g(the)h(mean)e(\014eld)i(free)456 2941 y(energy)26 b(densit)n(y)h(is)h(giv)n(en)f(b)n(y)1097 3118 y Fr(F)12 b Fw(\()p Fr(s)p Fw(\))84 b(=)e Fq(\000)1585 3062 y Fw(1)p 1585 3099 42 4 v 1585 3175 a(2)1636 3118 y Fr(s)1675 3084 y Fs(2)1731 3118 y Fq(\000)18 b Fr(hs)g Fq(\000)g Fr(\014)2053 3084 y Fp(\000)p Fs(1)2143 3118 y Fr(i)p Fw(\()p Fr(s)p Fw(\))p Fr(;)97 b(s)23 b Fq(2)g Fw([)p Fq(\000)p Fw(1)p Fr(;)14 b Fw(1])p Fr(;)484 b Fw(\(1.1\))1134 3318 y Fr(i)p Fw(\()p Fr(s)p Fw(\))83 b(=)f Fq(\000)1585 3262 y Fw(1)18 b(+)g Fr(s)p 1585 3299 182 4 v 1655 3375 a Fw(2)1791 3318 y(log)1922 3262 y(1)g(+)g Fr(s)p 1922 3299 V 1992 3375 a Fw(2)2132 3318 y Fq(\000)2225 3262 y Fw(1)g Fq(\000)g Fr(s)p 2225 3299 V 2295 3375 a Fw(2)2431 3318 y(log)2562 3262 y(1)g Fq(\000)g Fr(s)p 2562 3299 V 2632 3375 a Fw(2)2753 3318 y Fr(;)498 b Fw(\(1.2\))456 3484 y(where)23 b Fr(\014)29 b Fw(is)24 b(the)g(in)n(v)n(erse)f(temp)r (erature,)i Fr(s)f Fw(the)g(a)n(v)n(erage)d(magnetization,)j(and)g Fr(h)g Fw(an)g(external)456 3584 y(p)r(ositiv)n(e)31 b(magnetic)f(\014eld.)49 b(The)31 b(quadratic)f(term)i Fq(\000)p Fr(s)2205 3554 y Fs(2)2241 3584 y Fr(=)p Fw(2)f(is)g(the)h (in)n(ternal)e(energy)h(densit)n(y)-7 b(,)456 3684 y Fq(\000)p Fr(hs)25 b Fw(the)h(energy)e(densit)n(y)i(due)g(to)g(the)g (\014eld)g Fr(h)p Fw(,)g(and)f Fr(i)p Fw(\()p Fr(s)p Fw(\))h(the)h(en)n(trop)n(y)-7 b(.)35 b(The)25 b(critical)h(p)r(oin)n (ts)456 3783 y(of)h Fr(F)12 b Fw(\()p Fr(s)p Fw(\))28 b(are)f(the)h(solutions)f(of)g(the)h(mean)g(\014eld)f(equation:)1596 3926 y Fr(s)c Fw(=)g(tanh)p Fq(f)p Fr(\014)t Fw([)p Fr(s)18 b Fw(+)g Fr(h)p Fw(])p Fq(g)p Fr(:)970 b Fw(\(1.3\))555 4069 y(Giv)n(en)26 b Fr(\014)i(>)22 b Fw(1,)k(there)g(is)g(an)g Fr(h)1502 4081 y Fo(\014)1569 4069 y Fr(>)d Fw(0)j(suc)n(h)f(that)i (for)e Fr(h)e Fq(2)h Fw([0)p Fr(;)14 b(h)2514 4081 y Fo(\014)2558 4069 y Fw(])26 b(Eq.)f(\(1.3\))h(has)g(three)g(and)456 4169 y(only)h(three)g(di\013eren)n(t)h(ro)r(ots,)f(denoted)g(b)n(y)1487 4312 y Fr(m)1560 4276 y Fp(\000)1560 4337 y Fo(\014)s(;h)1687 4312 y Fr(<)22 b(m)1847 4278 y Fs(0)1847 4332 y Fo(\014)s(;h)1974 4312 y Fq(\024)g Fw(0)h Fr(<)f(m)2286 4276 y Fs(+)2286 4337 y Fo(\014)s(;h)2390 4312 y Fr(:)861 b Fw(\(1.4\))456 4471 y(F)-7 b(or)27 b Fr(h)c(>)f Fw(0,)27 b Fq(j)p Fr(m)951 4436 y Fp(\000)951 4496 y Fo(\014)s(;h)1055 4471 y Fq(j)c Fr(<)g(m)1262 4436 y Fs(+)1262 4496 y Fo(\014)s(;h)1393 4471 y Fw(and)k Fr(m)1627 4441 y Fs(0)1627 4495 y Fo(\014)s(;h)1753 4471 y Fr(<)c Fw(0;)k(for)g Fr(h)c Fw(=)g(0,)k Fr(m)2384 4441 y Fs(0)2384 4495 y Fo(\014)s(;h)2511 4471 y Fw(=)22 b(0)27 b(and)h Fr(m)2902 4436 y Fs(+)2902 4496 y Fo(\014)s(;h)3028 4471 y Fw(=)23 b Fq(\000)p Fr(m)3254 4483 y Fo(\014)s(;h)3401 4424 y Fr(:)3380 4471 y Fw(=)456 4584 y Fr(m)529 4596 y Fo(\014)573 4584 y Fw(.)38 b(The)28 b(t)n(w)n(o)f(phases)h Fq(\006)p Fr(m)1365 4596 y Fo(\014)1437 4584 y Fw(are)f(thermo)r (dynamically)g(stable)g(at)h Fr(h)c Fw(=)f(0,)28 b(while)g Fr(m)3188 4554 y Fs(0)3188 4607 y Fo(\014)s(;h)3315 4584 y Fw(=)23 b(0)456 4698 y(is)29 b(unstable.)43 b(F)-7 b(or)29 b Fr(h)d(>)g Fw(0,)k Fr(m)1396 4663 y Fs(+)1396 4723 y Fo(\014)s(;h)1529 4698 y Fw(is)f(the)h(only)f(stable)g(phase,)h Fr(m)2515 4668 y Fs(0)2515 4722 y Fo(\014)s(;h)2648 4698 y Fw(is)f(still)h(unstable,)g(while)456 4814 y Fr(m)529 4779 y Fp(\000)529 4839 y Fo(\014)s(;h)660 4814 y Fw(b)r(ecomes)d (metastable.)555 4917 y(Ho)n(w)n(ev)n(er)19 b(these)i(are)f(static)g (considerations,)h(the)g(dynamics)f(of)h(p)r(ersistence)f(and)h(deca)n (y)f(of)456 5016 y(the)j(metastable)f(state)g(can)g(b)r(e)h(in)n(v)n (estigated)f(only)g(going)f(bac)n(k)h(to)h(the)g(sto)r(c)n(hastic)e(ev) n(olution)456 5116 y(of)k(the)g(underlying)f(spin)h(system,)h(b)n(y)f (c)n(haracterizing)d(the)j(tunneling)h(from)f(the)g(metastable)456 5216 y(to)i(the)h(stable)f(phase.)p eop %%Page: 3 3 3 2 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)291 b(3)555 450 y Fw(According)18 b(to)h(general)f(heuristic)h (argumen)n(ts)f(w)n(e)h(exp)r(ect)g(the)h(transition)e(o)r(ccurs)g (through)456 550 y(the)34 b Fu(n)n(ucleation)f Fw(of)g(a)h(su\016cien)n (tly)f(large)f(droplet)i(of)f(the)h(stable)g(phase,)h(whic)n(h)e(will)h (start)456 649 y(to)27 b(gro)n(w)f(undergoing)g(an)h(irrev)n(ersible)e (pro)r(cess)h(leading)h(to)g(the)h(stable)f(phase)g(ev)n(erywhere.)456 749 y(On)g(the)g(con)n(trary)-7 b(,)26 b(small)h(droplets)f(will)i(ha)n (v)n(e)e(a)h(tendency)h(to)f(shrink.)36 b(This)27 b(arguing)f(leads)456 849 y(to)d(b)r(eliev)n(e)h(that)g(the)g(transition)f(o)r(ccurs)g (through)g(the)h(formation)f(of)g(a)h(w)n(ell)f(de\014ned)h Fu(critical)456 948 y(droplet)p Fw(,)j(whic)n(h)g(breaks)g(the)h (spatial)f(homogeneit)n(y)f(in)i(the)g(metastable)f(state.)555 1048 y(A)e(sp)r(eci\014c)g(feature)f(of)g(sto)r(c)n(hastic)g(dynamics)g (with)h(Kac)f(p)r(oten)n(tials)g(is)g(its)h(almost)f(deter-)456 1147 y(ministic)31 b(b)r(eha)n(vior)e(for)h(small)g(v)-5 b(alues)30 b(of)g(the)h(scaling)e(parameter)g Fr(\015)35 b Fw(\(i.e.)c(when)g(the)g(range)456 1247 y(of)24 b(the)g(in)n (teraction)f(is)h(large\).)34 b(In)24 b(fact)g(in)h([8)o(])f(it)h(is)f (sho)n(wn)f(that)h(in)g(the)h(con)n(tin)n(uum)e(limit)i(the)456 1347 y(Ising)30 b(spin)i(system)f(with)h(Glaub)r(er)f(dynamics)g(and)g (Kac)f(p)r(oten)n(tials)h(giv)n(es)f(rise)h(to)g(a)g(lo)r(cal)456 1446 y(magnetization)f(densit)n(y)i Fr(m)e Fw(=)f Fr(m)p Fw(\()p Fr(t;)14 b(x)p Fw(\),)34 b(\()p Fr(t;)14 b(x)p Fw(\))30 b Fq(2)h Fn(R)2143 1458 y Fs(+)2225 1446 y Fq(\002)21 b Fn(R)2365 1416 y Fo(d)2409 1446 y Fw(,)33 b(whic)n(h)f(ev)n(olv)n(es) e(according)g(to)456 1546 y(the)e(non)f(lo)r(cal)g(ev)n(olution)g (equation:)1352 1687 y Fr(@)5 b(m)p 1352 1724 122 4 v 1374 1800 a(@)g(t)1507 1743 y Fw(=)23 b Fq(\000)p Fr(m)17 b Fw(+)h(tanh)q Fq(f)p Fr(\014)t Fw([)p Fr(J)26 b Fq(\003)18 b Fr(m)g Fw(+)g Fr(h)p Fw(])p Fq(g)p Fr(;)716 b Fw(\(1.5\))456 1930 y(where)18 b Fr(J)27 b Fw(is)19 b(a)f(non)h(negativ)n(e,)g(ev)n (en)g(function)g(whic)n(h)g(is)g(related)f(to)h(the)g(\(long)f(range\)) g(coupling)456 2029 y(of)28 b(the)i(spin-spin)e(in)n(teraction,)h(and)f Fr(J)f Fq(\003)19 b Fr(m)29 b Fw(denotes)g(the)g(con)n(v)n(olution)e(b) r(et)n(w)n(een)i Fr(J)37 b Fw(and)28 b Fr(m)p Fw(.)456 2129 y(In)f(agreemen)n(t)g(with)h(\(1.1\))f(w)n(e)g(also)g(assume)g (that)h Fr(J)35 b Fw(is)28 b(normalized)e(so)h(that)1686 2217 y Fm(Z)1769 2330 y Fr(dx)14 b(J)8 b Fw(\()p Fr(x)p Fw(\))24 b(=)f(1)p Fr(;)1059 b Fw(\(1.6\))456 2531 y(whic)n(h)25 b(implies)h(the)g(homogeneous)e(stationary)g(solutions)h(of)32 b(\(1.5\))26 b(to)f(coincide)h(with)g(those)456 2631 y(of)34 b(\(1.3\))o(.)555 2730 y(This)c(pap)r(er)f(is)g(concerned)f (with)i(the)g(metastable)f(b)r(eha)n(vior)f(of)h(the)h(deterministic)g (ev)n(o-)456 2830 y(lution)i(equation)f(\(1.5\))g(in)h(dimension)g Fr(d)f Fw(=)e(1.)49 b(W)-7 b(e)32 b(pro)n(v)n(e)f(the)h(existence)f(of) h(an)g(in)n(v)-5 b(arian)n(t)456 2929 y(manifold)28 b(connecting)f(the) i(stable)f(and)g(metastable)f(phases.)38 b(This)28 b(manifold)g(is)g (the)g(union)456 3029 y(of)k(t)n(w)n(o)g(hetero)r(clinic)g(orbits,)h (connecting)f(the)h(ab)r(o)n(v)n(e)f(phases)f(with)j(a)e (distinguished,)i(non)456 3129 y(homogeneous,)18 b(stationary)g (solution)g(of)25 b(\(1.5\),)c(whic)n(h)e(here)f(represen)n(ts)f(the)j (critical)e(droplet.)555 3228 y(Our)41 b(ultimate)h(purp)r(ose)f(is)h (the)g(complete)f(description)g(of)h(the)g(tunneling)g(from)f(the)456 3328 y(metastable)20 b(to)g(the)h(stable)f(phase)g(for)g(the)h (underlying)f(sto)r(c)n(hastic)g(spin)g(dynamics.)35 b(Accord-)456 3428 y(ing)d(to)g(the)h(path)n(wise)e(approac)n(h)g(to)h (metastabilit)n(y)g(in)h(the)f(case)g(of)g(rev)n(ersible)f(dynamics,) 456 3527 y([4)o(,)g(13)o(,)h(18)o(],)g(the)g(presen)n(t)e(analysis)g (will)h(pla)n(y)g(a)g(cen)n(tral)f(role)g(in)h(determining)g(the)h(t)n (ypical)456 3627 y(path)27 b(of)h(the)g(tunneling)g(transition,)f(whic) n(h)g(will)h(b)r(e)g(the)g(sub)5 b(ject)28 b(of)f(future)h(w)n(orks.) 555 3726 y(In)34 b([7)o(])g(it)f(is)g(pro)n(v)n(ed)f(the)i(existence,)g (for)f Fr(\014)j(>)c Fw(1)h(and)g Fr(h)g Fw(small)g(enough,)h(of)f(the) h(critical)456 3826 y(droplet,)23 b(also)g(called)g(the)g Fu(bump)p Fw(,)i(i.e.)f(a)f(spatially)f(non)h(homogeneous,)g(symmetric) g(solution)456 3926 y Fr(q)30 b Fw(of)e(the)g(non)f(lo)r(cal,)g(one)h (dimensional)f(equation:)1179 4084 y Fr(q)s Fw(\()p Fr(x)p Fw(\))d(=)e(tanh)q Fq(f)p Fr(\014)t Fw([\()p Fr(J)k Fq(\003)18 b Fr(q)s Fw(\)\()p Fr(x)p Fw(\))i(+)e Fr(h)p Fw(])p Fq(g)p Fr(;)179 b(x)24 b Fq(2)f Fn(R)p Fr(;)559 b Fw(\(1.7\))456 4243 y(with)28 b(asymptotic)f(conditions)1654 4367 y(lim)1607 4424 y Fp(j)p Fo(x)p Fp(j!1)1831 4367 y Fr(q)s Fw(\()p Fr(x)p Fw(\))d(=)e Fr(m)2166 4331 y Fp(\000)2166 4392 y Fo(\014)s(;h)2270 4367 y Fr(;)981 b Fw(\(1.8\))456 4559 y(whic)n(h)30 b(is)g(therefore)g(a)f(stationary)g(solution)h(of)37 b(\(1.5\))o(.)46 b(When)30 b Fr(h)h Fw(is)f(v)n(ery)f(small)h(the)h (region)456 4658 y(where)38 b Fr(q)k Fw(is)d(close)f(to)h(the)g(stable) g(phase)f(is)h(of)g(order)f Fq(j)14 b Fw(log)g Fr(h)p Fq(j)p Fw(,)42 b(while)d(the)g(length)g(of)g(the)456 4758 y(transition)24 b(la)n(y)n(ers)f(from)h(this)i(region)d(to)i(the)h (one)e(where)h(the)g(metastable)g(phase)f(dominates)456 4858 y(is)36 b(of)h(order)f(1.)64 b(The)37 b(la)n(y)n(ers)e(ma)n(y)h(b) r(e)h(describ)r(ed)g(appro)n(ximately)e(as)h(standing)h(w)n(a)n(v)n(es) e(of)456 4957 y(\(1.5\))29 b(with)g Fr(h)d Fw(=)f(0.)42 b(These)29 b(are)f(translations)g(and/or)f(re\015ections)i(of)g(the)h Fu(instan)n(ton)44 b Fw(\026)-58 b Fr(m)p Fw(\()p Fr(x)p Fw(\),)456 5057 y(a)27 b(strictly)g(increasing)f(and)i(an)n (tisymmetric)f(function)h(solving:)1259 5216 y(\026)-57 b Fr(m)o Fw(\()p Fr(x)p Fw(\))25 b(=)d(tanh)p Fq(f)p Fr(\014)t Fw(\()p Fr(J)27 b Fq(\003)34 b Fw(\026)-58 b Fr(m)p Fw(\)\()p Fr(x)p Fw(\))p Fq(g)p Fr(;)180 b(x)24 b Fq(2)f Fn(R)p Fr(;)624 b Fw(\(1.9\))p eop %%Page: 4 4 4 3 bop 456 251 a Fs(4)756 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fw(with)28 b(asymptotic)f(conditions)1635 586 y(lim)1582 635 y Fo(x)p Fp(!\0061)1833 586 y Fw(\026)-58 b Fr(m)p Fw(\()p Fr(x)p Fw(\))24 b(=)f Fq(\006)p Fr(m)2251 598 y Fo(\014)2295 586 y Fr(:)914 b Fw(\(1.10\))555 753 y(All)33 b(the)g(translations)d(of)j Fr(q)i Fw(are)c(stationary)g (solutions,)i(and)f(it)h(is)f(not)g(kno)n(wn)g(whether)456 852 y(these)c(are)f(the)i(only)f(\(spatially)g(non)g(homogeneous\))f (solutions)h(of)35 b(\(1.7\))28 b(with)h(asymptotic)456 952 y(conditions)22 b(\(1.8\).)36 b(In)23 b([7])g(the)h(existence)f(of) g(the)h(bump)g(is)f(pro)n(v)n(ed)f(b)n(y)h(applying)g(the)g(Newton)456 1051 y(metho)r(d)28 b(to)g(\014nd)g(the)g(zeros)e(of)i(the)g(map)g (de\014ned)g(\(on)g(the)g(space)f(of)h(con)n(tin)n(uous)e(and)i(sym-) 456 1151 y(metric)d(functions\))h(b)n(y)f(the)h(r.h.s.)f(of)32 b(\(1.5\).)k(Consequence)25 b(of)g(this)h(approac)n(h)e(is)h(the)h(lac) n(k)f(of)456 1251 y(an)n(y)h(uniqueness)i(result.)555 1350 y(W)-7 b(e)22 b(partially)f(\014ll)h(up)f(this)h(gap)f(here)g(b)n (y)g(giving)g(a)g(di\013eren)n(t)h(pro)r(of)f(whic)n(h)g(is)h(also)e (supplied)456 1450 y(with)25 b(a)f(lo)r(cal)f(uniqueness)i(result)f (\(in)h(the)g(space)e(of)i(symmetric)f(functions,)h(see)f(b)r(elo)n (w\).)36 b(W)-7 b(e)456 1550 y(also)32 b(giv)n(e)h(a)h(detailed)f (description)h(of)f(the)i(spatial)e(structure)g(of)h(the)g(bump,)i(b)n (y)e(sho)n(wing)456 1649 y(it)e(is)f(a)g(strictly)h(decreasing)e (function)i(for)f Fr(x)f(>)g Fw(0,)i(con)n(v)n(erging)d(exp)r(onen)n (tially)i(fast)h(to)f(the)456 1749 y(metastable)c(phase)g(as)g Fq(j)p Fr(x)p Fq(j)c(!)h Fw(+)p Fq(1)p Fw(.)555 1848 y(The)i(existence)g(of)g(an)f(in)n(v)-5 b(arian)n(t,)25 b(one)h(dimensional,)g(unstable)f(manifold)h Fq(W)33 b Fw(through)25 b Fr(q)s Fw(,)456 1948 y(follo)n(ws)k(in)h(a)g (standard)f(w)n(a)n(y)g(\(see)h(for)f(instance)h([14)o(]\))h(from)f (the)g(existence,)h(pro)n(v)n(ed)d(in)j([6)o(],)456 2048 y(of)i(an)f(isolated,)i(simple,)h(p)r(ositiv)n(e)d(eigen)n(v)-5 b(alue)32 b(of)h(the)h(op)r(erator)d(obtained)i(b)n(y)g(linearizing)456 2147 y(\(1.7\))39 b(around)g(the)h(critical)f(droplet)g Fr(q)s Fw(.)74 b(Nev)n(ertheless)38 b(w)n(e)i(giv)n(e)e(in)i(Sect.)g(6) g(an)f(explicit)456 2247 y(construction)33 b(needed)i(to)f(establish)g (the)g(other)g(results.)57 b(This)34 b(manifold)g(consists)g(of)g(the) 456 2347 y(union)22 b(of)h(t)n(w)n(o)f(branc)n(hes)f Fq(W)1344 2359 y Fp(\006)1400 2347 y Fw(.)35 b(The)23 b(p)r(oin)n(ts)g(on)f Fq(W)2061 2359 y Fp(\000)2140 2347 y Fw(\(resp.)g Fq(W)2448 2359 y Fs(+)2503 2347 y Fw(\))h(are)f (symmetric)g(functions,)456 2446 y(non)i(increasing)g(for)g Fr(x)f(>)g Fw(0,)i(strictly)g(smaller)f(\(resp.)g(larger\))f(than)i Fr(q)s Fw(.)36 b(W)-7 b(e)25 b(th)n(us)g(refer)f(to)h(the)456 2546 y(p)r(oin)n(ts)k(on)f Fq(W)905 2558 y Fp(\000)990 2546 y Fw(\(resp.)h Fq(W)1305 2558 y Fs(+)1360 2546 y Fw(\))h(as)e(the)h(sub-critical)g(\(resp.)f(sup)r(er-critical\))g (droplets.)41 b(In)29 b(the)456 2645 y(branc)n(h)23 b Fq(W)809 2657 y Fp(\000)890 2645 y Fw(where)h(the)h(length)g(of)f(the)h (droplets)f(is)h(shorter)e(than)i(that)f(of)h Fr(q)s Fw(,)g(the)g(ev)n(olution)456 2745 y(shrinks)18 b(it)h(further,)i (while)e(it)g(gro)n(ws)e(if)i(it)g(is)g(larger.)32 b(This)19 b(shrinking)f(\(resp.)h(gro)n(wing\))e(pro)r(cess)456 2845 y(go)r(es)24 b(on)h(inde\014nitely)h(and)g(w)n(e)f(actually)f(pro) n(v)n(e)g(the)i(branc)n(h)e Fq(W)2488 2857 y Fp(\000)2570 2845 y Fw(\(resp.)h Fq(W)2881 2857 y Fs(+)2936 2845 y Fw(\))h(connects)f(the)456 2944 y(bump)k(with)g(the)f(metastable)g (\(resp.)g(stable\))h(phase.)38 b(W)-7 b(e)29 b(pro)n(v)n(e)e(this)i (last)f(result)g(b)n(y)g(using)456 3044 y(the)35 b(comparison)e (theorem)h(whic)n(h)h(holds)f(for)g(Eq.)g(\(1.5\).)58 b(W)-7 b(e)35 b(construct)g(suitable)f(upp)r(er)456 3144 y(solutions)c(for)g(the)h(sub-critical)f(droplet)g(and)h(lo)n(w)n(er)e (solutions)h(for)g(the)h(sup)r(er-critical)f(one)456 3243 y(that)39 b(con)n(v)n(erge)f(to)h(the)h(metastable,)i(resp)r (ectiv)n(ely)d(stable,)k(phase)c(as)g Fr(t)g Fw(div)n(erges.)72 b(This)456 3343 y(metho)r(d)28 b(do)r(es)g(not)g(allo)n(w)f(to)h(catc)n (h)f(the)h(actual)g(sp)r(eed)g(of)g(con)n(v)n(ergence)e(to)h(the)i (metastable)456 3442 y(and)e(stable)g(phases,)g(whic)n(h)h(will)g(b)r (e)g(the)g(sub)5 b(ject)27 b(of)h(a)f(future)h(w)n(ork.)555 3542 y(Ho)n(w)n(ev)n(er)h(w)n(e)i(ha)n(v)n(e)f(a)h(detailed)g (description)g(of)g(large)f(part)g(of)h(the)h(relaxation)e(pro)r(cess.) 456 3642 y(The)k(situation)h(is)f(in)h(fact)g(similar)f(to)g(that)h(of) g(the)g(in)n(v)-5 b(arian)n(t)33 b(manifolds)i(for)f(metastable)456 3741 y(patterns)27 b(in)h(solutions)f(of)h(the)g(singularly)f(p)r (erturb)r(ed)h(Ginzburg-Landau)e(equation,)h([2,)h(3)o(,)456 3844 y(12)o(].)61 b(W)-7 b(e)36 b(sho)n(w)f(that,)j(for)d Fr(h)g Fw(v)n(ery)g(small,)i(a)e(large)f(part,)p 2385 3777 89 4 v 38 w Fq(W)6 b Fw(,)38 b(of)e(the)g(unstable)f(manifold)456 3943 y Fq(W)42 b Fw(consists)35 b(of)h(droplets)f(with)h(t)n(w)n(o)f(w) n(ell-separated)f(transition)h(la)n(y)n(ers,)h(whose)f(patterns)456 4043 y(are)30 b(describ)r(ed)i(appro)n(ximately)e(b)n(y)h(the)h (standing)g(w)n(a)n(v)n(es.)47 b(The)32 b(dynamics)f(on)p 3075 3976 V 32 w Fq(W)38 b Fw(is)32 b(then)456 4143 y(reduced)d(to)h (the)h(motion)f(of)g(the)g(la)n(y)n(er)e(lo)r(cations,)i Fq(\006)p Fr(\030)t Fw(,)h(whic)n(h)f(can)g(b)r(e)g(describ)r(ed,)h(to) f(high)456 4242 y(accuracy)-7 b(,)26 b(b)n(y)h(the)h(ordinary)e (di\013eren)n(tial)h(equation:)1491 4358 y(_)1473 4380 y Fr(\030)g Fw(=)c Fq(\000)p Fr(\026K)6 b(e)1855 4346 y Fp(\000)p Fs(2)p Fo(\013\030)2036 4380 y Fw(+)18 b(2)p Fr(m)2234 4392 y Fo(\014)2292 4380 y Fr(\026)c(h;)805 b Fw(\(1.11\))456 4515 y(where)28 b(the)h(p)r(ositiv)n(e)f(co)r (e\016cien)n(ts)g Fr(K)6 b Fw(,)28 b Fr(\013)p Fw(,)i(and)e Fr(\026)h Fw(dep)r(end)g(on)g Fr(J)36 b Fw(and)29 b Fr(\014)t Fw(.)40 b(Th)n(us)28 b(the)h(v)n(elo)r(cit)n(y)456 4615 y(turns)h(out)g(to)g(b)r(e)h(the)g(sum)f(of)g(t)n(w)n(o)g(terms.)44 b(The)31 b(\014rst)f(one)g(is)g(an)g(attractiv)n(e)f(term,)i(due)g(to) 456 4715 y(the)c(in)n(teraction)f(b)r(et)n(w)n(een)i(the)f(la)n(y)n (ers,)f(in)h(agreemen)n(t)f(to)h(the)h(analogous)d(e\013ect)i(pro)n(v)n (ed)f(for)456 4814 y(the)e(singularly)e(p)r(erturb)r(ed)j (Ginzburg-Landau)d(equation.)35 b(The)24 b(second)f(one)h(is)g(a)f (constan)n(t,)456 4914 y(p)r(ositiv)n(e)31 b(drift,)j(whic)n(h)e(is)g (linear)g(in)g(the)h(small)f(magnetic)f(\014eld)i Fr(h)p Fw(.)50 b(It)33 b(coincides)e(with)i(the)456 5014 y(\014rst)g(order)f (expansion)g(of)i(the)g(v)n(elo)r(cit)n(y)e(for)h(the)h(tra)n(v)n (eling)e(w)n(a)n(v)n(e)g(solutions)g(of)40 b(\(1.5\),)35 b(see)456 5113 y(\(2.10\))e(in)i(the)g(next)g(section.)58 b(Since)35 b(the)g(motion)f(is)h(v)n(ery)e(slo)n(w)h(\(for)g(large)g Fr(\030)t Fw(\),)j(follo)n(wing)456 5216 y([12)o(])g(w)n(e)f(refer)g (to)p 1065 5149 V 37 w Fq(W)43 b Fw(as)36 b(a)h Fu(slo)n(w)f(motion)g (manifold)p Fw(.)65 b(The)36 b(v)n(elo)r(cit)n(y)g(\014eld)h(has)g(a)f (zero)g(at)p eop %%Page: 5 5 5 4 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)291 b(5)456 450 y Fw(\(2)p Fr(\013)p Fw(\))615 420 y Fp(\000)p Fs(1)718 450 y Fw(log[)p Fr(K)6 b Fw(\(2)p Fr(m)1072 462 y Fo(\014)1116 450 y Fr(h)p Fw(\))1196 420 y Fp(\000)p Fs(1)1286 450 y Fw(])27 b(whic)n(h)g(giv)n(es,)g(at)g (the)h(lo)n(w)n(er)d(order,)i(\(one)g(half)g(of)6 b(\))28 b(the)g(length)g(of)456 550 y(the)g(critical)f(droplet.)555 649 y(Our)j(approac)n(h)f(is)i(based)f(on)h(the)g(geometric)f(metho)r (d)h(suggested)f(b)n(y)g(G.)i(F)-7 b(usco)30 b(and)h(J.)456 749 y(Hale,)22 b([12)o(],)g(and)f(dev)n(elop)r(ed)f(b)n(y)h(J.)g(Carr)e (and)i(R.)g(P)n(ego,)f([2,)h(3],)h(for)e(the)i(singularly)d(p)r(erturb) r(ed)456 849 y(Ginzburg-Landau)27 b(equation)i(\(see)g(also)f(the)h (more)f(recen)n(t)h(pap)r(er)f([11)o(])i(b)n(y)e(J.-P)-7 b(.)29 b(Ec)n(kmann)456 948 y(and)22 b(J.)h(Rougemon)n(t\).)34 b(By)23 b(restricting)e(our)h(analysis)g(to)g(the)h(space)f(of)h (symmetric)f(functions)456 1048 y(\(whic)n(h)29 b(is)f(an)h(in)n(v)-5 b(arian)n(t)28 b(set)g(for)h(the)g(dynamics\),)g(w)n(e)f(\014rst)h (construct)f(an)h(\\appro)n(ximately)456 1147 y(in)n(v)-5 b(arian)n(t")34 b(manifold)i Fq(M)g Fw(of)f(quasi-stationary)f(states)h Fr(n)2357 1159 y Fo(\030)2393 1147 y Fw(\()p Fr(x)p Fw(\))j Fq(\031)53 b Fw(\026)-58 b Fr(m)p Fw(\()p Fr(\030)28 b Fq(\000)c(j)p Fr(x)p Fq(j)p Fw(\),)39 b(for)c Fr(\030)41 b(>)c Fw(0)456 1247 y(in)d(a)g(suitable)h(in)n(terv)-5 b(al)34 b(and)g Fr(h)g Fw(small.)57 b(The)35 b(motion)f(near)g Fq(M)g Fw(is)g(describ)r(ed)h(in)f(terms)h(of)456 1347 y(co)r(ordinates)e(along)h(and)i(transv)n(ersal)c(to)j Fq(M)g Fw(\(the)h(F)-7 b(ermi)36 b(co)r(ordinates\).)58 b(This)35 b(manifold)456 1446 y(con)n(tains)27 b(the)i(essen)n(tial)f (dynamics:)39 b(due)29 b(to)f(a)g(strong)g(transv)n(erse)e(stabilit)n (y)-7 b(,)29 b(the)g(solutions)456 1546 y(near)i Fq(M)i Fw(are)e(squeezed)i(in)n(to)f(a)g Fu(thin)i(c)n(hannel)e Fw(whic)n(h)g(con)n(tains)g Fq(M)h Fw(and)f(where)g(the)h(la)n(y)n(er) 456 1646 y(motion)27 b(is)h(describ)r(ed)f(with)i(high)e(accuracy)f(b)n (y)j(\(1.11\))o(.)38 b(W)-7 b(e)28 b(next)g(pro)n(v)n(e)e(that)i(there) g(exists)456 1745 y(an)33 b(unique)h(stationary)e(solution)i(of)f(Eq.)h (\(1.5\))f(near)g Fq(M)p Fw(,)i(whic)n(h)f(is)f(then)i(iden)n(ti\014ed) f(with)456 1845 y(the)29 b(critical)f(droplet)g Fr(q)k Fw(and)d(it)g(turns)g(out)g(to)f(b)r(e)i(strictly)e(con)n(tained)g(in)h (the)g(thin)h(c)n(hannel.)456 1944 y(The)c(slo)n(w)f(motion)h(manifold) p 1428 1878 89 4 v 26 w Fq(W)34 b Fw(is)26 b(th)n(us)g(de\014ned)g(b)n (y)g(that)h(part)f(of)g Fq(W)33 b Fw(inside)26 b(the)h(c)n(hannel.)555 2044 y(In)38 b(the)f(next)g(section)g(w)n(e)g(state)g(the)g(main)g 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b(for)i(the)g(dynamics.)67 b(Analogously)36 b(there)456 3923 y(exists)29 b(a)h(unique)g(solution)g (of)g(the)g(Cauc)n(h)n(y)f(problem)h(in)g Fr(C)2374 3935 y Fs(0)2411 3923 y Fw(\()p Fn(R)q Fw(\),)37 b(the)30 b(space)g(of)f(con)n(tin)n(uous)456 4023 y(and)d(b)r(ounded)i (functions)f(on)g Fn(R)p Fw(,)33 b(and)27 b(the)g(set)h Fr(C)6 b Fw(\()p Fn(R)p Fr(;)14 b Fw([)p Fq(\000)p Fw(1)p Fr(;)g Fw(1]\))32 b(is)27 b(left)h(in)n(v)-5 b(arian)n(t.)35 b(W)-7 b(e)28 b(denote)456 4122 y(b)n(y)j Fr(S)626 4134 y Fo(t)655 4122 y Fw(\()p Fr(m)p Fw(\))h(the)g(\015o)n(w)e(solution)h (with)h(initial)g(datum)f Fr(m)p Fw(,)i(so)d(that)i Fr(S)2637 4134 y Fo(t)2698 4122 y Fw(de\014nes)f(a)g(semi-group)456 4222 y(on)c Fr(L)628 4234 y Fp(1)698 4222 y Fw(\()p Fn(R)p Fw(;)14 b([)p Fq(\000)p Fw(1)p Fr(;)g Fw(1]\))32 b(for)c(whic)n(h)f Fr(C)6 b Fw(\()p Fn(R)q Fr(;)14 b Fw([)p Fq(\000)p Fw(1)p Fr(;)g Fw(1]\))32 b(is)c(an)f(in)n(v)-5 b(arian)n(t)26 b(\(closed\))i(subspace.)555 4321 y(Since)41 b Fr(J)50 b Fw(is)40 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Fq(k)2513 4897 y Fp(1)2582 4885 y Fr(;)669 b Fw(\(2.3\))456 5050 y(where)27 b Fr(k)739 5062 y Fs(1)799 5050 y Fr(>)c Fw(0)k(is)h(the)f(Lipsc)n(hitz)h(co)r(e\016cien)n(t)f(of)h Fr(f)9 b Fw(,)27 b(i.e.)h(for)f(an)n(y)g Fr(m;)14 b(u)22 b Fq(2)i Fr(L)2853 5062 y Fp(1)2923 5050 y Fw(\()p Fn(R)p Fw(\),)1353 5216 y Fq(k)p Fr(f)9 b Fw(\()p Fr(m)18 b Fw(+)g Fr(u)p Fw(\))g Fq(\000)h Fr(f)9 b Fw(\()p Fr(m)p Fw(\))p Fq(k)2062 5228 y Fp(1)2154 5216 y Fq(\024)23 b Fr(k)2285 5228 y Fs(1)2323 5216 y Fq(k)p Fr(u)p Fq(k)2455 5228 y Fp(1)2523 5216 y Fr(:)728 b Fw(\(2.4\))p eop %%Page: 6 6 6 5 bop 456 251 a Fs(6)756 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fw(F)e(or)27 b(a)g(more)g(re\014ned)g(b)r(ound,)h (let)g Fr(L)1610 462 y Fo(m)1700 450 y Fw(b)r(e)g(the)g(linear)f(op)r (erator)f(de\014ned)i(b)n(y)1064 633 y Fr(L)1121 645 y Fo(m)1184 633 y Fr(u)22 b Fw(=)h Fq(\000)p Fr(u)17 b Fw(+)h Fr(p)1597 645 y Fo(m)1660 633 y Fr(J)27 b Fq(\003)18 b Fr(u;)179 b(p)2085 645 y Fo(m)2192 586 y Fr(:)2171 633 y Fw(=)2510 577 y Fr(\014)p 2269 614 535 4 v 2269 698 a Fw(cosh)2426 661 y Fs(2)2463 698 y Fq(f)p Fr(\014)t(J)27 b Fq(\003)17 b Fr(m)p Fq(g)2813 633 y Fr(:)438 b Fw(\(2.5\))456 834 y(Then)27 b Fr(L)729 846 y Fo(m)p Fs(+)p Fo(h)909 834 y Fw(is)h(the)g(deriv)-5 b(ativ)n(e)27 b Fr(D)r(f)9 b Fq(j)1661 846 y Fo(m)1751 834 y Fw(of)28 b Fr(f)9 b Fw(\()p Fr(m)p Fw(\))27 b(at)h Fr(m)p Fw(,)f(so)g(that:)497 1037 y Fr(f)9 b Fw(\()p Fr(m)t Fw(+)t Fr(u)p Fw(\))t Fq(\000)t Fr(f)g Fw(\()p Fr(m)p Fw(\))t Fq(\000)t Fr(L)1195 1049 y Fo(m)p Fs(+)p Fo(h)1343 1037 y Fr(u)23 b Fw(=)f Fr(\014)1552 1003 y Fs(2)1590 1037 y Fw(\()p Fr(J)12 b Fq(\003)t Fr(u)p Fw(\))1806 1003 y Fs(2)1856 924 y Fm(Z)1939 945 y Fs(1)1902 1113 y(0)1976 1037 y Fr(ds)i Fw(\(1)t Fq(\000)t Fr(s)p Fw(\))g(tanh)2469 1000 y Fp(00)2511 1037 y Fq(f)p Fr(\014)t Fw([)p Fr(J)e Fq(\003)t Fw(\()p Fr(m)t Fw(+)t Fr(su)p Fw(\))t(+)t Fr(h)p Fw(])p Fq(g)p Fr(;)37 b Fw(\(2.6\))456 1223 y(from)27 b(whic)n(h)g(it)h(follo)n(ws)f (that)h(there)f(exists)h Fr(k)1910 1235 y Fs(2)1970 1223 y Fr(>)23 b Fw(0)k(suc)n(h)g(that:)1174 1367 y Fq(k)p Fr(f)9 b Fw(\()p Fr(m)18 b Fw(+)g Fr(u)p Fw(\))g Fq(\000)g Fr(f)9 b Fw(\()p Fr(m)p Fw(\))19 b Fq(\000)f Fr(L)1999 1379 y Fo(m)p Fs(+)p Fo(h)2151 1367 y Fr(u)p Fq(k)2241 1379 y Fp(1)2334 1367 y Fq(\024)k Fr(k)2464 1379 y Fs(2)2502 1367 y Fq(k)p Fr(u)p Fq(k)2634 1333 y Fs(2)2634 1388 y Fp(1)2703 1367 y Fr(:)548 b Fw(\(2.7\))456 1534 y Fz(Stationary)44 b(solutions.)74 b(\()p Fw([5,)38 b(6)o(,)g(7)o(,)g(9)o(,)g(10)o(])p Fz(\).)67 b Fw(Giv)n(en)37 b Fr(\014)44 b(>)39 b Fw(1,)h(there)d (exists)g(a)g(unique)456 1633 y(\(mo)r(dulo)25 b(translations\))f (solution)41 b(\026)-58 b Fr(m)p Fw(\()p Fr(x)p Fw(\))26 b(of)32 b(\(1.9\))25 b(with)g(asymptotes)g(\(1.10\))o(,)h(whic)n(h)f(w) n(e)g(call)456 1733 y(the)32 b Fu(instan)n(ton)p Fw(.)48 b(It)32 b(is)g(a)f Fr(C)1334 1703 y Fp(1)1405 1733 y 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Fr(h)p Fw(\))p 1143 3289 156 4 v 1197 3365 a Fr(h)1332 3308 y Fw(=)c Fq(\000)p Fw(2)p Fr(m)1600 3320 y Fo(\014)1657 3308 y Fr(\026;)180 b(\026)2004 3261 y(:)1983 3308 y Fw(=)2071 3191 y Fm(\024)2115 3195 y(Z)2198 3308 y Fr(dy)2442 3252 y Fw(\026)-58 b Fr(m)2499 3222 y Fp(0)2522 3252 y Fw(\()p Fr(y)s Fw(\))2630 3222 y Fs(2)p 2308 3289 478 4 v 2308 3365 a Fr(\014)t Fw(\(1)19 b Fq(\000)34 b Fw(\026)-58 b Fr(m)p Fw(\()p Fr(y)s Fw(\))2716 3341 y Fs(2)2753 3365 y Fw(\))2796 3191 y Fm(\025)2839 3205 y Fp(\000)p Fs(1)2942 3308 y Fr(:)267 b Fw(\(2.10\))456 3498 y(On)23 b(a)h(ph)n(ysical)f(bac)n (kground,)g Fr(\026)h Fw(is)g(the)g(linear)g(transp)r(ort)f(co)r (e\016cien)n(t)h(whic)n(h)g(represen)n(ts)e(the)456 3597 y Fu(mobilit)n(y)g Fw(of)h(an)g(in)n(terface)f(separating)g(the)h (stable)g(phases,)g(see)g(e.g.)f([1])h(and)g([19)o(])g(for)g(details.) 456 3697 y(W)-7 b(e)28 b(refer)e(to)i([5])f(for)g(stabilit)n(y)h(and)f (other)g(prop)r(erties)g(of)g(fron)n(ts.)555 3796 y(When)39 b Fr(h)i(>)g 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Fk(Given)h Fr(\014)f(>)27 b Fw(1)p Fk(,)33 b(ther)l(e)f(is)h Fr(h)2115 4550 y Fs(0)2179 4538 y Fr(>)28 b Fw(0)j Fk(and,)j(for)f(any)g Fr(h)27 b Fq(2)h Fw(\(0)p Fr(;)14 b(h)3149 4550 y Fs(0)3186 4538 y Fw(])p Fk(,)33 b(ther)l(e)456 4638 y(is)i Fr(q)h Fq(2)d Fr(C)776 4608 y Fs(sym)896 4638 y Fw(\()p Fn(R)p Fw(;)14 b([)p Fq(\000)p Fw(1)p Fr(;)g Fw(1]\))40 b Fk(which)c(solves)43 b Fw(\(1.7\))35 b Fk(with)g(asymptotes)h(as)f(in)42 b Fw(\(1.8\))p Fk(.)54 b(Mor)l(e)l(over)456 4738 y(ther)l(e)29 b(is)i Fr(\030)792 4707 y Fp(\003)853 4738 y Fw(=)23 b Fr(\030)981 4707 y Fp(\003)1019 4738 y Fw(\()p Fr(h)p Fw(\))30 b Fk(such)g(that:)1590 4882 y Fw(lim)1595 4936 y Fo(h)p Fp(#)p Fs(0)1719 4882 y Fq(k)p Fr(q)21 b Fq(\000)34 b Fw(\026)-58 b Fr(m)1975 4894 y Fo(\030)2007 4877 y Fj(\003)2046 4882 y Fq(k)2088 4894 y Fp(1)2181 4882 y Fw(=)23 b(0)921 b(\(2.12\))456 5066 y Fk(and)1464 5175 y Fw(lim)1469 5229 y Fo(h)p Fp(#)p Fs(0)1593 5175 y Fr(e)1632 5141 y Fs(2)p Fo(\013\030)1740 5116 y Fj(\003)1775 5141 y Fs(\()p Fo(h)p Fs(\))1870 5175 y Fr(h)23 b Fw(=)f(\(2)p Fr(m)2175 5187 y Fo(\014)2220 5175 y Fw(\))2252 5141 y Fp(\000)p Fs(1)2341 5175 y Fr(K)q(;)796 b Fw(\(2.13\))p eop %%Page: 7 7 7 6 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)291 b(7)456 450 y Fk(wher)l(e)1117 608 y Fr(K)1237 561 y(:)1217 608 y Fw(=)1481 552 y Fr(a)p 1314 589 377 4 v 1314 665 a(\014)t Fw(\(1)19 b Fq(\000)f Fr(m)1614 636 y Fs(2)1614 690 y Fo(\014)1659 665 y Fw(\))1715 495 y Fm(Z)1798 608 y Fr(dx)1912 541 y(e)1951 510 y Fo(\013x)2051 541 y Fw(\026)-57 b Fr(m)2109 510 y Fp(0)2132 541 y Fw(\()p Fr(x)p Fw(\)\()p Fr(m)2348 510 y Fs(2)2348 564 y Fo(\014)2412 541 y Fq(\000)34 b Fw(\026)-58 b Fr(m)2568 510 y Fs(2)2606 541 y Fw(\()p Fr(x)p Fw(\)\))p 1912 589 838 4 v 2148 665 a(1)18 b Fq(\000)34 b Fw(\026)-58 b Fr(m)2364 641 y Fs(2)2402 665 y Fw(\()p Fr(x)p Fw(\))2760 608 y Fr(;)449 b Fw(\(2.14\))456 804 y Fk(with)30 b Fr(\013)g Fk(de\014ne)l(d)g(by)38 b Fw(\(2.8\))29 b Fk(and)h Fr(a)g Fk(as)g(in)36 b Fw(\(2.9\))p Fk(.)456 986 y Fz(The)k(quasi-in)m(v)-5 b(arian)m(t)41 b(manifold.)56 b Fw(W)-7 b(e)35 b(no)n(w)f(fo)r(cus)h(on)g(the)g (dynamics)f(\(2.1\))h(for)f Fr(h)h(>)456 1086 y Fw(0)i(small,)k(when)d (restricted)g(to)g(the)g(class)f Fr(L)1924 1056 y Fs(sym)1924 1106 y Fp(1)2044 1086 y Fw(\()p Fn(R)p Fw(;)14 b([)p Fq(\000)p Fw(1)p Fr(;)g Fw(1]\).)73 b(W)-7 b(e)39 b(are)e(in)n (terested)h(in)g(the)456 1185 y(b)r(eha)n(vior)21 b(of)h(solutions)g (whic)n(h)h(consist)f(of)h(t)n(w)n(o)f(w)n(ell)g(separated)f(la)n(y)n (ers,)h(i.e.)h(whic)n(h)f(b)r(elong)h(to)456 1285 y(a)g(neigh)n(b)r (orho)r(o)r(d)g(of)h Fq(f)15 b Fw(\026)-57 b Fr(m)1245 1297 y Fo(\030)1281 1285 y Fw(;)14 b Fr(\030)27 b(>)c(`)p Fq(g)g Fw(\()16 b(\026)-58 b Fr(m)1674 1297 y Fo(\030)1734 1285 y Fw(de\014ned)25 b(in)f(\(2.11\))o(\))h(for)e Fr(`)h Fw(large)e(enough.)35 b(According)456 1385 y(to)e(\(2.13\),)i(w)n(e)f (ma)n(y)f(exp)r(ect)i(the)f(in)n(teresting)f(part)h(of)g(the)g (relaxation)f(pro)r(cess)f(is)i(caugh)n(t)456 1484 y(b)n(y)27 b(restricting)h(our)f(analysis)g(to)h(patterns)f(with)i(la)n(y)n(ers)d (at)i(a)g(distance)g(no)f(to)r(o)h(larger)e(than)456 1584 y(\(2)p Fr(\013)p Fw(\))615 1554 y Fp(\000)p Fs(1)704 1584 y Fq(j)14 b Fw(log)g Fr(h)p Fq(j)p Fw(.)67 b(In)38 b(fact)g(la)n(y)n(ers)e(far)h(apart)f(from)i(eac)n(h)f(other)g(in)n (teract)g(to)r(o)g(w)n(eakly)g(and)456 1684 y(the)25 b(e\013ect)h(of)f(the)h(magnetic)e(\014eld)i(predominates)e(\(one)h (could)g(pro)n(v)n(e)f(they)h(simply)g(go)g(a)n(w)n(a)n(y)456 1783 y(with)19 b(an)h(almost)e(constan)n(t)h(v)n(elo)r(cit)n(y)-7 b(,)20 b(eac)n(h)f(one)g(ev)n(en)n(tually)f(approac)n(hing)f(a)i(tra)n (v)n(eling)f(w)n(a)n(v)n(e\).)456 1883 y(With)28 b(this)g(in)g(mind,)g (for)f(an)n(y)g Fr(`)c(>)f Fw(1,)27 b Fr(h)c(<)g(e)1883 1853 y Fp(\000)p Fs(2)p Fo(\013`)2043 1883 y Fw(,)k(and)h Fr(\024)23 b(>)f Fw(0,)28 b(w)n(e)f(in)n(tro)r(duce)g(the)h(in)n(terv) -5 b(al)1240 2032 y(\000)1292 2044 y Fo(`;\024)1427 1985 y Fr(:)1406 2032 y Fw(=)23 b Fq(f)p Fr(\030)40 b Fw(:)d Fr(`)23 b(<)f(\030)28 b(<)22 b Fw(\(2)p Fr(\013)p Fw(\))2127 1998 y Fp(\000)p Fs(1)2217 2032 y Fq(j)14 b Fw(log)g Fr(h)p Fq(j)k Fw(+)g Fr(\024)p Fq(g)p Fr(:)572 b Fw(\(2.15\))456 2181 y(Observ)n(e)26 b(that)h(for)g Fr(\030)h Fq(2)23 b Fw(\000)1275 2193 y Fo(`;\024)1393 2181 y Fw(one)k(has:)1679 2337 y Fr(h)c(<)f(e)1876 2303 y Fs(2)p Fo(\013\024)1995 2337 y Fr(e)2034 2303 y Fp(\000)p Fs(2)p Fo(\013\030)2198 2337 y Fr(;)1011 b Fw(\(2.16\))456 2486 y(i.e.)27 b Fr(h)c Fw(=)g Fr(O)r Fw(\()p Fr(e)884 2456 y Fp(\000)p Fs(2)p Fo(\013\030)1049 2486 y Fw(\).)456 2644 y Fz(Theorem)32 b(2.2.)42 b Fk(Given)32 b Fr(\014)f(>)26 b Fw(1)p Fk(,)32 b(ther)l(e)g(is)g Fr(`)1902 2656 y Fs(0)1965 2644 y Fr(>)26 b Fw(1)31 b Fk(such)h(that,)h(for)f(any)g Fr(h)26 b(<)h(e)3018 2614 y Fp(\000)p Fs(2)p Fo(\013`)3174 2622 y Fl(0)3210 2644 y Fk(,)32 b(ther)l(e)456 2744 y(exists)d(a)h Fr(C)820 2714 y Fs(1)858 2744 y Fk({map:)1280 2858 y Fw(\000)1332 2870 y Fo(`)1360 2878 y Fl(0)1392 2870 y Fo(;)p Fs(+)p Fp(1)1556 2858 y Fq(3)24 b Fr(\030)j Fq(7\000)-14 b(!)23 b Fr(n)1905 2870 y Fo(\030)1964 2858 y Fq(2)h Fr(C)2108 2824 y Fs(sym)2228 2858 y Fw(\()p Fn(R)p Fw(;)14 b([)p Fq(\000)p Fw(1)p Fr(;)g Fw(1]\))617 b(\(2.17\))456 2990 y Fk(with)26 b(the)f(fol)t(lowing)j(pr)l(op)l(erties.)38 b(F)-6 b(or)26 b(some)f Fr(\016)1922 3002 y Fs(0)1983 2990 y Fr(>)d Fw(0)j Fk(and)h(any)g Fr(\024)c(>)h Fw(0)p Fk(,)j(ther)l(e)g(is)f Fr(c)3023 3002 y Fs(0)3084 2990 y Fw(=)d Fr(c)3207 3002 y Fs(0)3244 2990 y Fw(\()p Fr(\024)p Fw(\))i Fr(>)456 3089 y Fw(0)29 b Fk(such)h(that,)g(for)g(any)h Fr(\030)c Fq(2)c Fw(\000)1396 3101 y Fo(`)1424 3109 y Fl(0)1456 3101 y Fo(;\024)1519 3089 y Fk(,)1332 3245 y Fq(k)p Fr(n)1424 3257 y Fo(\030)1459 3245 y Fw(\()p Fr(x)p Fw(\))d Fq(\000)33 b Fw(\026)-57 b Fr(m)1746 3257 y Fo(\030)1782 3245 y Fw(\()p Fr(x)p Fw(\))p Fq(k)1935 3257 y Fp(1)2089 3245 y Fq(\024)82 b Fr(c)2272 3257 y Fs(0)2310 3245 y Fr(e)2349 3211 y Fp(\000)p Fo(\013\030)2480 3245 y Fr(;)729 b Fw(\(2.18\))1251 3377 y Fq(k)p Fr(@)1337 3389 y Fo(\030)1373 3377 y Fr(n)1423 3389 y Fo(\030)1459 3377 y Fw(\()p Fr(x)p Fw(\))20 b Fq(\000)33 b Fw(\026)-57 b Fr(m)1746 3343 y Fp(0)1746 3398 y Fo(\030)1782 3377 y Fw(\()p Fr(x)p Fw(\))p Fq(k)1935 3389 y Fp(1)2089 3377 y Fq(\024)82 b Fr(c)2272 3389 y Fs(0)2310 3377 y Fr(e)2349 3343 y Fp(\000)p Fo(\013\030)2480 3377 y Fr(;)729 b Fw(\(2.19\))1199 3520 y Fq(k)o Fr(f)9 b Fw(\()p Fr(n)1372 3532 y Fo(\030)1408 3520 y Fw(\))19 b Fq(\000)f Fr(V)h Fw(\()p Fr(\030)t Fw(\))14 b Fr(@)1771 3532 y Fo(\030)1808 3520 y Fr(n)1858 3532 y Fo(\030)1894 3520 y Fq(k)1936 3548 y Fp(1)2089 3520 y Fq(\024)82 b Fr(c)2272 3532 y Fs(0)2310 3520 y Fr(e)2349 3486 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)2584 3494 y Fl(0)2616 3486 y Fs(\))p Fo(\030)2678 3520 y Fr(;)531 b Fw(\(2.20\))456 3669 y Fk(with)1419 3783 y Fr(V)19 b Fw(\()p Fr(\030)t Fw(\))1634 3736 y Fr(:)1613 3783 y Fw(=)k Fq(\000)p Fr(\026K)6 b(e)1932 3749 y Fp(\000)p Fs(2)p Fo(\013\030)2114 3783 y Fw(+)18 b(2)p Fr(m)2312 3795 y Fo(\014)2370 3783 y Fr(\026)c(h)750 b Fw(\(2.21\))456 3915 y Fk(and)30 b Fr(\013)p Fk(,)g Fr(\026)p Fk(,)h(and)f Fr(K)35 b Fk(as)30 b(in)36 b Fw(\(2.8\))p Fk(,)30 b Fw(\(2.10\))o Fk(,)g(and)39 b Fw(\(2.14\))29 b Fk(r)l(esp)l(e)l(ctively.)555 4073 y Fw(This)g(theorem)f(will)h(b)r (e)f(pro)n(v)n(ed)f(in)i(Sect.)g(8)f(where)g(the)h(function)g Fr(n)2745 4085 y Fo(\030)2810 4073 y Fw(is)g(explicitly)f(giv)n(en)456 4172 y(in)f(De\014nition)i(8.2.)36 b(F)-7 b(or)27 b(an)n(y)g Fr(`)22 b Fq(\025)h Fr(`)1591 4184 y Fs(0)1655 4172 y Fw(and)28 b Fr(\024)23 b(>)f Fw(0)28 b(w)n(e)f(refer)g(to)1538 4386 y Fq(M)1638 4398 y Fo(`;\024)1772 4339 y Fr(:)1752 4386 y Fw(=)22 b Fq(f)p Fr(n)1931 4398 y Fo(\030)1990 4386 y Fw(:)h Fr(\030)k Fq(2)d Fw(\000)2230 4398 y Fo(`;\024)2320 4386 y Fq(g)870 b Fw(\(2.22\))456 4518 y(as)37 b(the)h Fu(quasi-in)n(v)-5 b(arian)n(t)35 b(manifold)p Fw(:)58 b(b)n(y)38 b(\(2.20\))o(,)j Fr(f)9 b Fw(\()p Fr(n)2225 4530 y Fo(\030)2261 4518 y Fw(\),)41 b Fr(n)2407 4530 y Fo(\030)2483 4518 y Fq(2)f(M)2678 4530 y Fo(`;\024)2768 4518 y Fw(,)h(is)c(appro)n(ximately)456 4618 y(tangen)n(t)27 b(to)g Fq(M)959 4630 y Fo(`;\024)1077 4618 y Fw(and)h(v)n(ery)e(small.) 555 4717 y(In)d(the)h(sequel)f(w)n(e)f(shall)h(need)g(a)g(go)r(o)r(d)f (con)n(trol)g(on)h(the)g(sp)r(ectral)g(prop)r(erties)f(of)h(the)g (linear)456 4817 y(op)r(erator)i Fr(D)r(f)9 b Fq(j)934 4829 y Fo(n)975 4838 y Fi(\030)1035 4817 y Fw(=)23 b Fr(L)1180 4829 y Fo(n)1221 4838 y Fi(\030)1253 4829 y Fs(+)p Fo(h)1347 4817 y Fw(,)k(where)h Fr(L)1695 4829 y Fo(m)1785 4817 y Fw(is)f(de\014ned)h(in)g(\(2.5\).)555 4917 y(In)23 b([6)o(])g(sev)n(eral)d(prop)r(erties)i(ab)r(out)g(the)h (sp)r(ectrum)f(of)h(the)f(linear)g(op)r(erator)f Fr(L)2984 4929 y Fo(m)3069 4917 y Fw(are)g(pro)n(v)n(ed)456 5016 y(when)e Fr(m)g Fw(is)g(close)f(to)h([a)g(sligh)n(tly)f(mo)r (di\014cation)h(of)6 b(])20 b(the)f(re\015ected)g(instan)n(ton)34 b(\026)-57 b Fr(m)2955 5028 y Fo(\030)2991 5016 y Fw(\()p Fr(x)p Fw(\))20 b(\()p Fr(\030)k Fw(large\).)456 5116 y(W)-7 b(e)26 b(shall)g(use)h(these)f(results)g(to)h(pro)n(v)n(e)d (Theorem)i(2.2)g(and)g(Prop)r(osition)f(2.3)g(b)r(elo)n(w,)i(details) 456 5216 y(are)f(giv)n(en)h(in)h(Sect.)g(8.)p eop %%Page: 8 8 8 7 bop 456 251 a Fs(8)756 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fz(Prop)s(osition)31 b(2.3.)41 b Fk(L)l(et)31 b Fr(n)1359 462 y Fo(\030)1395 450 y Fk(,)g Fr(\030)f Fq(2)25 b Fw(\000)1649 462 y Fo(`)1677 470 y Fl(0)1710 462 y Fo(;\024)1772 450 y Fk(,)32 b(b)l(e)f(as)g(in)g(The)l(or)l(em)h (2.2)g(and)g(set)f Fr(L)2980 462 y Fo(\030)3061 403 y Fr(:)3041 450 y Fw(=)24 b Fr(L)3187 462 y Fo(n)3228 471 y Fi(\030)3261 462 y Fs(+)p Fo(h)3380 450 y Fw(=)456 552 y Fr(D)r(f)9 b Fq(j)600 564 y Fo(n)641 573 y Fi(\030)677 552 y Fk(,)29 b Fr(p)773 564 y Fo(\030)853 505 y Fr(:)832 552 y Fw(=)23 b Fr(p)962 564 y Fo(n)1003 573 y Fi(\030)1035 564 y Fs(+)p Fo(h)1129 552 y Fk(.)38 b(F)-6 b(or)28 b(any)h Fr(\024)23 b(>)f Fw(0)28 b Fk(ther)l(e)g(ar)l(e)h Fr(`)2111 564 y Fs(1)2171 552 y Fq(\025)22 b Fr(`)2293 564 y Fs(0)2330 552 y Fk(,)29 b Fr(c)2420 564 y Fs(1)2480 552 y Fr(>)23 b Fw(1)p Fk(,)28 b(and)h Fr(!)2875 564 y Fs(1)2935 552 y Fr(>)22 b Fw(0)28 b Fk(such)g(that,)456 660 y(for)i(any)g Fr(h)23 b(<)g(e)945 630 y Fp(\000)p Fs(2)p Fo(\013`)1101 638 y Fl(1)1167 660 y Fk(and)30 b Fr(\030)d Fq(2)d Fw(\000)1522 672 y Fo(`)1550 680 y Fl(1)1582 672 y Fo(;\024)1644 660 y Fk(,)31 b(the)e(fol)t(lowing)k(holds.)555 760 y(1\))d(Ther)l(e)h (exist)e Fr(\025)1141 772 y Fo(\030)1201 760 y Fr(>)23 b Fw(0)29 b Fk(and)h(strictly)g(p)l(ositive)h(functions)f Fr(v)2500 772 y Fo(\030)2537 760 y Fr(;)14 b(v)2617 729 y Fp(\003)2614 783 y Fo(\030)2678 760 y Fq(2)23 b Fr(C)2821 720 y Fs(sym)2815 782 y(0)2942 760 y Fw(\()p Fn(R)p Fw(\))36 b Fk(so)30 b(that:)1112 921 y Fr(L)1169 933 y Fo(\030)1205 921 y Fr(v)1245 933 y Fo(\030)1305 921 y Fw(=)22 b Fr(\025)1440 933 y Fo(\030)1477 921 y Fr(v)1517 933 y Fo(\030)1554 921 y Fr(;)183 b(v)1803 887 y Fp(\003)1800 942 y Fo(\030)1842 921 y Fr(L)1899 933 y Fo(\030)1958 921 y Fw(=)22 b Fr(\025)2093 933 y Fo(\030)2130 921 y Fr(v)2173 887 y Fp(\003)2170 942 y Fo(\030)2211 921 y Fr(;)184 b(v)2461 887 y Fp(\003)2458 942 y Fo(\030)2522 921 y Fw(=)23 b Fr(p)2652 933 y Fo(\030)2688 921 y Fr(v)2728 933 y Fo(\030)2765 921 y Fr(:)444 b Fw(\(2.23\))456 1077 y Fk(Mor)l(e)l(over)1416 1127 y Fm(\015)1416 1177 y(\015)1462 1197 y Fr(e)1501 1163 y Fo(L)1547 1172 y Fi(\030)1579 1163 y Fo(t)1609 1127 y Fm(\015)1609 1177 y(\015)1655 1231 y Fp(1)1748 1197 y Fq(\024)23 b Fr(c)1872 1209 y Fs(1)1909 1197 y Fr(e)1948 1163 y Fo(\025)1987 1172 y Fi(\030)2019 1163 y Fo(t)2218 1197 y Fq(8)14 b Fr(t)22 b Fq(\025)h Fw(0)p Fr(:)748 b Fw(\(2.24\))555 1340 y Fk(2\))30 b(We)g(have:)1495 1465 y Fr(c)1531 1429 y Fp(\000)p Fs(1)1531 1487 y(1)1620 1465 y Fr(e)1659 1430 y Fp(\000)p Fs(2)p Fo(\013\030)1846 1465 y Fq(\024)23 b Fr(\025)1982 1477 y Fo(\030)2041 1465 y Fq(\024)g Fr(c)2165 1477 y Fs(1)2202 1465 y Fr(e)2241 1430 y Fp(\000)p Fs(2)p Fo(\013\030)3232 1465 y Fw(\(2.25\))456 1606 y Fk(and,)30 b(for)h Fr(\013)828 1576 y Fp(0)875 1606 y Fw(=)22 b Fr(\013)1015 1576 y Fp(0)1039 1606 y Fw(\()p Fr(\030)t Fw(\))1188 1559 y Fr(:)1167 1606 y Fw(=)g Fr(\013)d Fq(\000)f Fr(c)1445 1618 y Fs(1)1482 1606 y Fr(e)1521 1576 y Fp(\000)p Fs(2)p Fo(\013\030)1686 1606 y Fk(,)1345 1773 y Fr(v)1385 1785 y Fo(\030)1421 1773 y Fw(\()p Fr(x)p Fw(\))24 b Fq(\024)f Fr(c)1680 1785 y Fs(1)1717 1773 y Fr(e)1756 1739 y Fp(\000)p Fo(\013)1851 1714 y Fj(0)1873 1739 y Fp(j)p Fo(\030)r Fp(\000)p Fo(x)p Fp(j)2208 1773 y Fq(8)14 b Fr(x)23 b Fq(2)g Fn(R)2471 1785 y Fs(+)2532 1773 y Fr(:)677 b Fw(\(2.26\))456 1929 y Fk(Mor)l(e)l(over,)31 b(c)l(al)t(ling)1032 2085 y Fw(~)-58 b Fr(m)p Fw(\()p Fr(x)p Fw(\))1245 2037 y Fr(:)1224 2085 y Fw(=)1312 2029 y Fq(p)p 1381 2029 51 4 v 56 x Fr(\026)29 b Fw(\026)-57 b Fr(m)o Fw(\()p Fr(x)p Fw(\))p Fr(;)201 b Fw(~)-58 b Fr(m)1909 2097 y Fo(z)1947 2085 y Fw(\()p Fr(x)p Fw(\))2103 2037 y Fr(:)2082 2085 y Fw(=)39 b(~)-58 b Fr(m)p Fw(\()p Fr(z)22 b Fq(\000)c(j)p Fr(x)p Fq(j)p Fw(\))p Fr(;)100 b(z)26 b(>)c Fw(0)p Fr(;)348 b Fw(\(2.27\))456 2240 y Fk(\()p Fr(\026)29 b Fk(as)h(in)37 b Fw(\(2.10\))o Fk(\))29 b(we)h(also)h(have:)733 2333 y Fm(\014)733 2382 y(\014)761 2403 y Fr(v)801 2415 y Fo(\030)838 2403 y Fw(\()p Fr(x)p Fw(\))19 b Fq(\000)34 b Fw(~)-58 b Fr(m)1124 2369 y Fp(0)1124 2424 y Fo(\030)1160 2403 y Fw(\()p Fr(x)p Fw(\))1271 2333 y Fm(\014)1271 2382 y(\014)1323 2403 y Fq(\024)23 b Fr(c)1447 2415 y Fs(1)1484 2403 y Fr(e)1523 2369 y Fp(\000)p Fs(2)p Fo(\013\030)r Fs(+)p Fo(\013)p Fp(j)p Fo(\030)r Fp(\000)p Fo(x)p Fp(j)1942 2403 y Fr(\030)1982 2369 y Fs(4)2274 2403 y Fk(for)86 b Fq(j)p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fq(j)23 b(\024)g Fr(\030)t(=)p Fw(2)p Fr(:)277 b Fw(\(2.28\))555 2562 y Fk(3\))30 b(Assume)f Fr(v)1011 2574 y Fo(\030)1078 2562 y Fk(and)h Fr(v)1282 2531 y Fp(\003)1279 2585 y Fo(\030)1350 2562 y Fk(ar)l(e)g(normalize)l (d)h(in)f(such)g(a)g(way)h(that)1247 2671 y Fm(Z)1330 2692 y Fp(1)1293 2860 y Fs(0)1400 2784 y Fr(dx)1515 2728 y(v)1555 2740 y Fo(\030)1591 2728 y Fw(\()p Fr(x)p Fw(\))1702 2698 y Fs(2)p 1515 2765 226 4 v 1532 2841 a Fr(p)1574 2853 y Fo(\030)1611 2841 y Fw(\()p Fr(x)p Fw(\))1773 2784 y Fq(\021)1861 2671 y Fm(Z)1944 2692 y Fp(1)1907 2860 y Fs(0)2014 2784 y Fr(dx)14 b(v)2161 2750 y Fp(\003)2158 2805 y Fo(\030)2200 2784 y Fw(\()p Fr(x)p Fw(\))p Fr(v)2351 2796 y Fo(\030)2389 2784 y Fw(\()p Fr(x)p Fw(\))24 b(=)f(1)578 b(\(2.29\))456 2992 y Fk(and)30 b(de\014ne)g(the)g(line)l(ar)g (functional)g Fr(\031)1668 3004 y Fo(\030)1735 2992 y Fk(on)g Fr(L)1911 2962 y Fs(sym)1911 3012 y Fp(1)2030 2992 y Fw(\()p Fn(R)p Fw(\))36 b Fk(as)1470 3197 y Fr(\031)1517 3209 y Fo(\030)1554 3197 y Fw(\()p Fr( )s Fw(\))1719 3150 y Fr(:)1699 3197 y Fw(=)1786 3084 y Fm(Z)1869 3105 y Fp(1)1832 3273 y Fs(0)1940 3197 y Fr(dx)14 b(v)2087 3163 y Fp(\003)2084 3218 y Fo(\030)2126 3197 y Fw(\()p Fr(x)p Fw(\))p Fr( )s Fw(\()p Fr(x)p Fw(\))p Fr(:)804 b Fw(\(2.30\))456 3400 y Fk(We)29 b(have,)j(for)e(any)h Fr( )26 b Fq(2)d Fr(L)1330 3369 y Fs(sym)1330 3420 y Fp(1)1449 3400 y Fw(\()p Fn(R)q Fw(\))p Fk(,)1616 3555 y Fq(j)p Fr(\031)1686 3567 y Fo(\030)1722 3555 y Fw(\()p Fr( )s Fw(\))p Fq(j)h(\024)f Fr(c)2014 3567 y Fs(1)2051 3555 y Fq(k)p Fr( )s Fq(k)2192 3567 y Fp(1)2261 3555 y Fr(;)948 b Fw(\(2.31\))456 3711 y Fk(and,)30 b(for)h(any)f Fr(w)c Fq(2)d Fr(L)1154 3681 y Fs(sym)1154 3731 y Fp(1)1274 3711 y Fw(\()p Fn(R)p Fw(\))36 b Fk(such)30 b(that)f Fr(\031)1833 3723 y Fo(\030)1870 3711 y Fw(\()p Fr(w)r Fw(\))24 b(=)f(0)p Fk(,)1280 3802 y Fm(\015)1280 3852 y(\015)1326 3873 y Fr(e)1365 3839 y Fo(L)1411 3848 y Fi(\030)1442 3839 y Fo(t)1472 3873 y Fr(w)1533 3802 y Fm(\015)1533 3852 y(\015)1579 3906 y Fp(1)1673 3873 y Fq(\024)g Fr(c)1797 3885 y Fs(1)1834 3873 y Fr(e)1873 3839 y Fp(\000)p Fo(!)1967 3847 y Fl(1)1998 3839 y Fo(t)2042 3873 y Fq(k)o Fr(w)r Fq(k)2186 3898 y Fp(1)2355 3873 y Fq(8)14 b Fr(t)22 b Fq(\025)h Fw(0)p Fr(:)611 b Fw(\(2.32\))555 4049 y(In)28 b(particular)e(the)i(in)n(v)n(erse)e Fr(L)1517 4014 y Fp(\000)p Fs(1)1517 4074 y Fo(\030)1634 4049 y Fw(exists)h(and,)g(for)h(some)f(constan)n(t)h(^)-44 b Fr(c)2753 4061 y Fs(1)2814 4049 y Fw(=)24 b(^)-44 b Fr(c)2937 4061 y Fs(1)2974 4049 y Fw(\()p Fr(\024)p Fw(\),)1628 4223 y Fq(k)p Fr(L)1727 4187 y Fp(\000)p Fs(1)1727 4248 y Fo(\030)1815 4223 y Fq(k)1857 4235 y Fp(1)1950 4223 y Fq(\024)25 b Fw(^)-44 b Fr(c)2074 4235 y Fs(1)2111 4223 y Fr(\025)2159 4187 y Fp(\000)p Fs(1)2159 4248 y Fo(\030)2249 4223 y Fr(;)960 b Fw(\(2.33\))456 4386 y(while,)27 b(if)i Fr(w)c Fq(2)f Fr(L)992 4356 y Fs(sym)992 4406 y Fp(1)1111 4386 y Fw(\()p Fn(R)p Fw(\))34 b(is)28 b(suc)n(h)f(that)h Fr(\031)1761 4398 y Fo(\030)1798 4386 y Fw(\()p Fr(w)r Fw(\))c(=)f(0,)k(then)1017 4591 y Fr(L)1074 4556 y Fp(\000)p Fs(1)1074 4616 y Fo(\030)1163 4591 y Fr(w)f Fw(=)c Fq(\000)1414 4478 y Fm(Z)1497 4499 y Fp(1)1460 4667 y Fs(0)1567 4591 y Fr(dt)14 b(e)1693 4557 y Fo(L)1739 4566 y Fi(\030)1771 4557 y Fo(t)1800 4591 y Fr(w)r(;)181 b Fq(k)p Fr(L)2164 4556 y Fp(\000)p Fs(1)2164 4616 y Fo(\030)2252 4591 y Fr(w)r Fq(k)2355 4603 y Fp(1)2448 4591 y Fq(\024)2554 4535 y Fr(c)2590 4547 y Fs(1)p 2546 4572 89 4 v 2546 4648 a Fr(!)2598 4660 y Fs(1)2645 4591 y Fq(k)p Fr(w)r Fq(k)2790 4603 y Fp(1)2860 4591 y Fr(:)349 b Fw(\(2.34\))456 4817 y Fz(F)-8 b(ermi)24 b(co)s(ordinates.)34 b Fw(W)-7 b(e)23 b(in)n(tro)r(duce)g(tubular)f(co)r(ordinates)g(in)h(a)f(neigh)n (b)r(orho)r(o)r(d)f(of)i Fq(M)3354 4829 y Fo(`;\024)456 4917 y Fw(in)d(the)g(follo)n(wing)f(w)n(a)n(y)-7 b(.)34 b(The)20 b(estimates)g(\(2.19\))f(and)h(\(2.28\))f(suggest)g(that)i (for)e Fr(\030)24 b Fw(large)19 b(enough)456 5016 y(the)32 b(v)n(ectors)e Fr(v)929 5028 y Fo(\030)998 5016 y Fw(and)i Fr(@)1208 5028 y Fo(\030)1244 5016 y Fr(n)1294 5028 y Fo(\030)1362 5016 y Fw(are)f(almost)h(parallel.)48 b(Therefore)31 b(w)n(e)h(can)f(use)h(the)g(pro)5 b(jection)456 5116 y(\(2.30\))38 b(to)h(de\014ne)g(a)f(transv)n(erse)f(direction)h(to)h Fq(M)2126 5128 y Fo(`;\024)2216 5116 y Fw(.)71 b(T)-7 b(o)39 b(this)g(end)g(w)n(e)g(shall)f(need)h(the)456 5216 y(follo)n(wing)26 b(lemma,)i(pro)n(v)n(ed)e(in)i(Sect.)g(8.)p eop %%Page: 9 9 9 8 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)291 b(9)456 450 y Fz(Lemma)31 b(2.4.)41 b Fk(Ther)l(e)32 b(ar)l(e)g Fr(\016)1407 462 y Fs(1)1470 450 y Fr(>)25 b Fw(0)31 b Fk(and)h Fr(\013)1849 462 y Fs(1)1912 450 y Fr(>)26 b(\013)31 b Fk(and,)i(for)f(any)g Fr(\024)25 b(>)h Fw(0)p Fk(,)31 b(ther)l(e)h(ar)l(e)f Fr(`)3219 462 y Fs(2)3282 450 y Fq(\025)26 b Fr(`)3408 462 y Fs(1)456 550 y Fk(and)k Fr(c)653 562 y Fs(2)713 550 y Fr(>)23 b Fw(0)29 b Fk(such)h(that,)g(for)h(any)f Fr(h)23 b(<)f(e)1745 520 y Fp(\000)p Fs(2)p Fo(\013`)1901 528 y Fl(2)1967 550 y Fk(and)30 b Fr(\030)e Fq(2)23 b Fw(\000)2322 562 y Fo(`)2350 570 y Fl(2)2382 562 y Fo(;\024)2445 550 y Fk(,)30 b(the)g(fol)t(lowing)i(holds.)555 649 y(1\))e(R)l(e)l(c)l(al)t (ling)38 b Fw(\(2.27\))o Fk(,)902 726 y Fm(\014)902 776 y(\014)930 797 y Fr(@)974 809 y Fo(\030)1010 797 y Fr(v)1050 809 y Fo(\030)1087 797 y Fw(\()p Fr(x)p Fw(\))19 b Fq(\000)34 b Fw(~)-58 b Fr(m)1373 763 y Fp(00)1373 818 y Fo(\030)1416 797 y Fw(\()p Fr(x)p Fw(\))1527 726 y Fm(\014)1527 776 y(\014)1638 797 y Fq(\024)83 b Fr(c)1822 809 y Fs(2)1859 797 y Fr(e)1898 763 y Fp(\000)p Fo(\013\030)2029 797 y Fr(\030)2069 763 y Fs(4)2304 797 y Fk(for)j Fq(j)p Fr(\030)22 b Fq(\000)d Fr(x)p Fq(j)k(\024)g Fr(\030)t(=)p Fw(2)p Fr(;)247 b Fw(\(2.35\))1240 934 y Fq(j)p Fr(@)1307 946 y Fo(\030)1344 934 y Fr(v)1384 946 y Fo(\030)1420 934 y Fw(\()p Fr(x)p Fw(\))p Fq(j)84 b(\024)f Fr(c)1822 946 y Fs(2)1859 934 y Fr(\030)1899 899 y Fs(2)1937 934 y Fr(v)1977 946 y Fo(\030)2013 934 y Fw(\()p Fr(x)p Fw(\))256 b Fq(8)14 b Fr(x)22 b Fq(2)i Fn(R)p Fr(;)566 b Fw(\(2.36\))1352 991 y Fm(\014)1352 1041 y(\014)1352 1090 y(\014)1352 1140 y(\014)1390 1055 y Fr(d\025)1481 1067 y Fo(\030)p 1390 1092 128 4 v 1412 1168 a Fr(d\030)1528 991 y Fm(\014)1528 1041 y(\014)1528 1090 y(\014)1528 1140 y(\014)1638 1111 y Fq(\024)83 b Fr(c)1822 1123 y Fs(2)1859 1111 y Fr(e)1898 1077 y Fp(\000)p Fo(\013)1993 1085 y Fl(1)2025 1077 y Fo(\030)2062 1111 y Fr(:)1147 b Fw(\(2.37\))456 1304 y Fk(Mor)l(e)l(over,)31 b(for)g(any)f Fr( )c Fq(2)d Fr(L)1353 1274 y Fs(sym)1353 1325 y Fp(1)1473 1304 y Fw(\()p Fn(R)p Fw(\))p Fk(,)1575 1448 y Fq(j)p Fr(@)1642 1460 y Fo(\030)1679 1448 y Fr(\031)1726 1460 y Fo(\030)1763 1448 y Fw(\()p Fr( )s Fw(\))p Fq(j)g(\024)g Fr(c)2054 1460 y Fs(2)2091 1448 y Fq(k)p Fr( )s Fq(k)2232 1460 y Fp(1)2302 1448 y Fr(;)907 b Fw(\(2.38\))456 1592 y Fk(wher)l(e)1390 1727 y Fr(@)1434 1739 y Fo(\030)1470 1727 y Fr(\031)1517 1739 y Fo(\030)1554 1727 y Fw(\()p Fr( )s Fw(\))1719 1679 y Fr(:)1699 1727 y Fw(=)1786 1614 y Fm(Z)1869 1634 y Fp(1)1832 1802 y Fs(0)1940 1727 y Fr(dx)14 b(@)2088 1739 y Fo(\030)2125 1727 y Fr(v)2168 1692 y Fp(\003)2165 1747 y Fo(\030)2206 1727 y Fw(\()p Fr(x)p Fw(\))p Fr( )s Fw(\()p Fr(x)p Fw(\))p Fr(:)724 b Fw(\(2.39\))555 1899 y Fk(2\))30 b(We)g(have:)1278 2047 y Fq(k)p Fr(@)1364 2059 y Fo(\030)1400 2047 y Fr(n)1450 2059 y Fo(\030)1505 2047 y Fq(\000)18 b Fr(\031)1635 2059 y Fo(\030)1672 2047 y Fw(\()p Fr(@)1748 2059 y Fo(\030)1784 2047 y Fr(n)1834 2059 y Fo(\030)1871 2047 y Fw(\))p Fr(v)1943 2059 y Fo(\030)1979 2047 y Fq(k)2021 2075 y Fp(1)2174 2047 y Fq(\024)83 b Fr(c)2358 2059 y Fs(2)2395 2047 y Fr(e)2434 2013 y Fp(\000)p Fo(\016)2516 2021 y Fl(1)2548 2013 y Fo(\030)2585 2047 y Fr(;)624 b Fw(\(2.40\))1411 2112 y Fm(\014)1411 2162 y(\014)1439 2183 y Fr(\031)1486 2195 y Fo(\030)1523 2183 y Fw(\()p Fr(@)1599 2195 y Fo(\030)1635 2183 y Fr(n)1685 2195 y Fo(\030)1722 2183 y Fw(\))1754 2148 y Fp(\000)p Fs(1)1861 2183 y Fq(\000)1944 2127 y(p)p 2014 2127 51 4 v 2014 2183 a Fr(\026)2064 2112 y Fm(\014)2064 2162 y(\014)2174 2183 y Fq(\024)83 b Fr(c)2358 2195 y Fs(2)2395 2183 y Fr(e)2434 2148 y Fp(\000)p Fo(\016)2516 2156 y Fl(1)2548 2148 y Fo(\030)2585 2183 y Fr(;)624 b Fw(\(2.41\))1650 2320 y Fq(j)p Fr(@)1717 2332 y Fo(\030)1753 2320 y Fr(\031)1800 2332 y Fo(\030)1837 2320 y Fw(\()p Fr(@)1913 2332 y Fo(\030)1950 2320 y Fr(n)2000 2332 y Fo(\030)2036 2320 y Fw(\))p Fq(j)83 b(\024)g Fr(c)2358 2332 y Fs(2)2395 2320 y Fr(e)2434 2285 y Fp(\000)p Fo(\016)2516 2293 y Fl(1)2548 2285 y Fo(\030)2585 2320 y Fr(:)624 b Fw(\(2.42\))555 2477 y(Giv)n(en)28 b Fr(")22 b(>)h Fw(0,)k Fr(\024)c(>)g Fw(0,)k Fr(`)c Fq(\025)f Fr(`)1468 2489 y Fs(1)1505 2477 y Fw(,)28 b(and)f Fr(h)c(<)g(e)1915 2447 y Fp(\000)p Fs(2)p Fo(\013`)2075 2477 y Fw(,)k(let)1030 2666 y Fq(B)1085 2678 y Fo(`;\024;")1249 2666 y Fw(:=)1359 2549 y Fm(\032)1422 2666 y Fr(m)c Fq(2)g Fr(L)1653 2632 y Fs(sym)1653 2686 y Fp(1)1772 2666 y Fw(\()p Fn(R)q Fw(\))43 b(:)86 b(inf)1994 2720 y Fo(\030)r Fp(2)p Fs(\000)2112 2729 y Fi(`;\024)2208 2666 y Fq(k)p Fr(m)18 b Fq(\000)g Fr(n)2474 2678 y Fo(\030)2510 2666 y Fq(k)2552 2678 y Fp(1)2645 2666 y Fr(<)23 b(")2772 2549 y Fm(\033)2847 2666 y Fr(:)362 b Fw(\(2.43\))456 2855 y(A)40 b(standard)g(application)g(of)g(the)h(Implicit)g(F)-7 b(unction)41 b(Theorem)f(implies)h(the)g(follo)n(wing)456 2955 y(result.)456 3109 y Fz(Theorem)32 b(2.5.)41 b Fk(F)-6 b(or)31 b(any)h Fr(\024)25 b(>)h Fw(0)k Fk(let)i Fr(`)1761 3121 y Fs(2)1829 3109 y Fk(b)l(e)f(as)h(in)f(L)l(emma)g(2.4.)45 b(Then)32 b(ther)l(e)f(ar)l(e)g Fr(")3224 3121 y Fs(0)3287 3109 y Fr(>)25 b Fw(0)p Fk(,)457 3209 y Fw(\026)-43 b Fr(c)23 b Fq(\025)f Fw(1)p Fk(,)31 b Fw(\026)-45 b Fr(\024)23 b(>)f(\024)p Fk(,)28 b(and)f Fr(`)1149 3221 y Fs(3)1209 3209 y Fr(>)1303 3187 y Fw(\026)1297 3209 y Fr(`)c Fq(\025)f Fr(`)1477 3221 y Fs(2)1541 3209 y Fk(such)27 b(that,)h(for)g(any)f Fr(h)c(<)g(e)2404 3179 y Fp(\000)p Fs(2)p Fo(\013`)2560 3187 y Fl(3)2596 3209 y Fk(,)28 b(ther)l(e)f(exists)f(a)i Fr(C)3212 3179 y Fs(1)3249 3209 y Fk({map)456 3309 y Fw(\004)23 b(:)g Fq(B)635 3321 y Fo(`)663 3329 y Fl(3)695 3321 y Fo(;\024;")805 3329 y Fl(0)864 3309 y Fq(!)g Fw(\000)1026 3314 y Fs(\026)1022 3329 y Fo(`;)s Fs(\026)-36 b Fo(\024)1142 3309 y Fk(such)30 b(that,)g(for)h Fr(\030)c Fw(=)c(\004\()p Fr(m)p Fw(\))p Fk(,)1096 3455 y Fr(\031)1143 3467 y Fo(\030)1179 3455 y Fw(\()p Fr(m)c Fq(\000)f Fr(n)1436 3467 y Fo(\030)1472 3455 y Fw(\))84 b(=)e(0)p Fr(;)1432 b Fw(\(2.44\))1091 3579 y Fq(k)p Fr(m)17 b Fq(\000)h Fr(n)1356 3591 y Fo(\030)1393 3579 y Fq(k)1435 3591 y Fp(1)1588 3579 y Fq(\024)84 b Fw(\026)-44 b Fr(c)28 b Fw(inf)7 b Fq(fk)p Fr(m)17 b Fq(\000)h Fr(n)2207 3591 y Fo(\020)2245 3579 y Fq(k)2287 3591 y Fp(1)2380 3579 y Fw(:)23 b Fr(\020)30 b Fq(2)23 b Fw(\000)2622 3591 y Fo(`)2650 3599 y Fl(3)2682 3591 y Fo(;\024)2745 3579 y Fq(g)p Fr(:)422 b Fw(\(2.45\))456 3723 y Fk(Mor)l(e)l(over,)31 b(if)g Fr(\030)963 3735 y Fs(0)1023 3723 y Fq(2)24 b Fw(\000)1154 3735 y Fo(`)1182 3743 y Fl(3)1214 3735 y Fo(;\024)1306 3723 y Fk(and)31 b Fq(k)p Fr(m)17 b Fq(\000)h Fr(n)1733 3735 y Fo(\030)1763 3743 y Fl(0)1800 3723 y Fq(k)1842 3735 y Fp(1)1935 3723 y Fr(<)23 b(")2062 3735 y Fs(0)2099 3723 y Fk(,)30 b(then:)1512 3866 y Fq(j)p Fr(\030)23 b Fq(\000)18 b Fr(\030)1713 3878 y Fs(0)1751 3866 y Fq(j)23 b(\024)h Fw(\026)-44 b Fr(c)p Fq(k)p Fr(m)18 b Fq(\000)g Fr(n)2186 3878 y Fo(\030)2216 3886 y Fl(0)2253 3866 y Fq(k)2295 3878 y Fp(1)2365 3866 y Fr(:)844 b Fw(\(2.46\))555 4021 y(The)37 b(pro)r(of,)h(whic)n(h)f(is)f(sk)n(etc)n(hed)g(at)h(the)g(end)g(of)f (Sect.)h(8,)i(is)d(a)g(standard)g(application)456 4120 y(of)e(the)h(Con)n(traction)d(Mapping)i(Principle,)i(see)e(also)f(the)h (analogous)e(result)i(in)h([2)o(].)58 b(As)34 b(a)456 4220 y(corollary)25 b(w)n(e)i(get)g(the)h(existence)g(of)f(tubular)h (co)r(ordinates.)35 b(Let)907 4343 y(\026)888 4364 y Fq(S)938 4376 y Fo(`;\024;")1124 4317 y Fr(:)1103 4364 y Fw(=)23 b Fq(f)o Fw(\()p Fr(\030)t(;)14 b(')p Fw(\))24 b Fq(2)f Fw(\000)1581 4376 y Fo(`;\024)1690 4364 y Fq(\002)18 b Fr(L)1830 4329 y Fs(sym)1830 4384 y Fp(1)1950 4364 y Fw(\()p Fn(R)p Fw(\))43 b(:)37 b Fr(\031)2218 4376 y Fo(\030)2255 4364 y Fw(\()p Fr(')p Fw(\))24 b(=)e(0)p Fr(;)28 b Fq(k)p Fr(')p Fq(k)2715 4376 y Fp(1)2807 4364 y Fr(<)23 b(")p Fq(g)13 b Fr(:)220 b Fw(\(2.47\))456 4508 y(F)-7 b(or)27 b(\()p Fr(\030)t(;)14 b(')p Fw(\))24 b Fq(2)921 4487 y Fw(\026)902 4508 y Fq(S)952 4520 y Fo(`;\024;")1122 4508 y Fw(de\014ne)j Fr(M)9 b Fw(\()p Fr(\030)t(;)14 b(')p Fw(\))1691 4461 y Fr(:)1670 4508 y Fw(=)23 b Fr(n)1808 4520 y Fo(\030)1862 4508 y Fw(+)18 b Fr(')28 b Fw(and)g Fq(S)2239 4520 y Fo(`;\024;")2424 4461 y Fr(:)2404 4508 y Fw(=)22 b Fr(M)9 b Fw(\()2632 4487 y(\026)2613 4508 y Fq(S)2663 4520 y Fo(`;\024;")2805 4508 y Fw(\).)37 b(Then:)456 4662 y Fz(Corollary)k(2.6.)k Fk(F)-6 b(or)37 b(any)g Fr(\024)e(>)g Fw(0)p Fk(,)k(let)d Fr(")1864 4674 y Fs(0)1938 4662 y Fk(and)h Fr(`)2141 4674 y Fs(3)2214 4662 y Fk(b)l(e)g(as)g(in)g(The)l(or)l(em)g(2.6.)61 b(Then,)39 b(for)456 4762 y(any)31 b Fr(h)25 b(<)g(e)818 4732 y Fp(\000)p Fs(2)p Fo(\013`)974 4740 y Fl(3)1010 4762 y Fk(,)31 b(the)g(map)h Fr(M)48 b Fw(:)1599 4741 y(\026)1580 4762 y Fq(S)1630 4774 y Fo(`)1658 4782 y Fl(3)1690 4774 y Fo(;\024;")1800 4782 y Fl(0)1862 4762 y Fq(!)25 b(S)2020 4774 y Fo(`)2048 4782 y Fl(3)2080 4774 y Fo(;\024;")2190 4782 y Fl(0)2257 4762 y Fk(is)32 b(di\013er)l(entiable,)h(one)e(to)g(one,)h(and)456 4862 y Fw(\004\()p Fr(M)9 b Fw(\()p Fr(\030)t(;)14 b(')p Fw(\)\))24 b(=)f Fr(\030)t Fk(.)39 b(Mor)l(e)l(over)31 b Fq(S)1491 4874 y Fo(`)1519 4882 y Fl(3)1551 4874 y Fo(;\024;")1661 4882 y Fl(0)1727 4862 y Fk(is)f(op)l(en)g(in)g Fr(L)2170 4832 y Fs(sym)2170 4882 y Fp(1)2290 4862 y Fw(\()p Fn(R)p Fw(\))p Fk(.)555 5016 y Fw(Corollary)d(2.6)h(th)n(us)h(giv)n(es)f(the)h (existence)g(of)g(tubular)f(co)r(ordinates)g(in)h(a)g(neigh)n(b)r(orho) r(o)r(d)456 5116 y(of)c(the)h(manifold)g Fq(M)1131 5128 y Fo(`;\024)1221 5116 y Fw(.)37 b(W)-7 b(e)26 b(p)r(oin)n(t)f(out)h (that)g(the)g(size)f(of)h(the)g(ab)r(o)n(v)n(e)e(neigh)n(b)r(orho)r(o)r (d)h(is)g(not)456 5216 y(v)-5 b(anishing)27 b(as)g Fr(h)c Fq(#)f Fw(0.)p eop %%Page: 10 10 10 9 bop 456 251 a Fs(10)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fz(Equations)24 b(of)h(motion.)33 b Fw(Consider)20 b(a)i(solution)f(of)28 b(\(2.1\))21 b Fr(S)2401 462 y Fo(t)2431 450 y Fw(\()p Fr(m)p Fw(\),)i Fr(t)g Fq(2)h Fw([0)p Fr(;)14 b(T)e Fw(],)21 b Fr(T)34 b(>)23 b Fw(0,)g(whic)n(h)456 550 y(lies)29 b(in)h(the)f(tubular)h (neigh)n(b)r(orho)r(o)r(d)e Fq(S)1715 562 y Fo(`)1743 570 y Fl(3)1775 562 y Fo(;\024;")1885 570 y Fl(0)1951 550 y Fw(with)i Fr(`)2177 562 y Fs(3)2243 550 y Fw(and)g Fr(")2446 562 y Fs(0)2512 550 y Fw(as)f(in)h(Corollary)d(2.6.)41 b(Then)456 649 y(w)n(e)27 b(can)g(decomp)r(ose)g Fr(S)1199 661 y Fo(t)1228 649 y Fw(\()p Fr(m)p Fw(\))d(=)f Fr(n)1527 664 y Fo(\030)r Fs(\()p Fo(t)p Fs(\))1658 649 y Fw(+)c Fr(')p Fw(\()p Fr(t)p Fw(\).)38 b(By)27 b(di\013eren)n(tiating)g(the)h (iden)n(tit)n(y)g Fr(\031)3113 661 y Fo(\030)3150 649 y Fw(\()p Fr(')p Fw(\))c(=)f(0,)456 749 y(from)k(\(2.1\))g(w)n(e)g (obtain)h(the)g(equations)e(of)i(motion)f(in)h(the)g(F)-7 b(ermi)28 b(co)r(ordinates)e(\()p Fr(\030)t(;)14 b(')p Fw(\):)1084 910 y([)p Fr(\031)1154 922 y Fo(\030)1205 910 y Fw(\()p Fr(@)1281 922 y Fo(\030)1318 910 y Fr(n)1368 922 y Fo(\030)1404 910 y Fw(\))19 b Fq(\000)f Fr(@)1582 922 y Fo(\030)1618 910 y Fr(\031)1665 922 y Fo(\030)1702 910 y Fw(\()p Fr(')p Fw(\)])1875 888 y(_)1857 910 y Fr(\030)88 b Fw(=)82 b Fr(\031)2175 922 y Fo(\030)2226 910 y Fw(\()p Fr(f)9 b Fw(\()p Fr(n)2390 922 y Fo(\030)2445 910 y Fw(+)18 b Fr(')p Fw(\)\))c Fr(;)549 b Fw(\(2.48\))1866 1044 y(_)-46 b Fr(')84 b Fw(=)e Fr(f)9 b Fw(\()p Fr(n)2260 1056 y Fo(\030)2315 1044 y Fw(+)18 b Fr(')p Fw(\))h Fq(\000)f Fr(@)2630 1056 y Fo(\030)2666 1044 y Fr(n)2716 1056 y Fo(\030)2770 1023 y Fw(_)2752 1044 y Fr(\030)5 b(;)416 b Fw(\(2.49\))456 1208 y(where)28 b(\()747 1186 y(_)729 1208 y Fr(\030)t(;)37 b Fw(_)-46 b Fr(')q Fw(\))29 b(denotes)f(the)i (time)f(deriv)-5 b(ativ)n(e)28 b(of)h(\()p 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b(e)21 b(ma)n(y)f(assume)f Fr(")1182 2881 y Fs(1)1240 2869 y Fw(so)h(small)g(and)g Fr(`)1734 2881 y Fs(4)1791 2869 y Fw(so)g(large)f(that)i(\(1)t(+)t Fr(Q)p Fw(\))p Fr(")2539 2881 y Fs(1)2598 2869 y Fr(<)i(")2725 2881 y Fs(0)2782 2869 y Fw(and)d Fr(e)2975 2839 y Fp(\000)p Fs(3)p Fo(\013`)3131 2847 y Fl(4)3163 2839 y Fo(=)p Fs(2)3258 2869 y Fr(<)i(")3384 2881 y Fs(1)3421 2869 y Fw(.)456 2968 y(By)28 b(\(2.51\))f(w)n(e)g(th)n(us)h(see)f(that)h(the)f(tub)r(e) i Fq(S)1830 2980 y Fo(`)1858 2988 y Fl(4)1890 2980 y Fo(;\024;")2000 2988 y Fl(1)2064 2968 y Fw(is)f(exp)r(onen)n(tially)e (attracted)h(to)n(w)n(ards)f(the)456 3068 y(c)n(hannel)31 b Fq(Z)822 3080 y Fo(`)850 3088 y Fl(4)883 3080 y Fo(;\024;)p Fs(1)998 3068 y Fw(.)51 b(Moreo)n(v)n(er,)30 b(for)i Fr(m)e Fq(2)h(Z)1846 3080 y Fo(`;\024;Q)2041 3068 y Fw(there)h(are)f (only)h(t)n(w)n(o)f(p)r(ossibilities:)46 b(either)456 3168 y Fr(S)507 3180 y Fo(t)536 3168 y Fw(\()p Fr(m)p Fw(\))28 b(sta)n(ys)f(forev)n(er)g(in)h Fq(Z)1343 3180 y Fo(`;\024;)p Fs(1)1487 3168 y Fw(,)g(or)f(there)h(is)g(a)g(\014nite)h (time)f Fr(t)2439 3137 y Fp(\003)2505 3168 y Fw(for)g(whic)n(h)g Fr(\030)t Fw(\()p Fr(t)2973 3137 y Fp(\003)3012 3168 y Fw(\))g(b)r(elongs)g(to)456 3267 y(the)i(b)r(oundary)g(of)g(\000)1127 3279 y Fo(`)1155 3287 y Fl(4)1187 3279 y Fo(;\024)1250 3267 y Fw(.)44 b(The)31 b(dynamics)e(in)i(the)g(c)n(hannel)e Fq(Z)2470 3279 y Fo(`)2498 3287 y Fl(4)2531 3279 y Fo(;\024;)p Fs(1)2677 3267 y Fw(is)h(essen)n(tially)f(reduced)456 3367 y(to)e(the)h(motion)f(of)h(the)g(co)r(ordinate)e Fr(\030)t Fw(\()p Fr(t)p Fw(\),)j(for)e(whic)n(h)g(w)n(e)h(ha)n(v)n(e)e (an)h(explicit)h(form)n(ula:)456 3527 y Fz(Theorem)g(2.8.)39 b Fk(F)-6 b(or)28 b(any)h Fr(\024)23 b(>)g Fw(0)k Fk(ther)l(e)i(is)g Fr(\016)1921 3497 y Fp(\003)1982 3527 y Fr(>)23 b Fw(0)k Fk(such)i(that,)g(for)g(any)g Fr(`)22 b(>)h(`)2990 3539 y Fs(3)3055 3527 y Fk(su\016ciently)456 3627 y(lar)l(ge,)31 b(as)f(long)g(as)g Fr(S)1123 3639 y Fo(t)1152 3627 y Fw(\()p Fr(m)p Fw(\))24 b(=)e Fr(n)1450 3642 y Fo(\030)r Fs(\()p Fo(t)p Fs(\))1582 3627 y Fw(+)c Fr(')p Fw(\()p Fr(t)p Fw(\))24 b Fq(2)f(Z)1975 3639 y Fo(`;\024;)p Fs(1)2119 3627 y Fk(,)30 b(one)g(has:)1462 3793 y Fw(_)1444 3815 y Fr(\030)d Fw(=)c Fr(V)c Fw(\()p Fr(\030)t Fw(\))g(+)f Fr(O)1947 3722 y Fm(\020)1997 3815 y Fr(e)2036 3780 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)2273 3755 y Fj(\003)2307 3780 y Fs(\))p Fo(\030)2370 3722 y Fm(\021)2433 3815 y Fr(;)776 b Fw(\(2.52\))456 3993 y Fk(with)30 b Fr(V)19 b Fw(\()p Fr(\030)t Fw(\))30 b Fk(as)g(in)37 b Fw(\(2.21\))o Fk(.)555 4154 y Fw(Theorems)27 b(2.7)g(and)g(2.8)g (will)h(b)r(e)g(pro)n(v)n(ed)e(in)h(Sect.)38 b(3.)456 4281 y Fz(The)d(critical)h(droplet.)45 b Fw(Our)30 b(next)h(result)g (concerns)e(the)i(existence)g(and)f(uniqueness)h(of)456 4380 y(the)25 b(critical)g(droplet)g(in)h(a)e(neigh)n(b)r(orho)r(o)r(d) h(of)g Fq(M)2028 4392 y Fo(`;\024)2118 4380 y Fw(.)36 b(As)26 b(already)e(men)n(tioned)h(Theorem)f(2.1)456 4480 y(do)r(es)i(not)g(pro)n(vide)f(an)n(y)h(uniqueness)g(result:)36 b(the)27 b(follo)n(wing)e(theorem)h(is)h(pro)n(v)n(ed)d(in)j(Sect.)g (4.)456 4641 y Fz(Theorem)e(2.9.)37 b Fk(Ther)l(e)27 b(is)f Fr(\024)1411 4653 y Fs(0)1474 4641 y Fk(such)h(that)f(the)g(fol) t(lowing)i(holds.)39 b(F)-6 b(or)27 b(any)f Fr(\024)d Fq(\025)g Fr(\024)3067 4653 y Fs(0)3130 4641 y Fk(ther)l(e)j(ar)l(e)456 4740 y Fr(")495 4752 y Fs(2)560 4740 y Fq(2)k Fw(\(0)p Fr(;)14 b(")795 4752 y Fs(0)831 4740 y Fw(])p Fk(,)34 b Fr(`)948 4752 y Fs(5)1014 4740 y Fq(\025)28 b Fr(`)1142 4752 y Fs(3)1179 4740 y Fk(,)34 b(and)g Fr(h)1451 4752 y Fs(0)1517 4740 y Fr(<)28 b(e)1649 4710 y Fp(\000)p Fs(2)p Fo(\013`)1805 4718 y Fl(5)1874 4740 y Fk(such)33 b(that,)h(for)g(any)f Fr(h)c Fq(2)g Fw(\(0)p Fr(;)14 b(h)2883 4752 y Fs(0)2920 4740 y Fw(])p Fk(,)34 b(the)f(e)l(quation)456 4840 y Fr(f)9 b Fw(\()p Fr(m)p Fw(\))23 b(=)f(0)30 b Fk(in)f Fq(S)976 4852 y Fo(`)1004 4860 y Fl(5)1037 4852 y Fo(;\024;")1147 4860 y Fl(2)1213 4840 y Fk(has)h(a)g(unique)g (solution)g Fr(m)23 b Fw(=)f Fr(n)2255 4845 y Fs(\026)2248 4860 y Fo(\030)2303 4840 y Fw(+)31 b(\026)-55 b Fr(')p Fk(.)39 b(Mor)l(e)l(over:)1103 5017 y Fw(lim)1108 5071 y Fo(h)p Fp(#)p Fs(0)1232 5017 y Fr(e)1271 4983 y Fs(2)p Fo(\013)1354 4967 y Fs(\026)1347 4983 y Fo(\030)1384 5017 y Fr(h)22 b Fw(=)h(\(2)p Fr(m)1689 5029 y Fo(\014)1734 5017 y Fw(\))1766 4983 y Fp(\000)p Fs(1)1855 5017 y Fr(K)q(;)183 b Fw(lim)2138 5071 y Fo(h)p Fp(#)p Fs(0)2263 5017 y Fr(e)2302 4983 y Fs(2)p Fo(\013)2385 4967 y Fs(\026)2378 4983 y Fo(\030)2414 5017 y Fq(k)13 b Fw(\026)-55 b Fr(')p Fq(k)2552 5029 y Fp(1)2645 5017 y Fw(=)22 b(0)p Fr(;)435 b Fw(\(2.53\))456 5216 y Fk(with)30 b Fr(K)35 b Fk(as)30 b(in)36 b Fw(\(2.14\))p Fk(.)p eop %%Page: 11 11 11 10 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(11)555 450 y Fw(F)-7 b(rom)27 b(\(2.53\))o(,)h(\(2.18\))o (,)f(\(2.12\),)g(and)g(\(2.13\))g(this)g(solution)g(to)g Fr(f)9 b Fw(\()p Fr(m)p Fw(\))23 b(=)g(0)k(has)g(to)g(coincide)456 550 y(with)h(the)g(critical)f(droplet)g(of)g(Theorem)g(2.1,)g(i.e.)h Fr(q)e Fw(=)c Fr(n)2294 555 y Fs(\026)2287 570 y Fo(\030)2342 550 y Fw(+)31 b(\026)-55 b Fr(')p Fw(.)555 649 y(W)-7 b(e)32 b(ha)n(v)n(e)e(a)h(detailed)g(information)g(on)f(the)i(spatial)f (structure)f(of)i(the)f(critical)g(droplet,)456 749 y(whic)n(h)c(is)h (the)g(con)n(ten)n(t)f(of)g(the)h(follo)n(wing)f(prop)r(osition,)f(pro) n(v)n(ed)h(in)g(Sect.)h(5.)456 901 y Fz(Prop)s(osition)40 b(2.10.)46 b Fk(Given)39 b Fr(\014)j(>)c Fw(1)p Fk(,)j(ther)l(e)d(is)h Fr(h)2138 871 y Fp(\003)2214 901 y Fq(2)g Fw(\(0)p Fr(;)14 b(h)2467 913 y Fs(0)2504 901 y Fw(])38 b Fk(\()p Fr(h)2647 913 y Fs(0)2722 901 y Fk(as)h(in)f(The)l(or)l(em)h(2.1\))456 1001 y(such)29 b(that)h(for)h(any)f Fr(h)23 b Fq(2)h Fw(\(0)p Fr(;)14 b(h)1415 971 y Fp(\003)1452 1001 y Fw(])30 b Fk(the)g(bump)g Fr(q)s Fw(\()p Fr(x)p Fw(\))h Fk(is)f(a)g(strictly)g (de)l(cr)l(e)l(asing)h(function)e(on)h Fn(R)3383 1013 y Fs(+)456 1100 y Fk(\(actual)t(ly)g Fr(q)841 1070 y Fp(0)865 1100 y Fw(\()p Fr(x)p Fw(\))24 b Fr(<)e Fw(0)30 b Fk(for)g(al)t(l)h Fr(x)24 b(>)e Fw(0)p Fk(\).)38 b(Mor)l(e)l(over,)32 b(letting)e Fr(\015)d(>)c Fw(0)29 b Fk(b)l(e)h(such)g(that)1318 1282 y Fr(\014)1383 1190 y Fm(h)1422 1282 y Fw(1)18 b Fq(\000)g Fw(\()p Fr(m)1670 1246 y Fp(\000)1670 1307 y Fo(\014)s(;h)1774 1282 y Fw(\))1806 1248 y Fs(2)1843 1190 y Fm(i)1896 1169 y(Z)1979 1282 y Fr(dz)g(J)8 b Fw(\()p Fr(z)t Fw(\))p Fr(e)2279 1248 y Fp(\000)p Fo(\015)t(z)2430 1282 y Fw(=)23 b(1)p Fr(;)649 b Fw(\(2.54\))456 1463 y Fk(and)31 b Fr(\030)654 1475 y Fo(q)721 1463 y Fk(b)l(e)g(the)f (\(unique\))g(p)l(ositive)i(zer)l(o)f(of)h Fr(q)s Fw(\()p Fr(x)p Fw(\))p Fk(,)g(ther)l(e)e(ar)l(e)h Fr(A)25 b(>)f Fw(0)p Fk(,)31 b Fr(\016)c(>)d Fw(0)p Fk(,)31 b(and)g Fr(C)g(>)24 b Fw(0)30 b Fk(so)456 1563 y(that,)g(for)g(al)t(l)h Fr(x)24 b Fq(\025)e Fr(\030)1096 1575 y Fo(q)1133 1563 y Fk(,)973 1642 y Fm(\014)973 1692 y(\014)1001 1712 y Fr(q)s Fw(\()p Fr(x)p Fw(\))d Fq(\000)f Fw(\()p Fr(m)1359 1677 y Fp(\000)1359 1737 y Fo(\014)s(;h)1481 1712 y Fw(+)g Fr(Ae)1665 1678 y Fp(\000)p Fo(\015)t Fs(\()p Fo(x)p Fp(\000)p Fo(\030)1902 1686 y Fi(q)1934 1678 y Fs(\))1964 1712 y Fw(\))1996 1642 y Fm(\014)1996 1692 y(\014)2043 1712 y Fw(+)2126 1642 y Fm(\014)2126 1692 y(\014)2153 1712 y Fr(q)2193 1678 y Fp(0)2217 1712 y Fw(\()p Fr(x)p Fw(\))h(+)f Fr(\015)5 b(Ae)2579 1678 y Fp(\000)p Fo(\015)t Fs(\()p Fo(x)p Fp(\000)p Fo(\030)2816 1686 y Fi(q)2848 1678 y Fs(\))2878 1642 y Fm(\014)2878 1692 y(\014)1502 1860 y Fw(+)1585 1789 y Fm(\014)1585 1839 y(\014)1612 1860 y Fr(q)1652 1825 y Fp(00)1695 1860 y Fw(\()p Fr(x)p Fw(\))19 b Fq(\000)f Fr(\015)1956 1825 y Fs(2)1993 1860 y Fr(Ae)2094 1825 y Fp(\000)p Fo(\015)t Fs(\()p Fo(x)p Fp(\000)p Fo(\030)2331 1833 y Fi(q)2363 1825 y Fs(\))2393 1789 y Fm(\014)2393 1839 y(\014)2444 1860 y Fq(\024)23 b Fr(C)6 b(e)2636 1825 y Fp(\000)p Fs(\()p Fo(\015)t Fs(+)p Fo(\016)r Fs(\)\()p Fo(x)p Fp(\000)p Fo(\030)3008 1833 y Fi(q)3040 1825 y Fs(\))3070 1860 y Fr(:)139 b Fw(\(2.55\))456 1999 y Fk(Final)t(ly,)26 b(as)e Fr(h)f Fq(#)f Fw(0)p Fk(,)j Fr(A)p Fk(,)g Fr(\016)s Fk(,)g(and)f Fr(C)29 b Fk(r)l(emain)24 b(strictly)f(p)l(ositive)i(and)e(b)l(ounde)l (d,)j(while)e Fr(\030)3105 2011 y Fo(q)3165 1999 y Fq(!)f Fw(+)p Fq(1)p Fk(.)456 2171 y Fz(The)36 b(in)m(v)-5 b(arian)m(t)37 b(manifold.)45 b Fw(The)31 b(qualitativ)n(e)g(b)r(eha)n(vior)f(of)h (the)g(dynamics)g(around)f(the)456 2270 y(bump)25 b(follo)n(ws)f(from)g (the)h(sp)r(ectral)f(prop)r(erties)g(of)h(the)g(linear)f(op)r(erator)f Fr(L)2880 2223 y(:)2860 2270 y Fw(=)f Fr(D)r(f)9 b Fq(j)3091 2282 y Fo(q)3151 2270 y Fw(=)22 b Fr(L)3295 2282 y Fo(q)r Fs(+)p Fo(h)3421 2270 y Fw(.)456 2370 y(Since)27 b Fr(q)k Fw(satis\014es)c(\(1.7\))g(w)n(e)g(compute,)h(for)f(an)n(y)g Fr( )f Fq(2)e Fr(L)2226 2340 y Fs(sym)2226 2390 y Fp(1)2345 2370 y Fw(\()p Fn(R)p Fw(\))q(,)1599 2509 y Fr(L )i Fw(=)c Fq(\000)p Fr( )f Fw(+)e Fr(pJ)26 b Fq(\003)18 b Fr( )s(;)931 b Fw(\(2.56\))456 2648 y(where)1566 2752 y Fr(p)p Fw(\()p Fr(x)p Fw(\))1764 2705 y Fr(:)1743 2752 y Fw(=)22 b Fr(\014)1896 2685 y Fm(\002)1930 2752 y Fw(1)c Fq(\000)g Fr(q)s Fw(\()p Fr(x)p Fw(\))2224 2718 y Fs(2)2263 2685 y Fm(\003)2311 2752 y Fr(:)898 b Fw(\(2.57\))555 2874 y(Giv)n(en)28 b Fr(\020)h Fq(2)24 b Fn(R)p Fw(,)33 b(w)n(e)27 b(in)n(tro)r(duce)h (the)g(normed)f(spaces)742 3013 y Fr(X)811 3025 y Fo(\020)892 2966 y Fr(:)872 3013 y Fw(=)22 b Fq(f)p Fr(w)j Fw(:)f Fn(R)29 b Fq(!)23 b Fn(R)47 b Fw(measurable)26 b(and)i(symmetric)50 b(:)37 b Fq(k)p Fr(w)r Fq(k)2650 3025 y Fo(\020)2711 3013 y Fr(<)23 b Fq(1g)p Fr(;)285 b Fw(\(2.58\))456 3152 y(where)1532 3256 y Fq(k)p Fr(w)r Fq(k)1677 3268 y Fo(\020)1759 3209 y Fr(:)1738 3256 y Fw(=)46 b(sup)1826 3326 y Fo(x)p Fp(2)p Fh(R)1956 3334 y Fl(+)2011 3256 y Fr(e)2050 3222 y Fo(\020)t(x)2125 3256 y Fq(j)p Fr(w)r Fw(\()p Fr(x)p Fw(\))p Fq(j)p Fr(:)456 3443 y Fw(The)22 b(follo)n(wing)f(prop)r (osition)g(follo)n(ws)g(from)h(the)g(results)g(pro)n(v)n(ed)e(in)i([6,) g(7])g(and)g(for)f(the)i(reader)456 3542 y(con)n(v)n(enience)28 b(w)n(e)h(giv)n(e)g(detailed)h(references)f(on)g(where)g(the)i(pro)r (ofs)d(can)i(b)r(e)g(found)g(in)g(Sect.)456 3642 y(8.)456 3794 y Fz(Prop)s(osition)h(2.11.)41 b Fk(Given)32 b Fr(\014)e(>)25 b Fw(1)30 b Fk(let)h Fr(h)1861 3764 y Fp(\003)1930 3794 y Fk(b)l(e)g(as)h(in)f(Pr)l(op)l(osition)i(2.10.)44 b(Then)32 b(ther)l(e)f(ar)l(e)456 3893 y(c)l(onstants)e Fr(C)880 3905 y Fs(0)940 3893 y Fr(>)23 b Fw(1)29 b Fk(and)h Fr(C)1319 3905 y Fs(1)1380 3893 y Fr(>)23 b Fw(0)29 b Fk(such)h(that)f(for)i(any) f Fr(h)23 b Fq(2)g Fw(\(0)p Fr(;)14 b(h)2497 3863 y Fp(\003)2535 3893 y Fw(])30 b Fk(the)g(fol)t(lowing)i(holds.)555 3993 y(1\))e(Ther)l(e)h(ar)l(e)f Fr(\025)24 b(>)e Fw(0)29 b Fk(and)i(strictly)f(p)l(ositive)h(functions)f Fr(v)s(;)14 b(v)2495 3963 y Fp(\003)2556 3993 y Fq(2)23 b Fr(C)2699 3953 y Fs(sym)2693 4015 y(0)2820 3993 y Fw(\()p Fn(R)p Fw(\))36 b Fk(so)30 b(that:)1495 4132 y Fr(Lv)c Fw(=)c Fr(\025v)s(;)185 b(v)2047 4098 y Fp(\003)2085 4132 y Fr(L)23 b Fw(=)f Fr(\025v)2343 4098 y Fp(\003)2382 4132 y Fr(;)827 b Fw(\(2.59\))1413 4276 y Fr(v)1456 4242 y Fp(\003)1494 4276 y Fw(\()p Fr(x)p Fw(\))24 b(=)f Fr(p)p Fw(\()p Fr(x)p Fw(\))p Fr(v)s Fw(\()p Fr(x)p Fw(\))171 b Fq(8)14 b Fr(x)23 b Fq(2)g Fn(R)p Fr(;)751 b Fw(\(2.60\))456 4398 y Fk(and)1707 4468 y Fr(h)p 1683 4505 97 4 v 1683 4581 a(C)1742 4593 y Fs(0)1813 4524 y Fq(\024)22 b Fr(\025)i Fq(\024)e Fr(C)2118 4536 y Fs(0)2156 4524 y Fr(h:)1005 b Fw(\(2.61\))555 4686 y Fk(2\))30 b(Ther)l(e)h(is)f(a)g(unique)g Fr(\015)1372 4698 y Fo(v)1434 4686 y Fr(>)23 b(\015)k(>)c Fw(0)p Fk(,)30 b(\()p Fr(\015)k Fk(as)c(in)36 b Fw(\(2.54\))o Fk(\))30 b(such)g(that:)1217 4868 y Fr(\014)1282 4776 y Fm(\020)1331 4868 y Fw(1)18 b Fq(\000)g Fw(\()p Fr(m)1579 4832 y Fp(\000)1579 4893 y Fo(\014)s(;h)1683 4868 y Fw(\))1715 4833 y Fs(2)1752 4776 y Fm(\021)1816 4755 y(Z)1899 4868 y Fr(dz)f(J)8 b Fw(\()p Fr(z)t Fw(\))p Fr(e)2198 4833 y Fp(\000)p Fo(\015)2285 4841 y Fi(v)2320 4833 y Fo(z)2381 4868 y Fw(=)23 b(1)18 b(+)g Fr(\025;)549 b Fw(\(2.62\))456 5045 y Fk(and)30 b(ther)l(e)g(is)g Fr(M)994 5057 y Fo(v)1056 5045 y Fr(>)23 b Fw(0)29 b Fk(for)h(which)1605 5184 y Fw(lim)1553 5234 y Fo(x)p Fp(!)p Fs(+)p Fp(1)1787 5184 y Fr(e)1826 5149 y Fo(\015)1861 5157 y Fi(v)1897 5149 y Fo(x)1938 5184 y Fr(v)s Fw(\()p Fr(x)p Fw(\))25 b(=)d Fr(M)2285 5196 y Fo(v)2324 5184 y Fr(:)885 b Fw(\(2.63\))p eop %%Page: 12 12 12 11 bop 456 251 a Fs(12)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fk(Mor)l(e)l(over,)31 b(for)g(any)f Fr(\020)f Fq(\024)23 b Fr(\015)1334 462 y Fo(v)1403 450 y Fk(and)30 b Fr(t)23 b Fq(\025)g Fw(0)p Fk(,)1287 522 y Fm(\015)1287 572 y(\015)1333 592 y Fr(e)1372 558 y Fo(Lt)1447 592 y Fr(w)1508 522 y Fm(\015)1508 572 y(\015)1555 626 y Fo(\020)1616 592 y Fq(\024)f Fr(C)1762 604 y Fs(1)1800 592 y Fr(e)1839 558 y Fo(\025t)1907 592 y Fq(k)p Fr(w)r Fq(k)2052 604 y Fo(\020)2260 592 y Fq(8)14 b Fr(w)24 b Fq(2)g Fr(X)2552 604 y Fo(\020)2590 592 y Fr(:)619 b Fw(\(2.64\))555 750 y Fk(3\))30 b(Assume)f Fr(v)k Fk(and)d Fr(v)1248 720 y Fp(\003)1317 750 y Fk(ar)l(e)g(normalize)l(d)h(in)f (such)f(a)i(way)f(that)1269 833 y Fm(Z)1352 853 y Fp(1)1315 1021 y Fs(0)1422 946 y Fr(dx)1536 890 y(v)s Fw(\()p Fr(x)p Fw(\))1690 860 y Fs(2)p 1536 927 193 4 v 1556 1003 a Fr(p)p Fw(\()p Fr(x)p Fw(\))1762 946 y Fq(\021)1850 833 y Fm(Z)1933 853 y Fp(1)1896 1021 y Fs(0)2003 946 y Fr(dx)14 b(v)2150 912 y Fp(\003)2189 946 y Fw(\()p Fr(x)p Fw(\))p Fr(v)s Fw(\()p Fr(x)p Fw(\))25 b(=)e(1)p Fr(;)600 b Fw(\(2.65\))456 1132 y Fk(and)30 b(de\014ne)g(the)g(line)l(ar)g(functional)g Fr(\031)j Fk(on)d Fr(X)1889 1144 y Fo(\020)1927 1132 y Fk(,)g Fr(\020)g(<)22 b(\015)2178 1144 y Fo(v)2218 1132 y Fk(,)30 b(as:)1487 1324 y Fr(\031)s Fw(\()p Fr( )s Fw(\))1703 1277 y Fr(:)1682 1324 y Fw(=)1770 1211 y Fm(Z)1853 1232 y Fp(1)1816 1400 y Fs(0)1923 1324 y Fr(dx)14 b(v)2070 1290 y Fp(\003)2109 1324 y Fw(\()p Fr(x)p Fw(\))p Fr( )s Fw(\()p Fr(x)p Fw(\))p Fr(:)821 b Fw(\(2.66\))456 1513 y Fk(Then)30 b(ther)l(e)g(is)g Fr(!)c(>)c Fw(0)30 b Fk(so)g(that,)g (for)g(any)h Fr(w)25 b Fq(2)f Fr(L)2018 1483 y Fs(sym)2018 1534 y Fp(1)2137 1513 y Fw(\()p Fn(R)p Fw(\))36 b Fk(such)30 b(that)g Fr(\031)s Fw(\()p Fr(w)r Fw(\))24 b(=)f(0)29 b Fk(and)i Fr(t)23 b Fq(\025)f Fw(0)p Fk(,)1463 1589 y Fm(\015)1463 1638 y(\015)1509 1659 y Fr(e)1548 1625 y Fo(Lt)1622 1659 y Fr(w)1683 1589 y Fm(\015)1683 1638 y(\015)1730 1692 y Fp(1)1824 1659 y Fq(\024)g Fr(C)1970 1671 y Fs(1)2008 1659 y Fr(e)2047 1625 y Fp(\000)p Fo(!)r(t)2185 1659 y Fq(k)p Fr(w)r Fq(k)2330 1684 y Fp(1)2414 1659 y Fr(:)795 b Fw(\(2.67\))456 1802 y Fk(Mor)l(e)l(over)1731 1871 y Fw(1)p 1704 1908 97 4 v 1704 1984 a Fr(C)1763 1996 y Fs(0)1833 1927 y Fq(\024)23 b Fr(!)j Fq(\024)c Fr(C)2145 1939 y Fs(0)2183 1927 y Fr(:)1026 b Fw(\(2.68\))555 2093 y Fk(4\))30 b(Final)t(ly,)i(r)l(e)l(c)l(al)t(ling)38 b Fw(\(2.27\))p Fk(,)1577 2235 y Fw(lim)1581 2289 y Fo(h)p Fp(#)p Fs(0)1706 2235 y Fq(k)p Fr(v)21 b Fq(\000)34 b Fw(~)-58 b Fr(m)1965 2201 y Fp(0)1965 2256 y Fo(\030)1997 2239 y Fj(\003)2036 2235 y Fq(k)2078 2247 y Fp(1)2171 2235 y Fw(=)23 b(0)p Fr(;)908 b Fw(\(2.69\))456 2422 y Fk(with)30 b Fr(\030)676 2392 y Fp(\003)744 2422 y Fk(as)g(in)36 b Fw(\(2.13\))o Fk(.)555 2576 y Fw(As)29 b(w)n(e)g(are)f(going)g(to)h(see,)g(from)f(the)i(fact)f(that)g(the)g (linearized)g(op)r(erator)e(around)h Fr(q)k Fw(has)456 2676 y(only)23 b(one)g(p)r(ositiv)n(e)f(eigen)n(v)-5 b(alue,)24 b(it)g(follo)n(ws)e(the)i(existence,)g(for)e(an)n(y)h Fr(h)g Fw(small)g(enough,)h(of)f(t)n(w)n(o)456 2775 y(distinct,)36 b(one)e(dimensional)g(manifolds)g Fq(W)1875 2787 y Fp(\006)1966 2775 y Fq(\032)g Fr(C)2130 2745 y Fs(sym)2250 2775 y Fw(\()p Fn(R)p Fw(;)14 b([)p Fq(\000)p Fw(1)p Fr(;)g Fw(1]\).)62 b(W)-7 b(e)35 b(giv)n(e)e(the)i(precise)456 2875 y(statemen)n(t)27 b(in)h(the)g(next)g(theorem,)f(pro)n(v)n(ed)f (in)i(Sect.)g(6.)456 3029 y Fz(Theorem)j(2.12.)40 b Fk(Given)31 b Fr(\014)d(>)23 b Fw(1)30 b Fk(let)g Fr(h)1743 2999 y Fp(\003)1811 3029 y Fk(b)l(e)h(as)f(in)h(Pr)l(op)l(osition)g(2.10.)42 b(Then,)32 b(for)f(any)f Fr(h)24 b Fq(2)456 3128 y Fw(\(0)p Fr(;)14 b(h)615 3098 y Fp(\003)652 3128 y Fw(])p Fk(,)31 b(ther)l(e)e(ar)l(e)h(two)g(distinct,)h(one)f(dimensional)h(manifolds)h Fq(W)2635 3140 y Fp(\006)2714 3128 y Fq(\032)23 b Fr(C)2867 3098 y Fs(sym)2987 3128 y Fw(\()p Fn(R)p Fw(;)14 b([)p Fq(\000)p Fw(1)p Fr(;)g Fw(1]\))p Fk(,)1553 3271 y Fq(W)1635 3283 y Fp(\006)1714 3271 y Fw(=)1801 3204 y Fm(\010)1850 3271 y Fr(m)1923 3237 y Fp(\006)1923 3291 y Fo(s)2002 3271 y Fw(:)37 b Fr(s)23 b Fq(2)g Fn(R)2256 3204 y Fm(\011)2324 3271 y Fr(:)885 b Fw(\(2.70\))456 3413 y Fk(F)-6 b(or)30 b(any)g Fr(s)23 b Fq(2)g Fn(R)36 b Fk(and)30 b Fr(t)23 b Fq(\025)g Fw(0)p Fk(,)1653 3521 y Fr(S)1704 3533 y Fo(t)1747 3453 y Fm(\000)1785 3521 y Fr(m)1858 3486 y Fp(\006)1858 3541 y Fo(s)1914 3453 y Fm(\001)1975 3521 y Fw(=)g Fr(m)2136 3485 y Fp(\006)2136 3541 y Fo(s)p Fs(+)p Fo(t)3232 3521 y Fw(\(2.71\))456 3646 y Fk(and)1586 3753 y Fw(lim)1536 3803 y Fo(s)p Fp(!\0001)1765 3753 y Fq(k)p Fr(m)1880 3719 y Fp(\006)1880 3774 y Fo(s)1954 3753 y Fq(\000)18 b Fr(q)s Fq(k)2119 3765 y Fp(1)2212 3753 y Fw(=)23 b(0)p Fr(:)867 b Fw(\(2.72\))456 3914 y Fk(Mor)l(e)l(over,)31 b(given)g Fr(\026)e Fk(as)h(in)37 b Fw(\(2.27\))o Fk(,)30 b(we)g(have:)1308 4108 y Fw(lim)1258 4158 y Fo(s)p Fp(!\0001)1487 4108 y Fr(e)1526 4074 y Fp(\000)p Fo(\025s)1666 3988 y Fm(\015)1666 4037 y(\015)1666 4087 y(\015)1666 4137 y(\015)1723 4052 y Fr(dm)1839 4022 y Fp(\006)1839 4072 y Fo(s)p 1723 4089 173 4 v 1768 4165 a Fr(ds)1923 4108 y Fq(\007)18 b Fr(\025e)2093 4074 y Fo(\025s)2217 4052 y Fw(1)p 2178 4089 120 4 v 2178 4113 a Fq(p)p 2247 4113 51 4 v 52 x Fr(\026)2307 4108 y(v)2350 3988 y Fm(\015)2350 4037 y(\015)2350 4087 y(\015)2350 4137 y(\015)2397 4197 y Fp(1)2490 4108 y Fw(=)23 b(0)p Fr(:)589 b Fw(\(2.73\))456 4314 y Fk(Final)t(ly,)37 b(for)e(any)f Fr(s)d Fq(2)g Fn(R)p Fk(,)42 b(the)34 b(\(symmetric\))g(functions)g Fr(m)2405 4284 y Fp(\006)2405 4335 y Fo(s)2495 4314 y Fk(ar)l(e)h(non)e(incr)l(e)l(asing)i(on)f Fn(R)3383 4326 y Fs(+)456 4414 y Fk(and)c(satisfy:)1005 4556 y Fr(m)1078 4521 y Fp(\000)1078 4581 y Fo(\014)s(;h)1204 4556 y Fq(\024)23 b Fr(m)1365 4522 y Fp(\000)1365 4577 y Fo(s)1421 4556 y Fw(\()p Fr(x)p Fw(\))h Fq(\024)f Fr(q)s Fw(\()p Fr(x)p Fw(\))h Fq(\024)e Fr(m)1979 4522 y Fs(+)1979 4577 y Fo(s)2034 4556 y Fw(\()p Fr(x)p Fw(\))i Fq(\024)f Fr(m)2330 4521 y Fs(+)2330 4581 y Fo(\014)s(;h)2603 4556 y Fq(8)14 b Fr(x)23 b Fq(2)g Fn(R)p Fr(:)343 b Fw(\(2.74\))555 4717 y(Th)n(us)31 b(the)h(one)f(dimensional)g(manifolds)h Fq(W)1996 4729 y Fp(\006)2083 4717 y Fw(originates)e(at)i Fr(s)d Fw(=)g Fq(\0001)i Fw(from)h Fr(q)s Fw(,)g(\(2.72\))o(,)456 4817 y(and)27 b(are)g(time)h(in)n(v)-5 b(arian)n(t,)27 b(\(2.71\))o(.)38 b(Eac)n(h)26 b(one)i(of)g(them)g(is)g(therefore)f (describ)r(ed)g(b)n(y)h(a)f(single)456 4917 y(orbit)i(of)h Fr(S)809 4929 y Fo(t)868 4917 y Fw(with)h(time)f(going)f(from)h Fq(\0001)f Fw(to)h(+)p Fq(1)p Fw(.)44 b(The)30 b(t)n(w)n(o)f(orbits)h (are)f(denoted)h(b)n(y)f Fr(m)3388 4887 y Fp(\006)3388 4937 y Fo(s)456 5016 y Fw(and)22 b(the)g(parameter)f Fr(s)h Fw(is)g(iden)n(ti\014ed)g(with)h(time.)35 b(Of)23 b(course)d(the)j(origin)e(of)h(time)h(is)f(arbitrary)456 5116 y(and)29 b(this)i(can)e(b)r(e)i(exploited)f(to)g(\014x)g(up)g(the) h(constan)n(ts)e(in)h(suc)n(h)g(a)f(w)n(a)n(y)g(that)h(\(2.73\))g (holds,)456 5216 y(w)n(e)d(refer)g(to)g(Sect.)h(6)f(for)g(details)h(on) f(this)h(p)r(oin)n(t.)p eop %%Page: 13 13 13 12 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(13)555 450 y Fw(By)28 b(in)n(tegrating)f(\(2.73\))g(from)g Fq(\0001)g Fw(to)h Fr(s)f Fw(w)n(e)h(get)1593 643 y Fr(m)1666 609 y Fp(\006)1666 664 y Fo(s)1745 643 y Fq(\031)23 b Fr(q)e Fq(\006)d Fr(e)2013 609 y Fo(\025s)2137 587 y Fw(1)p 2098 624 120 4 v 2098 649 a Fq(p)p 2167 649 51 4 v 51 x Fr(\026)2241 643 y(v)s(:)925 b Fw(\(2.75\))456 854 y(Therefore,)26 b(since)h(from)h(\(2.69\))f(w)n(e)g(ha)n(v)n(e)1312 1008 y(lim)1316 1062 y Fo(h)p Fp(#)p Fs(0)1441 1008 y Fw(sup)1442 1078 y Fo(x)p Fp(\024)p Fs(0)1580 937 y Fm(\014)1580 987 y(\014)1607 1008 y Fr(v)s Fw(\()p Fr(x)p Fw(\))20 b Fq(\000)1864 952 y(p)p 1933 952 V 56 x Fr(\026)30 b Fw(\026)-58 b Fr(m)2070 974 y Fp(0)2093 1008 y Fw(\()p Fr(x)20 b Fw(+)e Fr(\030)2315 974 y Fp(\003)2353 1008 y Fw(\))2385 937 y Fm(\014)2385 987 y(\014)2436 1008 y Fw(=)23 b(0)p Fr(;)643 b Fw(\(2.76\))456 1222 y(b)n(y)28 b(\(2.12\))f(and)g(\(2.69\),)g(for)g Fr(h)h Fw(small)f(and)g Fr(x)d Fq(\024)f Fw(0,)934 1380 y Fr(m)1007 1346 y Fp(\006)1007 1401 y Fo(s)1063 1380 y Fw(\()p Fr(x)p Fw(\))h Fq(\031)38 b Fw(\026)-58 b Fr(m)p Fw(\()p Fr(x)19 b Fw(+)f Fr(\030)1579 1346 y Fp(\003)1618 1380 y Fw(\))g Fq(\006)g Fr(e)1790 1346 y Fo(\025s)1881 1380 y Fw(\026)-58 b Fr(m)1938 1346 y Fp(0)1961 1380 y Fw(\()p Fr(x)19 b Fw(+)f Fr(\030)2182 1346 y Fp(\003)2221 1380 y Fw(\))23 b Fq(\031)38 b Fw(\026)-57 b Fr(m)p Fw(\()p Fr(x)19 b Fw(+)f Fr(\030)2658 1346 y Fp(\003)2714 1380 y Fq(\006)h Fr(e)2837 1346 y Fo(\025s)2911 1380 y Fw(\))p Fr(:)266 b Fw(\(2.77\))555 1535 y(By)32 b(symmetry)f(the)h(result)f(extends)h(to)f Fr(x)g Fq(\025)e Fw(0.)49 b(Th)n(us)31 b(the)h(p)r(oin)n(ts)g(in)f(a)h(neigh)n(b)r(orho) r(o)r(d)456 1634 y(of)g Fr(q)k Fw(that)d(are)f(in)h Fq(W)1141 1646 y Fs(+)1229 1634 y Fw(are)f(\\droplets)f(longer")g(than)i Fr(q)j Fw(while)d(those)f(in)h Fq(W)2934 1646 y Fp(\000)3023 1634 y Fw(are)f(shorter.)456 1734 y(Their)h(length)g(c)n(hanges)g(at)g (the)h(exp)r(onen)n(tial)f(rate)g Fr(\025)p Fw(,)j(whic)n(h)d(is)g (therefore)g(the)h(Ly)n(apuno)n(v)456 1833 y(exp)r(onen)n(t)28 b(at)g Fr(q)s Fw(,)h(with)g Fq(W)1280 1845 y Fp(\006)1365 1833 y Fw(the)g(corresp)r(onding)e(unstable)h(manifolds.)40 b(Since)28 b Fr(\025)e Fq(\031)e Fr(h)p Fw(,)29 b(for)f Fr(h)456 1933 y Fw(small,)c(there)f(is)g(a)g Fu(dorman)n(t)g (instabilit)n(y)p Fw(,)h(in)g(the)g(sense)f(that)h(for)f(small)g Fr(h)g Fw(\(whic)n(h)h(is)f(the)h(case)456 2033 y(of)i(in)n(terest)g (in)h(metastabilit)n(y\))f(ev)n(en)g(though)h(ultimately)g(unstable,)f (the)h(bump)g Fr(q)j Fw(seems)c(in)456 2132 y(fact)h(stable)h(for)f(v)n (ery)f(long)h(times)h(\(of)g(order)e Fr(h)1963 2102 y Fp(\000)p Fs(1)2052 2132 y Fw(\).)555 2232 y(By)g(\(2.53\))e(the)i (critical)e(droplet)g Fr(q)k Fw(b)r(elongs)d(to)g(the)g(in)n(terior)f (of)h(the)g(c)n(hannel)f Fq(Z)3101 2244 y Fo(`)3129 2252 y Fl(4)3162 2244 y Fo(;\024;Q)3296 2232 y Fw(,)i(for)456 2337 y Fr(\024)32 b Fw(large)e(and)i Fr(h)g Fw(small)g(enough.)50 b(Then,)33 b(according)e(to)h(Theorem)f(2.7,)h(a)g(large)f(part)p 3256 2271 89 4 v 31 w Fq(W)40 b Fw(of)456 2437 y(the)30 b(in)n(v)-5 b(arian)n(t)30 b(manifold)g Fq(W)37 b Fw(is)30 b(con)n(tained)g(in)h Fq(Z)2043 2449 y Fo(`)2071 2457 y Fl(4)2103 2449 y Fo(;\024;)p Fs(1)2219 2437 y Fw(:)42 b(it)31 b(th)n(us)f(consists)g(of)g(droplets)g(with)456 2537 y(t)n(w)n(o)24 b(w)n(ell-separated)f(transition)h(la)n(y)n(ers)f (whose)h(lo)r(cations)g(ev)n(olv)n(e)f(according)g(to)i(Eq.)f(\(2.52\)) o(.)555 2636 y(Our)i(last)f(result)h(concerns)f(the)i(global)e (structure)g(of)h(the)h(in)n(v)-5 b(arian)n(t)25 b(manifold)h Fq(W)7 b Fw(.)36 b(More)456 2736 y(precisely)c(w)n(e)i(sho)n(w)f(that)h (the)g(sub-critical)e(and)i(sup)r(er-critical)e(droplets)h(will)h(ev)n (en)n(tually)456 2835 y(go)27 b(to)i(the)g(metastable)f(and)h(stable)f (phase)g(resp)r(ectiv)n(ely:)38 b(this)29 b(is)f(the)h(con)n(ten)n(t)g (of)f(the)h(next)456 2935 y(theorem)e(pro)n(v)n(ed)f(in)i(Sect.)g(7.) 456 3096 y Fz(Theorem)f(2.13.)37 b Fk(Given)27 b Fr(\014)h(>)22 b Fw(1)27 b Fk(ther)l(e)g(is)g Fr(h)1901 3066 y Fp(y)1959 3096 y Fq(2)c Fw(\(0)p Fr(;)14 b(h)2196 3066 y Fp(\003)2234 3096 y Fw(])27 b Fk(\()p Fr(h)2366 3066 y Fp(\003)2431 3096 y Fk(as)g(in)g(Pr)l(op)l(osition)i(2.10\))f(such)456 3195 y(that,)i(for)g(any)h Fr(h)22 b Fq(2)i Fw(\(0)p Fr(;)14 b(h)1251 3165 y Fp(y)1285 3195 y Fw(])p Fk(,)1513 3351 y Fw(lim)1463 3400 y Fo(s)p Fp(!)p Fs(+)p Fp(1)1692 3280 y Fm(\015)1692 3330 y(\015)1738 3351 y Fr(m)1811 3316 y Fp(\000)1811 3371 y Fo(s)1886 3351 y Fq(\000)k Fr(m)2042 3315 y Fp(\000)2042 3376 y Fo(\014)s(;h)2145 3280 y Fm(\015)2145 3330 y(\015)2191 3384 y Fp(1)2284 3351 y Fw(=)23 b(0)p Fr(;)795 b Fw(\(2.78\))1391 3570 y(lim)1341 3620 y Fo(s)p Fp(!)p Fs(+)p Fp(1)1570 3570 y Fr(m)1643 3536 y Fs(+)1643 3590 y Fo(s)1698 3570 y Fw(\()p Fr(x)p Fw(\))24 b(=)f Fr(m)1994 3534 y Fs(+)1994 3595 y Fo(\014)s(;h)2267 3570 y Fq(8)14 b Fr(x)22 b Fq(2)i Fn(R)p Fr(:)679 b Fw(\(2.79\))456 3794 y Fz(A)28 b(notation)f(w)m (arning:)35 b Fw(In)25 b(Sects.)f(3,)g(4,)h(and)f(8)f(w)n(e)h(shall)g (denote)g(b)n(y)g Fr(C)29 b Fw(=)23 b Fr(C)6 b Fw(\()p Fr(\024)p Fw(\))25 b(a)e(generic)456 3893 y(p)r(ositiv)n(e)k(function)h (of)g Fr(\024)f Fw(whose)g(n)n(umerical)g(v)-5 b(alue)27 b(ma)n(y)g(c)n(hange)f(from)i(line)g(to)f(line.)1437 4119 y(3.)41 b Fv(Local)31 b(a)-6 b(ttra)n(ctiveness)555 4269 y Fw(W)f(e)28 b(start)f(with)h(a)g(preliminary)e(lemma.)456 4429 y Fz(Lemma)g(3.1.)37 b Fk(F)-6 b(or)27 b(any)g Fr(\024)c(>)g Fw(0)j Fk(let)h Fr(`)1662 4441 y Fs(3)1726 4429 y Fk(and)g Fr(")1923 4441 y Fs(0)1987 4429 y Fk(b)l(e)g(as)g(in)g(The)l(or)l(em)h (2.5.)39 b(Then,)29 b(ther)l(e)e(exists)456 4529 y Fr(G)g Fw(=)h Fr(G)p Fw(\()p Fr(\024)p Fw(\))33 b Fk(such)f(that,)h(for)g(any) g Fr(h)27 b(<)h(e)1744 4499 y Fp(\000)p Fs(2)p Fo(\013`)1900 4507 y Fl(3)1936 4529 y Fk(,)33 b Fr(N)j(>)28 b Fw(0)p Fk(,)k(and)h Fr(m)28 b Fw(=)f Fr(n)2696 4541 y Fo(\030)2752 4529 y Fw(+)20 b Fr(')28 b Fq(2)g(S)3052 4541 y Fo(`)3080 4549 y Fl(3)3113 4541 y Fo(;\024;")3223 4549 y Fl(0)3259 4529 y Fk(,)33 b(one)456 4629 y(has:)573 4783 y Fq(k)p Fr(S)666 4795 y Fo(t)695 4783 y Fw(\()p Fr(m)p Fw(\))19 b Fq(\000)f Fr(n)984 4795 y Fo(\030)1020 4783 y Fq(k)1062 4795 y Fp(1)1155 4783 y Fq(\024)1243 4716 y Fm(\000)1281 4783 y Fr(c)1317 4795 y Fs(1)1354 4783 y Fr(e)1393 4749 y Fp(\000)p Fo(!)1487 4757 y Fl(1)1519 4749 y Fo(t)1566 4783 y Fw(+)g Fr(N)9 b Fq(k)p Fr(')p Fq(k)1863 4795 y Fp(1)1933 4716 y Fm(\001)1985 4783 y Fq(k)p Fr(')p Fq(k)2123 4795 y Fp(1)2211 4783 y Fw(+)18 b Fr(N)9 b(e)2409 4749 y Fp(\000)p Fs(2)p Fo(\013\030)2742 4783 y Fq(8)14 b Fr(t)22 b(<)h(T)2992 4795 y Fo(m;N)3133 4783 y Fr(;)118 b Fw(\(3.1\))456 4937 y Fk(with)30 b Fr(c)672 4949 y Fs(1)739 4937 y Fk(and)g Fr(!)952 4949 y Fs(1)1019 4937 y Fk(as)g(in)g(Pr)l(op)l(osition)h(2.3)g(and)1067 5137 y Fr(T)1116 5149 y Fo(m;N)1301 5090 y Fr(:)1280 5137 y Fw(=)22 b(min)1520 5020 y Fm(\032)1582 5137 y Fr(e)1621 5102 y Fs(2)p Fo(\013\030)1733 5137 y Fw(;)2214 5081 y Fr(N)p 1780 5118 945 4 v 1780 5194 a(G)p Fw([1)c(+)g Fr(N)2087 5170 y Fs(2)2124 5194 y Fw(\()p Fr(e)2195 5170 y Fp(\000)p Fs(2)p Fo(\013\030)2378 5194 y Fw(+)g Fq(k)p Fr(')p Fq(k)2599 5170 y Fs(2)2599 5214 y Fp(1)2669 5194 y Fw(\)])2734 5020 y Fm(\033)2810 5137 y Fr(:)441 b Fw(\(3.2\))p eop %%Page: 14 14 14 13 bop 456 251 a Fs(14)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fz(Pro)s(of.)41 b Fw(The)28 b(function)g Fr( )1306 462 y Fo(t)1379 403 y Fr(:)1359 450 y Fw(=)22 b Fr(S)1497 462 y Fo(t)1526 450 y Fw(\()p Fr(m)p Fw(\))d Fq(\000)f Fr(n)1815 462 y Fo(\030)1879 450 y Fw(solv)n(es)1320 585 y Fr(d )1417 597 y Fo(t)p 1320 622 127 4 v 1346 698 a Fr(dt)1479 641 y Fw(=)23 b Fr(L)1624 653 y Fo(\030)1660 641 y Fr( )1714 653 y Fo(t)1761 641 y Fw(+)c([)p Fr(f)9 b Fw(\()p Fr(n)2000 653 y Fo(\030)2054 641 y Fw(+)18 b Fr( )2191 653 y Fo(t)2220 641 y Fw(\))h Fq(\000)f Fr(L)2411 653 y Fo(\030)2447 641 y Fr( )2501 653 y Fo(t)2530 641 y Fw(])c Fr(;)456 809 y Fw(whic)n(h)27 b(implies,)h(since)f Fr( )1255 821 y Fs(0)1316 809 y Fw(=)22 b Fr(')p Fw(,)1071 1007 y Fr( )1125 1019 y Fo(t)1178 1007 y Fw(=)g Fr(e)1304 973 y Fo(L)1350 982 y Fi(\030)1382 973 y Fo(t)1411 1007 y Fr(')d Fw(+)1567 894 y Fm(Z)1650 915 y Fo(t)1613 1083 y Fs(0)1679 1007 y Fr(ds)14 b(e)1814 973 y Fo(L)1860 982 y Fi(\030)1892 973 y Fs(\()p Fo(t)p Fp(\000)p Fo(s)p Fs(\))2070 1007 y Fw([)p Fr(f)9 b Fw(\()p Fr(n)2225 1019 y Fo(\030)2280 1007 y Fw(+)18 b Fr( )2417 1019 y Fo(s)2453 1007 y Fw(\))g Fq(\000)g Fr(L)2643 1019 y Fo(\030)2679 1007 y Fr( )2733 1019 y Fo(s)2769 1007 y Fw(])c Fr(:)456 1199 y Fw(By)28 b(\(2.24\))f(w)n(e)g(then)h(get:)828 1401 y Fq(k)p Fr( )924 1413 y Fo(t)971 1401 y Fq(\000)18 b Fr(e)1093 1367 y Fo(L)1139 1376 y Fi(\030)1171 1367 y Fo(t)1200 1401 y Fr(')p Fq(k)1296 1413 y Fp(1)1449 1401 y Fq(\024)83 b Fr(c)1633 1413 y Fs(1)1684 1288 y Fm(Z)1767 1309 y Fo(t)1730 1477 y Fs(0)1796 1401 y Fr(ds)14 b(e)1931 1367 y Fo(\025)1970 1376 y Fi(\030)2003 1367 y Fs(\()p Fo(t)p Fp(\000)p Fo(s)p Fs(\))2167 1401 y Fq(k)p Fr(f)9 b Fw(\()p Fr(n)2341 1413 y Fo(\030)2395 1401 y Fw(+)18 b Fr( )2532 1413 y Fo(s)2568 1401 y Fw(\))g Fq(\000)g Fr(L)2758 1413 y Fo(\030)2794 1401 y Fr( )2848 1413 y Fo(s)2884 1401 y Fq(k)2926 1413 y Fp(1)1449 1638 y Fq(\024)83 b Fr(c)1633 1650 y Fs(1)1684 1525 y Fm(Z)1767 1546 y Fo(t)1730 1714 y Fs(0)1796 1638 y Fr(ds)14 b(e)1931 1604 y Fo(\025)1970 1613 y Fi(\030)2003 1604 y Fs(\()p Fo(t)p Fp(\000)p Fo(s)p Fs(\))2181 1571 y Fm(\000)2219 1638 y Fq(k)p Fr(f)9 b Fw(\()p Fr(n)2393 1650 y Fo(\030)2428 1638 y Fw(\))p Fq(k)2502 1650 y Fp(1)2591 1638 y Fw(+)18 b Fr(k)2717 1650 y Fs(2)2754 1638 y Fq(k)p Fr( )2850 1650 y Fo(s)2886 1638 y Fq(k)2928 1604 y Fs(2)2928 1659 y Fp(1)2997 1571 y Fm(\001)3049 1638 y Fr(;)456 1832 y Fw(where)35 b(in)g(the)h(last)f(inequalit)n(y)g(w)n(e)g(used)h (\(2.7\))f(\(recall)g(that)g Fr(L)2557 1844 y Fo(\030)2650 1785 y Fr(:)2629 1832 y Fw(=)h Fr(L)2787 1844 y Fo(n)2828 1853 y Fi(\030)2860 1844 y Fs(+)p Fo(h)2954 1832 y Fw(\).)61 b(By)36 b(\(2.32\))o(,)456 1939 y(since)30 b Fr(\031)709 1951 y Fo(\030)746 1939 y Fw(\()p Fr(')p Fw(\))f(=)e(0,)k Fq(k)p Fr( )1177 1951 y Fo(s)1212 1939 y Fq(k)1254 1951 y Fp(1)1352 1939 y Fq(\024)d Fr(c)1481 1951 y Fs(1)1518 1939 y Fq(k)p Fr(')p Fq(k)1656 1951 y Fp(1)1746 1939 y Fw(+)20 b Fq(k)p Fr( )1927 1951 y Fo(s)1982 1939 y Fq(\000)g Fr(e)2106 1909 y Fo(L)2152 1918 y Fi(\030)2184 1909 y Fo(s)2220 1939 y Fr(')p Fq(k)2316 1951 y Fp(1)2386 1939 y Fw(.)46 b(Moreo)n(v)n(er,)28 b(b)n(y)k(\(2.16\))o(,)f(\(2.20\)) 456 2042 y(and)d(\(2.21\),)h Fq(k)p Fr(f)9 b Fw(\()p Fr(n)1057 2054 y Fo(\030)1093 2042 y Fw(\))p Fq(k)1167 2054 y Fp(1)1262 2042 y Fq(\024)25 b Fr(C)6 b(e)1456 2012 y Fp(\000)p Fs(2)p Fo(\013\030)1621 2042 y Fw(.)41 b(Finally)-7 b(,)29 b(b)n(y)h(\(2.25\))o(,)g(if)f Fr(t)d Fq(\024)f Fr(e)2630 2012 y Fs(2)p Fo(\013\030)2771 2042 y Fw(then)30 b Fr(\025)3010 2054 y Fo(\030)3046 2042 y Fr(t)c Fq(\024)f Fr(c)3228 2054 y Fs(1)3265 2042 y Fw(.)41 b(W)-7 b(e)456 2141 y(conclude)27 b(that)h(there)f(exists)g Fr(G)d Fw(=)e Fr(G)p Fw(\()p Fr(\024)p Fw(\))i Fr(>)f Fw(0)k(suc)n(h)g(that:)497 2344 y Fq(k)p Fr( )593 2356 y Fo(t)638 2344 y Fq(\000)16 b Fr(e)758 2310 y Fo(L)804 2319 y Fi(\030)836 2310 y Fo(t)865 2344 y Fr(')p Fq(k)961 2356 y Fp(1)1054 2344 y Fq(\024)23 b Fr(G)1221 2231 y Fm(Z)1304 2251 y Fo(t)1267 2419 y Fs(0)1333 2344 y Fr(ds)1443 2277 y Fm(\000)1481 2344 y Fr(e)1520 2310 y Fp(\000)p Fs(2)p Fo(\013\030)1702 2344 y Fw(+)18 b Fq(k)p Fr(')p Fq(k)1923 2310 y Fs(2)1923 2364 y Fp(1)2012 2344 y Fw(+)g Fq(k)p Fr( )2191 2356 y Fo(s)2244 2344 y Fq(\000)g Fr(e)2366 2310 y Fo(L)2412 2319 y Fi(\030)2444 2310 y Fo(t)2473 2344 y Fr(')p Fq(k)2569 2310 y Fs(2)2569 2364 y Fp(1)2639 2277 y Fm(\001)2857 2344 y Fq(8)c Fr(t)22 b Fq(\024)h Fr(e)3097 2310 y Fs(2)p Fo(\013\030)3209 2344 y Fr(:)42 b Fw(\(3.3\))456 2531 y(Giv)n(en)27 b Fr(N)32 b(>)22 b Fw(0,)28 b(\014x)f Fr(\034)33 b(<)23 b(T)1298 2543 y Fo(m;N)1467 2531 y Fw(and)k(let)h Fr(T)34 b Fq(\024)23 b Fr(\034)37 b Fw(b)r(e)28 b(the)g(\014rst)f(time)h(when)g(the)g (inequalit)n(y)1299 2683 y Fq(k)p Fr( )1395 2695 y Fo(t)1442 2683 y Fq(\000)18 b Fr(e)1564 2648 y Fo(L)1610 2657 y Fi(\030)1642 2648 y Fo(t)1671 2683 y Fr(')p Fq(k)1767 2695 y Fp(1)1861 2683 y Fq(\024)k Fr(N)9 b Fw(\()p Fr(e)2095 2648 y Fp(\000)p Fs(2)p Fo(\013\030)2278 2683 y Fw(+)18 b Fq(k)p Fr(')p Fq(k)2499 2648 y Fs(2)2499 2703 y Fp(1)2569 2683 y Fw(\))673 b(\(3.4\))456 2827 y(b)r(ecomes)27 b(an)g(equalit)n(y) -7 b(.)36 b(Then,)28 b(b)n(y)h(\(3.3\))o(,)645 2997 y Fr(N)9 b Fw(\()p Fr(e)792 2963 y Fp(\000)p Fs(2)p Fo(\013\030)974 2997 y Fw(+)18 b Fq(k)p Fr(')p Fq(k)1195 2963 y Fs(2)1195 3018 y Fp(1)1265 2997 y Fw(\))83 b Fq(\024)g Fr(GT)1667 2905 y Fm(h)1706 2997 y Fr(e)1745 2963 y Fp(\000)p Fs(2)p Fo(\013\030)1928 2997 y Fw(+)18 b Fq(k)p Fr(')p Fq(k)2149 2963 y Fs(2)2149 3018 y Fp(1)2237 2997 y Fw(+)g Fr(N)2396 2963 y Fs(2)2447 2930 y Fm(\000)2485 2997 y Fr(e)2524 2963 y Fp(\000)p Fs(2)p Fo(\013\030)2707 2997 y Fw(+)g Fq(k)p Fr(')p Fq(k)2928 2963 y Fs(2)2928 3018 y Fp(1)2997 2930 y Fm(\001)3035 2943 y Fs(2)3072 2905 y Fm(i)1380 3176 y Fq(\024)83 b Fr(N)1617 3109 y Fm(\000)1656 3176 y Fr(e)1695 3142 y Fp(\000)p Fs(2)p Fo(\013\030)1877 3176 y Fw(+)18 b Fq(k)p Fr(')p Fq(k)2098 3142 y Fs(2)2098 3197 y Fp(1)2168 3109 y Fm(\001)2302 3120 y Fr(\034)p 2230 3157 191 4 v 2230 3233 a(T)2279 3245 y Fo(m;N)2453 3176 y Fr(<)k(N)2630 3109 y Fm(\000)2668 3176 y Fr(e)2707 3142 y Fp(\000)p Fs(2)p Fo(\013\030)2890 3176 y Fw(+)c Fq(k)p Fr(')p Fq(k)3111 3142 y Fs(2)3111 3197 y Fp(1)3180 3109 y Fm(\001)3232 3176 y Fr(:)456 3373 y Fw(W)-7 b(e)34 b(ha)n(v)n(e)f(th)n(us)i(reac)n(hed)e(a)h(con)n(tradiction)f(and)h (hence)g(\(3.4\))g(is)g(v)-5 b(alid)34 b(for)g(all)g Fr(t)g Fq(\024)g Fr(\034)9 b Fw(.)58 b(By)456 3472 y(\(2.32\))34 b(and)h(\(3.4\))f(the)i(inequalit)n(y)e(in)h(\(3.1\))g(holds)f(for)h (an)n(y)f Fr(t)h Fq(\024)g Fr(\034)9 b Fw(.)60 b(The)35 b(lemma)g(is)f(th)n(us)456 3572 y(pro)n(v)n(ed.)2657 b Fg(\003)456 3694 y Fz(Pro)s(of)31 b(of)h(Theorem)e(2.7.)36 b Fw(W)-7 b(e)28 b(\014x)f(the)h(parameter)f Fr(N)36 b Fw(of)28 b(Lemma)f(3.1)g(suc)n(h)g(that)1241 3877 y(4)r(\026)-44 b Fr(c)o(c)1354 3889 y Fs(1)1391 3877 y Fr(e)1430 3843 y Fs(3)p Fo(\013)r Fs(\026)-35 b Fo(c")1567 3851 y Fl(0)1600 3843 y Fo(=)p Fs(2)1671 3877 y Fr(e)1710 3843 y Fp(\000)p Fo(!)1804 3851 y Fl(1)1836 3843 y Fo(T)1875 3851 y Fi(N)1955 3877 y Fw(=)23 b(1)p Fr(;)179 b(T)2336 3889 y Fo(N)2443 3830 y Fr(:)2422 3877 y Fw(=)2535 3821 y Fr(N)p 2520 3858 107 4 v 2520 3934 a Fw(2)p Fr(G)2636 3877 y(;)615 b Fw(\(3.5\))456 4045 y(with)27 b Fr(c)680 4057 y Fs(1)717 4045 y Fw(,)g Fr(!)819 4057 y Fs(1)882 4045 y Fw(as)f(in)h(Prop)r (osition)e(2.3,)i(\026)-43 b Fr(c)p Fw(,)26 b Fr(")1805 4057 y Fs(0)1869 4045 y Fw(as)g(in)h(Theorem)e(2.5,)h(and)h Fr(G)f Fw(as)g(in)h(Lemma)f(3.1.)456 4145 y(W)-7 b(e)28 b(next)f(de\014ne:)875 4308 y Fr(Q)985 4261 y(:)964 4308 y Fw(=)1062 4252 y(1)p 1062 4289 42 4 v 1062 4365 a(2)1127 4241 y Fm(\000)1165 4308 y Fw(1)18 b(+)g Fr(e)1347 4274 y Fo(!)1389 4282 y Fl(1)1421 4274 y Fo(T)1460 4282 y Fi(N)1518 4241 y Fm(\001)1579 4308 y Fw(=)1676 4252 y(1)p 1676 4289 V 1676 4365 a(2)1746 4308 y(+)g(2)r(\026)-44 b Fr(cc)1943 4320 y Fs(1)1980 4308 y Fr(e)2019 4274 y Fs(3)p Fo(\013)r Fs(\026)-35 b Fo(c)o(")2155 4282 y Fl(0)2188 4274 y Fo(=)p Fs(2)2259 4308 y Fr(;)180 b(b)2542 4261 y(:)2521 4308 y Fw(=)2619 4252 y(1)p 2619 4289 V 2619 4365 a(2)2670 4308 y Fr(e)2709 4274 y Fp(\000)p Fs(3)p Fo(\013)r Fs(\026)-35 b Fo(c")2898 4282 y Fl(0)2930 4274 y Fo(=)p Fs(2)3002 4308 y Fr(:)249 b Fw(\(3.6\))456 4481 y(\(note)39 b(that)g Fr(Q)i Fq(\025)h Fw(1)c(and)h Fr(b)i(<)h Fw(1)p Fr(=)p Fw(2\).)69 b(W)-7 b(e)40 b(\014nally)e(c)n(ho)r(ose)g Fr(`)2503 4493 y Fs(4)2582 4481 y Fr(>)j(`)2723 4493 y Fs(3)2799 4481 y Fw(large)c(enough)i(and)456 4580 y Fr(")495 4592 y Fs(1)555 4580 y Fq(2)23 b Fw(\(0)p Fr(;)14 b(")783 4592 y Fs(0)820 4580 y Fw(])27 b(small)h(enough)f(that:)583 4729 y Fr(N)659 4699 y Fs(2)696 4729 y Fw(\()p Fr(e)767 4699 y Fp(\000)p Fs(2)p Fo(\013`)923 4707 y Fl(4)978 4729 y Fw(+)18 b Fr(")1100 4699 y Fs(2)1100 4750 y(1)1137 4729 y Fw(\))24 b Fr(<)e Fw(1)p Fr(;)138 b(N)9 b(e)1598 4695 y Fp(\000)p Fs(2)p Fo(\013`)1754 4703 y Fl(4)1813 4729 y Fr(<)23 b Fw(2)p Fr(G;)180 b(c)2247 4741 y Fs(1)2284 4729 y Fr(")2323 4741 y Fs(1)2378 4729 y Fw(+)18 b Fr(N)9 b Fw(\()p Fr(")2608 4695 y Fs(2)2608 4750 y(1)2664 4729 y Fw(+)18 b Fr(e)2786 4695 y Fp(\000)p Fs(2)p Fo(\013`)2942 4703 y Fl(4)2978 4729 y Fw(\))23 b Fr(<)g(")3160 4741 y Fs(0)3197 4729 y Fr(;)54 b Fw(\(3.7\))746 4861 y(\026)-44 b Fr(cN)9 b(")895 4873 y Fs(1)955 4861 y Fr(<)22 b(b=)p Fw(2)p Fr(;)300 b Fw(\026)-44 b Fr(cN)9 b(e)1634 4827 y Fp(\000)p Fo(\013`)1757 4835 y Fl(4)1789 4827 y Fo(=)p Fs(2)1883 4861 y Fr(<)23 b(b=Q;)178 b(e)2355 4827 y Fp(\000)p Fs(3)p Fo(\013`)2511 4835 y Fl(4)2543 4827 y Fo(=)p Fs(2)2638 4861 y Fr(<)22 b(")2764 4873 y Fs(1)2801 4861 y Fr(=)p Fw(2)p Fr(:)366 b Fw(\(3.8\))456 5006 y(Let)29 b(no)n(w)f Fr(m)e Fw(=)f Fr(n)1019 5018 y Fo(\030)1074 5006 y Fw(+)19 b Fr(')26 b Fq(2)g(S)1369 5018 y Fo(`)1397 5026 y Fl(4)1429 5018 y Fo(;\024;")1539 5026 y Fl(1)1575 5006 y Fw(.)42 b(By)28 b(the)i(\014rst)f(and)f(the)i(second)e(inequalit)n(y)h(in)g (\(3.7\))g(w)n(e)456 5106 y(ha)n(v)n(e:)1404 5216 y Fr(T)1453 5228 y Fo(m;N)1617 5216 y Fr(>)23 b(T)1754 5228 y Fo(N)1982 5216 y Fq(8)14 b Fr(m)22 b Fq(2)h(S)2266 5228 y Fo(`)2294 5236 y Fl(4)2327 5228 y Fo(;\024;")2437 5236 y Fl(1)2473 5216 y Fr(;)p eop %%Page: 15 15 15 14 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(15)456 450 y Fw(with)40 b Fr(T)706 462 y Fo(m;N)888 450 y Fw(as)f(in)i(\(3.2\))o(.)75 b(By)41 b(\(3.1\))f(and)g(the)h(third)f(inequalit)n(y)g(in)h(\(3.7\))f(w)n(e)f (th)n(us)i(get)456 550 y Fq(k)p Fr(S)549 562 y Fo(t)577 550 y Fw(\()p Fr(m)p Fw(\))27 b Fq(\000)g Fr(n)883 562 y Fo(\030)919 550 y Fq(k)961 562 y Fp(1)1075 550 y Fr(<)43 b(")1222 562 y Fs(0)1299 550 y Fw(for)d(all)f Fr(t)44 b Fq(2)g Fw([0)p Fr(;)14 b(T)1890 562 y Fo(N)1952 550 y Fw(].)75 b(Then,)43 b(b)n(y)d(Theorem)f(2.5,)j(the)f(function)456 649 y Fr(\030)t Fw(\()p Fr(t)p Fw(\))23 b(=)g(\004\()p Fr(S)839 661 y Fo(t)869 649 y Fw(\()p Fr(m)p Fw(\)\))28 b(is)g(w)n(ell)f(de\014ned)h(for)f(all)g Fr(t)c Fq(2)h Fw([0)p Fr(;)14 b(T)2129 661 y Fo(N)2191 649 y Fw(])28 b(and)873 800 y Fq(j)p Fr(\030)t Fw(\()p Fr(t)p Fw(\))20 b Fq(\000)e Fr(\030)t Fq(j)83 b(\024)h Fw(\026)-44 b Fr(c)p Fq(k)p Fr(S)1555 812 y Fo(t)1584 800 y Fw(\()p Fr(m)p Fw(\))19 b Fq(\000)f Fr(n)1873 812 y Fo(\030)1909 800 y Fq(k)1951 812 y Fp(1)2058 800 y Fq(\024)38 b Fw(\026)-44 b Fr(c")2234 812 y Fs(0)2271 800 y Fr(;)980 b Fw(\(3.9\))894 924 y Fq(k)p Fr(')p Fw(\()p Fr(t)p Fw(\))p Fq(k)1126 936 y Fp(1)1279 924 y Fq(\024)84 b Fw(\026)-44 b Fr(c)p Fq(k)p Fr(S)1555 936 y Fo(t)1584 924 y Fw(\()p Fr(m)p Fw(\))19 b Fq(\000)f Fr(n)1873 936 y Fo(\030)1909 924 y Fq(k)1951 936 y Fp(1)1279 1059 y Fq(\024)84 b Fw(\026)-44 b Fr(c)p Fw(\()p Fr(c)1530 1071 y Fs(1)1568 1059 y Fr(e)1607 1025 y Fp(\000)p Fo(!)1701 1033 y Fl(1)1733 1025 y Fo(t)1780 1059 y Fw(+)18 b Fr(N)9 b(")1978 1071 y Fs(1)2015 1059 y Fw(\))p Fq(k)p Fr(')p Fq(k)2185 1071 y Fp(1)2273 1059 y Fw(+)20 b(\026)-44 b Fr(cN)9 b(e)2507 1025 y Fp(\000)p Fo(\013`)2630 1033 y Fl(4)2662 1025 y Fo(=)p Fs(2)2733 1059 y Fr(e)2772 1025 y Fp(\000)p Fs(3)p Fo(\013\030)r(=)p Fs(2)3004 1059 y Fr(;)205 b Fw(\(3.10\))456 1209 y(where)31 b Fr(')p Fw(\()p Fr(t)p Fw(\))900 1162 y Fr(:)879 1209 y Fw(=)f Fr(S)1025 1221 y Fo(t)1054 1209 y Fw(\()p Fr(m)p Fw(\))22 b Fq(\000)e Fr(n)1348 1224 y Fo(\030)r Fs(\()p Fo(t)p Fs(\))1493 1209 y Fw(and)32 b(w)n(e)g(again)e(used)i(\(3.1\))g (for)f(the)h(last)g(b)r(ound)g(in)g(\(3.10\))o(.)456 1309 y(By)c(\(3.5\),)f(\(3.8\),)h(and)f(\(3.10\))g(it)h(easily)f(follo) n(ws)f(that:)1065 1503 y Fq(k)p Fr(')p Fw(\()p Fr(t)p Fw(\))p Fq(k)1297 1515 y Fp(1)1390 1503 y Fq(\024)1491 1447 y Fr(b)p 1488 1484 42 4 v 1488 1560 a Fw(2)1553 1411 y Fm(\020)1602 1503 y Fw(1)18 b(+)g Fr(e)1784 1469 y Fo(!)1826 1477 y Fl(1)1858 1469 y Fs(\()p Fo(T)1923 1477 y Fi(N)1977 1469 y Fp(\000)p Fo(t)p Fs(\))2084 1411 y Fm(\021)2147 1503 y Fq(k)p Fr(')p Fq(k)2285 1515 y Fp(1)2373 1503 y Fw(+)2481 1447 y Fr(b)p 2466 1484 66 4 v 2466 1560 a(Q)2542 1503 y(e)2581 1469 y Fp(\000)p Fs(3)p Fo(\013\030)r(=)p Fs(2)2812 1503 y Fr(:)456 1697 y Fw(Moreo)n(v)n(er,)25 b(b)n(y)j(\(3.9\),)1239 1855 y(2)p Fr(be)1356 1821 y Fp(\000)p Fs(3)p Fo(\013\030)r(=)p Fs(2)1610 1855 y Fq(\024)22 b Fr(e)1736 1821 y Fp(\000)p Fs(3)p Fo(\013\030)r Fs(\()p Fo(t)p Fs(\))p Fo(=)p Fs(2)2210 1855 y Fq(8)14 b Fr(t)22 b Fq(2)i Fw([0)p Fr(;)14 b(T)2553 1867 y Fo(N)2615 1855 y Fw(])p Fr(:)456 2005 y Fw(Then,)27 b(recalling)g(the)h(de\014nition)g(\(3.6\))f(of)h Fr(Q)f Fw(and)g(that)h Fr(Q)23 b Fq(\025)g Fw(1,)k(w)n(e)g(conclude)g(that:) 788 2181 y Fq(k)p Fr(')p Fw(\()p Fr(t)p Fw(\))p Fq(k)1020 2193 y Fp(1)1173 2181 y Fq(\024)83 b Fr(b)1371 2088 y Fm(\020)1420 2181 y Fr(Q)p Fq(k)p Fr(')p Fq(k)1624 2193 y Fp(1)1712 2181 y Fq(\000)18 b Fr(e)1834 2146 y Fp(\000)p Fs(3)p Fo(\013\030)r(=)p Fs(2)2065 2088 y Fm(\021)2133 2181 y Fw(+)g Fr(e)2255 2146 y Fp(\000)p Fs(3)p Fo(\013\030)r Fs(\()p Fo(t)p Fs(\))p Fo(=)p Fs(2)2729 2181 y Fq(8)c Fr(t)22 b Fq(2)i Fw([0)p Fr(;)14 b(T)3072 2193 y Fo(N)3134 2181 y Fw(])p Fr(;)52 b Fw(\(3.11\))706 2382 y Fq(k)p Fr(')p Fw(\()p Fr(T)883 2394 y Fo(N)946 2382 y Fw(\))p Fq(k)1020 2394 y Fp(1)1173 2382 y Fq(\024)1346 2325 y Fr(b)p 1331 2363 V 1331 2439 a(Q)1420 2289 y Fm(\020)1470 2382 y Fr(Q)p Fq(k)p Fr(')p Fq(k)1674 2394 y Fp(1)1762 2382 y Fq(\000)18 b Fr(e)1884 2347 y Fp(\000)p Fs(3)p Fo(\013\030)r(=)p Fs(2)2115 2289 y Fm(\021)2183 2382 y Fw(+)2288 2325 y(1)p 2276 2363 V 2276 2439 a Fr(Q)2351 2382 y(e)2390 2347 y Fp(\000)p Fs(3)p Fo(\013\030)r Fs(\()p Fo(T)2615 2355 y Fi(N)2669 2347 y Fs(\))p Fo(=)p Fs(2)2766 2382 y Fr(:)443 b Fw(\(3.12\))456 2576 y(W)-7 b(e)21 b(ha)n(v)n(e)e(th)n(us)i(pro)n(v)n(ed)f(that)h(if)g Fr(m)i Fq(2)g(S)1682 2588 y Fo(`)1710 2596 y Fl(4)1743 2588 y Fo(;\024;")1853 2596 y Fl(1)1910 2576 y Fw(then)e Fr(S)2143 2588 y Fo(t)2173 2576 y Fw(\()p Fr(m)p Fw(\))i Fq(2)h(B)2467 2588 y Fo(`)2495 2596 y Fl(4)2526 2588 y Fo(;\024;")2636 2596 y Fl(0)2693 2576 y Fw(for)d(all)f Fr(t)j Fq(2)h Fw([0)p Fr(;)14 b(T)3205 2588 y Fo(N)3267 2576 y Fw(])21 b(and)456 2676 y Fr(')p Fw(\()p Fr(t)p Fw(\))28 b(=)g Fr(S)776 2688 y Fo(t)805 2676 y Fw(\()p Fr(m)p Fw(\))21 b Fq(\000)f Fr(n)1098 2691 y Fo(\030)r Fs(\()p Fo(t)p Fs(\))1242 2676 y Fw(satis\014es)29 b(the)i(b)r(ounds)g(\(3.11\))e(and) i(\(3.12\))o(.)46 b(It)31 b(follo)n(ws)e(that)i Fr(S)3278 2688 y Fo(t)3307 2676 y Fw(\()p Fr(m)p Fw(\))456 2776 y(ma)n(y)i(lea)n(v)n(e)g Fq(S)904 2788 y Fo(`)932 2796 y Fl(4)964 2788 y Fo(;\024;")1074 2796 y Fl(0)1144 2776 y Fw(at)h(a)g(giv)n(en)f(time)i Fr(t)1777 2746 y Fp(\003)1848 2776 y Fq(2)g Fw([0)p Fr(;)14 b(T)2089 2788 y Fo(N)2151 2776 y Fw(])34 b(only)f(b)r(ecause)h Fr(\030)t Fw(\()p Fr(t)2812 2746 y Fp(\003)2850 2776 y Fw(\))h(b)r(elongs)e(to)h(the)456 2875 y(b)r(oundary)25 b(of)h(\000)973 2887 y Fo(`)1001 2895 y Fl(4)1033 2887 y Fo(;\024)1096 2875 y Fw(.)36 b(If)27 b(this)f(is)g(not)g(the)h(case,)f(since)g(the)g(third)g (inequalit)n(y)g(in)g(\(3.8\))g(implies)456 2975 y Fq(k)p Fr(')p Fw(\()p Fr(T)633 2987 y Fo(N)695 2975 y Fw(\))p Fq(k)769 2987 y Fp(1)868 2975 y Fr(<)h(")999 2987 y Fs(1)1036 2975 y Fw(,)32 b(then)f Fr(S)1334 2987 y Fo(T)1373 2995 y Fi(N)1431 2975 y Fw(\()p Fr(m)p Fw(\))e Fq(2)f(S)1730 2987 y Fo(`)1758 2995 y Fl(4)1791 2987 y Fo(;\024;")1901 2995 y Fl(1)1968 2975 y Fw(and)i(w)n(e)h(can)f(rep)r(eat)g(the)h(same)g (analysis)e(for)456 3075 y(the)d(solution)f(in)h(the)g(in)n(terv)-5 b(al)25 b([)p Fr(T)1515 3087 y Fo(N)1578 3075 y Fr(;)14 b Fw(2)p Fr(T)1706 3087 y Fo(N)1767 3075 y Fw(].)37 b(Let)26 b Fr(n)f Fw(b)r(e)h(the)g(largest)f(in)n(teger)f(suc)n(h)i(that)g Fr(S)3278 3087 y Fo(t)3307 3075 y Fw(\()p Fr(m)p Fw(\))456 3174 y(lea)n(v)n(es)d Fq(S)741 3186 y Fo(`)769 3194 y Fl(4)802 3186 y Fo(;\024;")912 3194 y Fl(0)973 3174 y Fw(at)i Fr(t)1102 3144 y Fp(\003)1164 3174 y Fq(\025)d Fr(t)1281 3186 y Fo(n)1370 3127 y Fr(:)1349 3174 y Fw(=)h Fr(nT)1536 3186 y Fo(N)1598 3174 y Fw(,)j(setting)g Fr(n)c Fw(=)h(+)p Fq(1)i Fw(if)h Fr(S)2376 3186 y Fo(t)2405 3174 y Fw(\()p Fr(m)p Fw(\))d Fq(2)h(S)2694 3186 y Fo(`)2722 3194 y Fl(4)2754 3186 y Fo(;\024;")2864 3194 y Fl(0)2926 3174 y Fw(forev)n(er.)34 b(Then,)456 3274 y(for)27 b(an)n(y)f(in)n (teger)h Fr(k)f Fq(\024)d Fr(n)k Fw(w)n(e)g(iterate)h(\(3.12\))e (getting)644 3480 y Fq(k)p Fr(')p Fw(\()p Fr(t)802 3492 y Fo(k)843 3480 y Fw(\))p Fq(k)917 3492 y Fp(1)1010 3480 y Fq(\024)1108 3424 y Fr(b)1144 3394 y Fo(k)p 1108 3461 77 4 v 1113 3537 a Fr(Q)1208 3388 y Fm(\020)1258 3480 y Fr(Q)p Fq(k)p Fr(')p Fw(\(0\))p Fq(k)1568 3492 y Fp(1)1656 3480 y Fq(\000)18 b Fr(e)1778 3446 y Fp(\000)p Fs(3)p Fo(\013\030)r Fs(\(0\))p Fo(=)p Fs(2)2094 3388 y Fm(\021)2162 3480 y Fw(+)2267 3424 y(1)p 2255 3461 66 4 v 2255 3537 a Fr(Q)2330 3480 y(e)2369 3446 y Fp(\000)p Fs(3)p Fo(\013\030)r Fs(\()p Fo(t)2580 3455 y Fi(k)2617 3446 y Fs(\))p Fo(=)p Fs(2)2714 3480 y Fr(;)180 b(t)2947 3492 y Fo(k)3032 3433 y Fr(:)3011 3480 y Fw(=)22 b Fr(k)s(T)3193 3492 y Fo(N)456 3675 y Fw(from)27 b(whic)n(h,)g(using)i(\(3.11\))e(with)h Fr(')23 b Fw(=)g Fr(')p Fw(\()p Fr(t)1841 3687 y Fo(k)1882 3675 y Fw(\),)28 b(w)n(e)f(\014nally)h(obtain:)620 3850 y Fq(k)p Fr(')p Fw(\()p Fr(t)p Fw(\))p Fq(k)852 3862 y Fp(1)945 3850 y Fq(\024)23 b Fr(b)1069 3816 y Fo(k)q Fs(+1)1207 3758 y Fm(\020)1257 3850 y Fr(Q)p Fq(k)p Fr(')p Fw(\(0\))p Fq(k)1567 3862 y Fp(1)1654 3850 y Fq(\000)18 b Fr(e)1776 3816 y Fp(\000)p Fs(3)p Fo(\013\030)r Fs(\(0\))p Fo(=)p Fs(2)2093 3758 y Fm(\021)2161 3850 y Fw(+)g Fr(e)2283 3816 y Fp(\000)p Fs(3)p Fo(\013\030)r Fs(\()p Fo(t)p Fs(\))p Fo(=)p Fs(2)2757 3850 y Fq(8)c Fr(t)22 b Fq(2)h Fw([)p Fr(t)3001 3862 y Fo(k)3042 3850 y Fr(;)14 b(t)3109 3862 y Fo(k)q Fs(+1)3234 3850 y Fw(])p Fr(;)456 4025 y Fw(whic)n(h)27 b(implies)h(\(2.51\))f(with)h Fr(\027)g Fw(=)23 b Fq(\000)p Fw(\(log)14 b Fr(b)p Fw(\))p Fr(=T)1938 4037 y Fo(N)2000 4025 y Fw(.)37 b(The)27 b(theorem)g(is)h(th)n(us)g (pro)n(v)n(ed.)293 b Fg(\003)456 4150 y Fz(Pro)s(of)31 b(of)h(Theorem)e(2.8.)36 b Fw(Recalling)28 b(\(2.6\),)924 4300 y Fr(f)9 b Fw(\()p Fr(n)1056 4312 y Fo(\030)1111 4300 y Fw(+)18 b Fr(')p Fw(\))24 b(=)e Fr(f)9 b Fw(\()p Fr(n)1523 4312 y Fo(\030)1559 4300 y Fw(\))19 b(+)f Fr(L)1750 4312 y Fo(\030)1786 4300 y Fr(')h Fw(+)f Fr(R)2005 4312 y Fo(\030)2041 4300 y Fw([)p Fr(')p Fw(])p Fr(;)1068 b Fw(\(3.13\))924 4487 y Fr(R)987 4499 y Fo(\030)1024 4487 y Fw([)p Fr(')p Fw(])23 b(=)g Fr(\014)1286 4453 y Fs(2)1323 4487 y Fw(\()p Fr(J)k Fq(\003)18 b Fr(')p Fw(\))1574 4453 y Fs(2)1626 4374 y Fm(Z)1709 4395 y Fs(1)1672 4563 y(0)1746 4487 y Fr(ds)c Fw(\(1)k Fq(\000)g Fr(s)p Fw(\))c(tanh)2268 4450 y Fp(00)2311 4487 y Fq(f)p Fr(\014)t Fw([)p Fr(J)26 b Fq(\003)18 b Fw(\()p Fr(n)2641 4499 y Fo(\030)2696 4487 y Fw(+)g Fr(s')p Fw(\))h(+)f Fr(h)p Fw(])p Fq(g)p Fr(;)90 b Fw(\(3.14\))456 4684 y(so)27 b(that,)g(since)h Fr(\031)1011 4696 y Fo(\030)1048 4684 y Fw(\()p Fr(L)1137 4696 y Fo(\030)1173 4684 y Fr(')p Fw(\))23 b(=)g Fr(\025)1418 4696 y Fo(\030)1455 4684 y Fr(\031)1502 4696 y Fo(\030)1538 4684 y Fw(\()p Fr(')p Fw(\))h(=)f(0,)k(Eq.)g(\(2.48\))g(b)r(ecomes:)1472 4863 y(_)1454 4885 y Fr(\030)g Fw(=)1615 4829 y Fr(\031)1662 4841 y Fo(\030)1699 4829 y Fw(\()p Fr(f)9 b Fw(\()p Fr(n)1863 4841 y Fo(\030)1899 4829 y Fw(\)\))19 b(+)f Fr(\031)2112 4841 y Fo(\030)2149 4829 y Fw(\()p Fr(R)2244 4841 y Fo(\030)2280 4829 y Fw([)p Fr(')p Fw(]\))p 1615 4866 799 4 v 1657 4942 a Fr(\031)1704 4954 y Fo(\030)1755 4942 y Fw(\()p Fr(@)1831 4954 y Fo(\030)1868 4942 y Fr(n)1918 4954 y Fo(\030)1954 4942 y Fw(\))g Fq(\000)g Fr(@)2131 4954 y Fo(\030)2168 4942 y Fr(\031)2215 4954 y Fo(\030)2252 4942 y Fw(\()p Fr(')p Fw(\))2423 4885 y Fr(:)456 5087 y Fw(By)38 b(\(2.7\),)i(\(2.20\))o(,)h(\(2.31\))o(,)f(\(2.38\))o(,)h (and)c(the)h(de\014nition)h(of)e Fq(Z)2526 5099 y Fo(`;\024;)p Fs(1)2670 5087 y Fw(,)j(\(2.52\))d(follo)n(ws)g(for)g(a)456 5187 y(suitable)27 b Fr(\016)805 5157 y Fp(\003)866 5187 y Fr(>)c Fw(0.)2361 b Fg(\003)p eop %%Page: 16 16 16 15 bop 456 251 a Fs(16)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)1445 450 y Fw(4.)41 b Fv(The)32 b(critical)g(dr)n(oplet)555 600 y Fw(In)d(this)g(section)f(w)n(e)g(pro)n(v)n(e)f(Theorem)h(2.9.)39 b(Recalling)29 b(\(2.48\))e(and)i(\(2.49\))f(w)n(e)g(ha)n(v)n(e)f(that) 456 699 y Fr(m)c Fw(=)f Fr(n)689 711 y Fo(\030)744 699 y Fw(+)c Fr(')23 b Fq(2)h(S)1033 711 y Fo(`)1061 719 y Fl(3)1093 711 y Fo(;\024;")1203 719 y Fl(0)1267 699 y Fw(is)j(a)h(solution)f(of)g Fr(f)9 b Fw(\()p Fr(m)p Fw(\))23 b(=)g(0)k(i\013)h(\()p Fr(\030)t(;)14 b(')p Fw(\))29 b(solv)n(e)976 835 y Fr(f)9 b Fw(\()p Fr(n)1108 847 y Fo(\030)1162 835 y Fw(+)18 b Fr(')p Fw(\))24 b(=)f(0)p Fr(;)179 b(\031)1734 847 y Fo(\030)1771 835 y Fw(\()p Fr(f)9 b Fw(\()p Fr(n)1935 847 y Fo(\030)1989 835 y Fw(+)18 b Fr(')p Fw(\)\))24 b(=)f(0)p Fr(;)179 b(\031)2593 847 y Fo(\030)2630 835 y Fw(\()p Fr(')p Fw(\))24 b(=)f(0)p Fr(:)456 970 y Fw(W)-7 b(e)30 b(rewrite)f(the)i(ab)r(o)n(v)n(e)d (equations)i(as)f(follo)n(ws.)43 b(F)-7 b(or)29 b Fr(\030)j Fq(2)27 b Fw(\000)2412 982 y Fo(`)2440 990 y Fl(3)2472 982 y Fo(;\024)2565 970 y Fw(w)n(e)j(de\014ne)g(the)g(pro)5 b(jection)456 1070 y Fr(T)505 1082 y Fo(\030)568 1070 y Fw(acting)27 b(on)h Fr(L)990 1040 y Fs(sym)990 1091 y Fp(1)1109 1070 y Fw(\()p Fn(R)p Fw(\))34 b(b)n(y:)1592 1171 y Fr(T)1641 1183 y Fo(\030)1677 1171 y Fr( )26 b Fw(=)d Fr( )e Fq(\000)d Fr(\031)2050 1183 y Fo(\030)2087 1171 y Fw(\()p Fr( )s Fw(\))p Fr(v)2248 1183 y Fo(\030)2285 1171 y Fr(:)966 b Fw(\(4.1\))456 1289 y(Observ)n(e)26 b(that)h(b)n(y)i(\(2.26\))e(and)g(\(2.31\))g(there)g(is)h Fr(C)34 b Fw(so)27 b(that)1262 1425 y Fq(k)p Fr(T)1353 1437 y Fo(\030)1388 1425 y Fr( )s Fq(k)1487 1437 y Fp(1)1580 1425 y Fq(\024)c Fr(C)6 b Fq(k)p Fr( )s Fq(k)1874 1437 y Fp(1)2110 1425 y Fq(8)p Fr( )25 b Fq(2)e Fr(L)2371 1391 y Fs(sym)2371 1445 y Fp(1)2491 1425 y Fw(\()p Fn(R)p Fw(\))p Fr(:)642 b Fw(\(4.2\))456 1560 y(Then,)25 b(using)e(the)i (expansion)e(\(3.13\))g(and)h(the)h(fact)f(that)g Fr(\031)2345 1572 y Fo(\030)2382 1560 y Fw(\()p Fr(L)2471 1572 y Fo(\030)2507 1560 y Fr(')p Fw(\))g(=)f(0)g(w)n(e)h(get)g(that)g Fr(f)9 b Fw(\()p Fr(n)3332 1572 y Fo(\030)3380 1560 y Fw(+)456 1660 y Fr(')p Fw(\))23 b(=)g(0)k(i\013)h(\()p Fr(\030)t(;)14 b(')p Fw(\))29 b(solv)n(e)1339 1796 y Fr(L)1396 1808 y Fo(\030)1432 1796 y Fr(')18 b Fw(+)g Fr(T)1636 1808 y Fo(\030)1686 1796 y Fw(\()p Fr(f)9 b Fw(\()p Fr(n)1850 1808 y Fo(\030)1886 1796 y Fw(\))19 b(+)f Fr(R)2083 1808 y Fo(\030)2120 1796 y Fw([)p Fr(')p Fw(]\))83 b(=)g(0)p Fr(;)726 b Fw(\(4.3\))1589 1920 y Fr(\031)1636 1932 y Fo(\030)1686 1920 y Fw(\()p Fr(f)9 b Fw(\()p Fr(n)1850 1932 y Fo(\030)1886 1920 y Fw(\))19 b(+)f Fr(R)2083 1932 y Fo(\030)2120 1920 y Fw([)p Fr(')p Fw(]\))83 b(=)g(0)p Fr(:)726 b Fw(\(4.4\))456 2056 y(W)-7 b(e)37 b(no)n(w)f(pro)r(ceed)g (in)h(the)g(follo)n(wing)e(w)n(a)n(y:)54 b(w)n(e)36 b(\014rst)h(sho)n (w)e(in)i(Lemma)g(4.1)f(b)r(elo)n(w,)i(that)456 2156 y(there)29 b(exists)f(a)h(unique)g(graph)f Fr(\030)i Fq(7!)c Fr(')1708 2168 y Fo(\030)1774 2156 y Fw(in)j(a)g(tubular)g (neigh)n(b)r(orho)r(o)r(d)e Fq(S)2812 2168 y Fo(`;\024;")2984 2156 y Fw(suc)n(h)h(that)i Fr(')3408 2168 y Fo(\030)456 2255 y Fw(is)j(solution)g(of)h(equation)f(\(4.3\).)55 b(Replacing)33 b Fr(')h Fw(b)n(y)f Fr(')2213 2267 y Fo(\030)2283 2255 y Fw(in)h(\(4.4\))g(the)g(problem)f(reduces)g(to)456 2355 y(study)27 b(an)h(equation)f(in)h(the)g(real)e(v)-5 b(ariable)27 b Fr(\030)t Fw(.)456 2505 y Fz(Lemma)32 b(4.1.)42 b Fk(F)-6 b(or)32 b(any)h Fr(\024)27 b(>)h Fw(0)j Fk(ther)l(e)i(ar)l(e)f Fr(")1941 2517 y Fs(2)2006 2505 y Fq(2)c Fw(\(0)p Fr(;)14 b(")2239 2517 y Fs(0)2276 2505 y Fw(])32 b Fk(and)h Fr(`)2530 2517 y Fs(6)2594 2505 y Fq(\025)27 b Fr(`)2721 2517 y Fs(3)2790 2505 y Fk(such)33 b(that,)g(for)g(any)456 2605 y Fr(h)22 b(<)h(e)653 2574 y Fp(\000)p Fs(2)p Fo(\013`)809 2582 y Fl(6)875 2605 y Fk(and)30 b Fr(\030)d Fq(2)d Fw(\000)1230 2617 y Fo(`)1258 2625 y Fl(6)1290 2617 y Fo(;\024)1353 2605 y Fk(,)30 b(the)g(map)g Fr(A)1790 2617 y Fo(\030)1850 2605 y Fw(:)23 b Fr(L)1953 2574 y Fs(sym)1953 2625 y Fp(1)2072 2605 y Fw(\()p Fn(R)q Fw(\))29 b Fq(!)23 b Fr(L)2383 2574 y Fs(sym)2383 2625 y Fp(1)2502 2605 y Fw(\()p Fn(R)q Fw(\))36 b Fk(de\014ne)l(d)30 b(by)1348 2746 y Fr(A)1410 2758 y Fo(\030)1447 2746 y Fw(\()p Fr(')p Fw(\))1610 2699 y Fr(:)1589 2746 y Fw(=)23 b Fq(\000)p Fr(L)1799 2710 y Fp(\000)p Fs(1)1799 2771 y Fo(\030)1887 2746 y Fr(T)1936 2758 y Fo(\030)1985 2746 y Fw(\()q Fr(f)9 b Fw(\()p Fr(n)2150 2758 y Fo(\030)2186 2746 y Fw(\))19 b(+)f Fr(R)2383 2758 y Fo(\030)2419 2746 y Fw([)p Fr(')p Fw(]\))723 b(\(4.5\))456 2890 y Fk(is)30 b(a)g(c)l(ontr)l(action)g(in)g Fr(Y)1201 2902 y Fo(")1232 2910 y Fl(2)1313 2843 y Fr(:)1292 2890 y Fw(=)22 b Fq(f)p Fr(')h Fq(2)h Fr(L)1634 2860 y Fs(sym)1634 2911 y Fp(1)1753 2890 y Fw(\()p Fn(R)p Fw(\))30 b(:)23 b Fq(k)p Fr(')p Fq(k)2085 2902 y Fp(1)2177 2890 y Fq(\024)g Fr(")2304 2902 y Fs(2)2341 2890 y Fq(g)p Fk(.)456 3059 y Fz(Pro)s(of.)41 b Fw(F)-7 b(rom)28 b(\(2.7\))f(and)g (\(2.34\))g(w)n(e)g(ha)n(v)n(e:)1213 3216 y Fq(k)p Fr(A)1317 3228 y Fo(\030)1353 3216 y Fr(')p Fq(k)1449 3228 y Fp(1)1542 3216 y Fq(\024)1648 3159 y Fr(c)1684 3171 y Fs(1)p 1640 3196 89 4 v 1640 3273 a Fr(!)1692 3285 y Fs(1)1753 3148 y Fm(\002)1787 3216 y Fq(k)p Fr(T)1878 3228 y Fo(\030)1914 3216 y Fr(f)9 b Fw(\()p Fr(n)2046 3228 y Fo(\030)2082 3216 y Fw(\))p Fq(k)2156 3228 y Fp(1)2244 3216 y Fw(+)18 b Fr(k)2370 3228 y Fs(2)2408 3216 y Fq(k)p Fr(')p Fq(k)2546 3181 y Fs(2)2546 3236 y Fp(1)2615 3148 y Fm(\003)2664 3216 y Fr(:)456 3386 y Fw(W)-7 b(e)28 b(write)1084 3487 y Fr(T)1133 3499 y Fo(\030)1168 3487 y Fr(f)9 b Fw(\()p Fr(n)1300 3499 y Fo(\030)1336 3487 y Fw(\))24 b(=)e Fr(T)1528 3499 y Fo(\030)1564 3487 y Fw(\()p Fr(f)9 b Fw(\()p Fr(n)1728 3499 y Fo(\030)1765 3487 y Fw(\))18 b Fq(\000)g Fr(V)h Fw(\()p Fr(\030)t Fw(\))p Fr(@)2113 3499 y Fo(\030)2150 3487 y Fr(n)2200 3499 y Fo(\030)2237 3487 y Fw(\))f(+)g Fr(V)h Fw(\()p Fr(\030)t Fw(\))p Fr(T)2590 3499 y Fo(\030)2627 3487 y Fr(@)2671 3499 y Fo(\030)2707 3487 y Fr(n)2757 3499 y Fo(\030)2793 3487 y Fr(:)456 3605 y Fw(F)-7 b(rom)27 b(\(2.20\))g(and)g(\(4.2\))g(it)h(follo)n(ws)f(that)1195 3748 y Fq(k)p Fr(T)1286 3760 y Fo(\030)1321 3748 y Fw(\()p Fr(f)9 b Fw(\()p Fr(n)1485 3760 y Fo(\030)1522 3748 y Fw(\))18 b Fq(\000)g Fr(V)h Fw(\()p Fr(\030)t Fw(\))p Fr(@)1870 3760 y Fo(\030)1907 3748 y Fr(n)1957 3760 y Fo(\030)1993 3748 y Fw(\))p Fq(k)2067 3760 y Fp(1)2161 3748 y Fq(\024)j Fr(C)6 b(e)2352 3714 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)2587 3722 y Fl(0)2620 3714 y Fs(\))p Fo(\030)2682 3748 y Fr(:)456 3887 y Fw(Since)27 b(for)g Fr(\030)h Fq(2)23 b Fw(\000)993 3899 y Fo(`)1021 3907 y Fl(3)1053 3899 y Fo(;\024)1144 3887 y Fw(and,)k(if)h Fr(h)23 b(<)g(e)1602 3856 y Fp(\000)p Fs(2)p Fo(\013`)1758 3864 y Fl(3)1794 3887 y Fw(,)28 b(then)g Fr(he)2121 3856 y Fp(\000)p Fs(2)p Fo(\013\030)2308 3887 y Fq(\024)22 b Fr(e)2434 3856 y Fs(2)p Fo(\013\024)2553 3887 y Fw(,)28 b(from)f(\(2.40\))g(w)n(e)g(get:)1355 4033 y Fq(j)p Fr(V)19 b Fw(\()p Fr(\030)t Fw(\))p Fq(jk)p Fr(T)1663 4045 y Fo(\030)1699 4033 y Fr(@)1743 4045 y Fo(\030)1779 4033 y Fr(n)1829 4045 y Fo(\030)1866 4033 y Fq(k)1908 4045 y Fp(1)2001 4033 y Fq(\024)j Fr(C)6 b(e)2192 3998 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)2427 4006 y Fl(1)2460 3998 y Fs(\))p Fo(\030)2522 4033 y Fr(:)456 4168 y Fw(By)27 b(the)h(ab)r(o)n(v)n(e)e(estimates)h(w)n(e)h(conclude)f(that)h(there)f (is)h Fr(\016)e(>)c Fw(0)28 b(so)f(that)1462 4307 y Fq(k)p Fr(T)1553 4319 y Fo(\030)1588 4307 y Fr(f)9 b Fw(\()p Fr(n)1720 4319 y Fo(\030)1756 4307 y Fw(\))p Fq(k)1830 4319 y Fp(1)1924 4307 y Fq(\024)22 b Fr(C)6 b(e)2115 4272 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)r Fs(\))p Fo(\030)2415 4307 y Fr(:)836 b Fw(\(4.6\))456 4445 y(Then,)27 b(for)h(an)n(y)e Fr(`)d Fq(\025)g Fr(`)1160 4457 y Fs(3)1196 4445 y Fw(,)28 b Fr(h)23 b(<)g(e)1445 4415 y Fp(\000)p Fs(2)p Fo(\013`)1604 4445 y Fw(,)28 b Fr(\030)f Fq(2)d Fw(\000)1849 4457 y Fo(`;\024)1939 4445 y Fw(,)k(and)f Fr(')d Fq(2)f Fr(Y)2355 4457 y Fo(")2391 4445 y Fw(,)1262 4609 y Fq(k)p Fr(A)1366 4621 y Fo(\030)1402 4609 y Fw(\()p Fr(')p Fw(\))p Fq(k)1562 4621 y Fp(1)1656 4609 y Fq(\024)1761 4553 y Fr(c)1797 4565 y Fs(1)p 1753 4590 V 1753 4666 a Fr(!)1805 4678 y Fs(1)1866 4517 y Fm(h)1905 4609 y Fr(C)6 b(e)2009 4574 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)r Fs(\))p Fo(`)2323 4609 y Fw(+)18 b Fr(k)2449 4621 y Fs(2)2486 4609 y Fr(")2525 4574 y Fs(2)2562 4517 y Fm(i)2615 4609 y Fr(:)636 b Fw(\(4.7\))456 4785 y(Moreo)n(v)n(er,)25 b(recalling)i(\(3.14\))o(,)h(there)f(is)h Fr(k)1779 4797 y Fs(3)1839 4785 y Fr(>)23 b Fw(0)k(for)g(whic)n(h,)h (if)g Fr(')2514 4797 y Fs(1)2551 4785 y Fr(;)14 b(')2642 4797 y Fs(2)2703 4785 y Fq(2)23 b Fr(Y)2829 4797 y Fo(")2865 4785 y Fw(,)28 b(then)1128 4920 y Fq(k)p Fr(R)1233 4932 y Fo(\030)1269 4920 y Fw([)p Fr(')1346 4932 y Fs(1)1384 4920 y Fw(])19 b Fq(\000)f Fr(R)1572 4932 y Fo(\030)1608 4920 y Fw([)p Fr(')1685 4932 y Fs(2)1723 4920 y Fw(])p Fq(k)1788 4932 y Fp(1)1880 4920 y Fq(\024)23 b Fr(k)2011 4932 y Fs(3)2049 4920 y Fr(")p Fw(\(1)18 b(+)g Fr(")p Fw(\))p Fq(k)p Fr(')2430 4932 y Fs(1)2485 4920 y Fq(\000)g Fr(')2622 4932 y Fs(2)2660 4920 y Fq(k)2702 4932 y Fp(1)456 5056 y Fw(so)27 b(that)1053 5179 y Fq(k)p Fr(A)1157 5191 y Fo(\030)1193 5179 y Fw(\()p Fr(')1279 5191 y Fs(1)1317 5179 y Fw(\))18 b Fq(\000)g Fr(A)1512 5191 y Fo(\030)1549 5179 y Fw(\()p Fr(')1635 5191 y Fs(2)1673 5179 y Fw(\))p Fq(k)1747 5191 y Fp(1)1840 5179 y Fq(\024)1938 5123 y Fr(c)1974 5135 y Fs(1)2011 5123 y Fr(k)2054 5135 y Fs(3)p 1938 5160 154 4 v 1970 5236 a Fr(!)2022 5248 y Fs(1)2101 5179 y Fr(")p Fw(\(1)g(+)g Fr(")p Fw(\))p Fq(k)p Fr(')2482 5191 y Fs(1)2538 5179 y Fq(\000)g Fr(')2675 5191 y Fs(2)2712 5179 y Fq(k)2754 5191 y Fp(1)2824 5179 y Fr(:)427 b Fw(\(4.8\))p eop %%Page: 17 17 17 16 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(17)555 450 y Fw(F)-7 b(rom)23 b(\(4.7\))g(and)g(\(4.8\))f (w)n(e)h(conclude)g(that,)h(b)n(y)f(c)n(ho)r(osing)f Fr(")2450 462 y Fs(2)2510 450 y Fw(=)g(2)p Fr(C)6 b(c)2740 462 y Fs(1)2777 450 y Fr(!)2832 415 y Fp(\000)p Fs(1)2829 472 y(1)2921 450 y Fr(e)2960 420 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)r Fs(\))p Fo(`)3251 428 y Fl(6)3311 450 y Fw(and)456 550 y Fr(`)491 562 y Fs(6)551 550 y Fq(\025)22 b Fr(`)673 562 y Fs(3)738 550 y Fw(large)k(enough,)h Fr(A)1312 562 y Fo(\030)1376 550 y Fw(is)h(a)f(con)n(traction)f(in)i Fr(Y)2110 562 y Fo(")2141 570 y Fl(2)2178 550 y Fw(.)1179 b Fg(\003)456 732 y Fz(Prop)s(osition)33 b(4.2.)42 b Fk(F)-6 b(or)33 b(any)g Fr(\024)c(>)f Fw(0)k Fk(ther)l(e)h(is)g Fr(`)2068 744 y Fs(7)2133 732 y Fr(>)28 b(`)2261 744 y Fs(6)2330 732 y Fk(such)33 b(that,)h(for)g(any)f Fr(h)28 b(<)g(e)3227 702 y Fp(\000)p Fs(2)p Fo(\013`)3383 710 y Fl(7)3419 732 y Fk(,)456 832 y(the)i(gr)l(aph)g Fr(\030)e Fq(!)23 b Fr(')1043 844 y Fo(\030)1109 832 y Fk(is)30 b(di\013er)l(entiable)i(and)1107 988 y Fq(k)p Fr(')1203 1000 y Fo(\030)1240 988 y Fq(k)1282 1000 y Fp(1)1375 988 y Fq(\024)22 b Fr(C)6 b(e)1566 954 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)1801 962 y Fl(2)1834 954 y Fs(\))p Fo(\030)1896 988 y Fr(;)184 b Fq(k)p Fr(@)2189 1000 y Fo(\030)2225 988 y Fr(')2279 1000 y Fo(\030)2315 988 y Fq(k)2357 1000 y Fp(1)2450 988 y Fq(\024)23 b Fr(C)6 b(e)2642 954 y Fp(\000)p Fo(\016)2724 962 y Fl(2)2756 954 y Fo(\030)3274 988 y Fw(\(4.9\))456 1134 y Fk(with)22 b Fr(\016)665 1146 y Fs(2)725 1134 y Fw(=)h(min)p Fq(f)p Fr(\016)1030 1146 y Fs(0)1067 1134 y Fr(;)14 b(\016)1141 1146 y Fs(1)1178 1134 y Fr(;)g Fw(2)p Fr(\013)p Fq(g)20 b Fk(\()p Fr(\016)1443 1146 y Fs(0)1502 1134 y Fk(and)i Fr(\016)1692 1146 y Fs(1)1751 1134 y Fk(as)f(in)h(The)l(or)l(em)h(2.2)f (and)g(L)l(emma)g(2.4)h(r)l(esp)l(e)l(ctively\).)456 1313 y Fz(Pro)s(of.)46 b Fw(Since)29 b Fr(')1033 1325 y Fo(\030)1099 1313 y Fw(is)f(the)i(\014xed)f(p)r(oin)n(t)g(of)g(the)g (map)g(\(4.5\))o(,)h(the)f(b)r(ound)g(for)f Fq(k)p Fr(')3022 1325 y Fo(\030)3059 1313 y Fq(k)3101 1325 y Fp(1)3199 1313 y Fw(follo)n(ws)456 1413 y(immediately)40 b(from)f(\(4.6\))h (\(with)g Fr(\016)1630 1425 y Fs(2)1711 1413 y Fw(=)j Fr(\016)s Fw(\).)74 b(Moreo)n(v)n(er,)41 b(since)e(the)i(r.h.s.)e(of)46 b(\(4.5\))40 b(is)g(a)456 1513 y(con)n(tin)n(uous)26 b(function)i(of)g Fr(\030)k Fw(and)27 b Fr(')p Fw(,)h(the)g(con)n(tin)n (uit)n(y)f(of)h Fr(\030)f Fq(7!)c Fr(')2478 1525 y Fo(\030)2542 1513 y Fw(is)28 b(also)e(straigh)n(tforw)n(ard.)555 1612 y(W)-7 b(e)31 b(next)g(study)g(the)g(equation)g(for)f Fr(@)1786 1624 y Fo(\030)1822 1612 y Fr(')1876 1624 y Fo(\030)1944 1612 y Fw(whic)n(h)h(is)f(obtained)h(b)n(y)f(di\013eren)n (tiating)i(\(4.3\))456 1712 y(\(for)23 b Fr(')g Fw(=)g Fr(')830 1724 y Fo(\030)867 1712 y Fw(\))h(w.r.t.)f Fr(\030)t Fw(.)36 b(W)-7 b(e)24 b(shall)f(pro)n(v)n(e)f(that)i(for)f(an)n(y)g Fr(\024)g(>)g Fw(0)g(there)g(is)h Fr(`)2787 1724 y Fs(7)2847 1712 y Fq(\025)e Fr(`)2969 1724 y Fs(6)3030 1712 y Fw(suc)n(h)h(that)h (it)456 1812 y(has)f(a)g(unique)h(solution)g(for)f(an)n(y)g Fr(\030)k Fq(2)d Fw(\000)1712 1824 y Fo(`;\024)1802 1812 y Fw(.)36 b(By)24 b(standard)e(argumen)n(ts)h(this)h(solution)f (de\014nes)456 1911 y(the)28 b(deriv)-5 b(ativ)n(e)26 b Fr(@)1023 1923 y Fo(\030)1060 1911 y Fr(')1114 1923 y Fo(\030)1150 1911 y Fw(.)555 2011 y(Let)i Fr(G)769 2023 y Fo(\030)849 1964 y Fr(:)828 2011 y Fw(=)23 b Fq(\000)p Fw(\()p Fr(f)9 b Fw(\()p Fr(n)1145 2023 y Fo(\030)1181 2011 y Fw(\))18 b(+)f Fr(R)1376 2023 y Fo(\030)1412 2011 y Fw([)p Fr(')1489 2023 y Fo(\030)1526 2011 y Fw(]\),)28 b(so)e(that)i Fr(')1967 2023 y Fo(\030)2031 2011 y Fw(solv)n(es)d Fr(L)2323 2023 y Fo(\030)2359 2011 y Fr(')2413 2023 y Fo(\030)2473 2011 y Fw(=)e Fr(T)2610 2023 y Fo(\030)2645 2011 y Fr(G)2710 2023 y Fo(\030)2747 2011 y Fw(.)37 b(By)27 b(di\013eren)n(tiating)456 2110 y(\(recall)g(\(2.39\))o(\))h(w)n(e)f (get:)819 2257 y Fr(L)876 2269 y Fo(\030)912 2257 y Fr(@)956 2269 y Fo(\030)993 2257 y Fr(')1047 2269 y Fo(\030)1102 2257 y Fw(+)18 b Fr(@)1229 2269 y Fo(\030)1265 2257 y Fr(p)1307 2269 y Fo(\030)1343 2257 y Fr(J)27 b Fq(\003)18 b Fr(')1530 2269 y Fo(\030)1649 2257 y Fw(=)83 b Fr(T)1846 2269 y Fo(\030)1882 2257 y Fr(@)1926 2269 y Fo(\030)1962 2257 y Fr(G)2027 2269 y Fo(\030)2082 2257 y Fq(\000)18 b Fr(@)2209 2269 y Fo(\030)2246 2257 y Fr(\031)2293 2269 y Fo(\030)2329 2257 y Fw(\()p Fr(G)2426 2269 y Fo(\030)2463 2257 y Fw(\))p Fr(v)2535 2269 y Fo(\030)2591 2257 y Fq(\000)g Fr(\031)2721 2269 y Fo(\030)2757 2257 y Fw(\()p Fr(G)2854 2269 y Fo(\030)2891 2257 y Fw(\))p Fr(@)2967 2269 y Fo(\030)3004 2257 y Fr(v)3044 2269 y Fo(\030)1649 2381 y Fw(=)83 b Fr(T)1846 2393 y Fo(\030)1882 2381 y Fr(@)1926 2393 y Fo(\030)1962 2381 y Fr(G)2027 2393 y Fo(\030)2082 2381 y Fq(\000)18 b Fr(\031)2212 2393 y Fo(\030)2249 2381 y Fw(\()p Fr(G)2346 2393 y Fo(\030)2383 2381 y Fw(\))p Fr(T)2464 2393 y Fo(\030)2500 2381 y Fr(@)2544 2393 y Fo(\030)2581 2381 y Fr(v)2621 2393 y Fo(\030)1797 2506 y Fq(\000)c Fw([)p Fr(\031)1946 2518 y Fo(\030)1982 2506 y Fw(\()p Fr(G)2079 2518 y Fo(\030)2116 2506 y Fw(\))p Fr(\031)2195 2518 y Fo(\030)2232 2506 y Fw(\()p Fr(@)2308 2518 y Fo(\030)2345 2506 y Fr(v)2385 2518 y Fo(\030)2421 2506 y Fw(\))19 b(+)f Fr(@)2599 2518 y Fo(\030)2635 2506 y Fr(\031)2682 2518 y Fo(\030)2719 2506 y Fw(\()p Fr(G)2816 2518 y Fo(\030)2853 2506 y Fw(\)])p Fr(v)2948 2518 y Fo(\030)2985 2506 y Fr(:)456 2652 y Fw(By)27 b(substituting)h Fr(G)1113 2664 y Fo(\030)1173 2652 y Fw(=)22 b Fr(L)1317 2664 y Fo(\030)1353 2652 y Fr(')1407 2664 y Fo(\030)1462 2652 y Fw(+)c Fr(\031)1592 2664 y Fo(\030)1629 2652 y Fw(\()p Fr(G)1726 2664 y Fo(\030)1763 2652 y Fw(\))p Fr(v)1835 2664 y Fo(\030)1899 2652 y Fw(in)28 b(the)g(term)g Fr(@)2382 2664 y Fo(\030)2418 2652 y Fr(\031)2465 2664 y Fo(\030)2502 2652 y Fw(\()p Fr(G)2599 2664 y Fo(\030)2636 2652 y Fw(\))g(w)n(e)f(ha)n(v)n(e:)793 2798 y Fr(L)850 2810 y Fo(\030)886 2798 y Fr(@)930 2810 y Fo(\030)966 2798 y Fr(')1020 2810 y Fo(\030)1140 2798 y Fw(=)82 b Fq(\000)p Fr(@)1396 2810 y Fo(\030)1432 2798 y Fr(p)1474 2810 y Fo(\030)1511 2798 y Fr(J)26 b Fq(\003)18 b Fr(')1697 2810 y Fo(\030)1752 2798 y Fw(+)g Fr(T)1884 2810 y Fo(\030)1920 2798 y Fr(@)1964 2810 y Fo(\030)2000 2798 y Fr(G)2065 2810 y Fo(\030)2120 2798 y Fq(\000)h Fr(\031)2251 2810 y Fo(\030)2287 2798 y Fw(\()p Fr(G)2384 2810 y Fo(\030)2421 2798 y Fw(\))p Fr(T)2502 2810 y Fo(\030)2538 2798 y Fr(@)2582 2810 y Fo(\030)2619 2798 y Fr(v)2659 2810 y Fo(\030)1287 2925 y Fq(\000)1366 2858 y Fm(\010)1414 2925 y Fr(\031)1461 2937 y Fo(\030)1498 2925 y Fw(\()p Fr(G)1595 2937 y Fo(\030)1632 2925 y Fw(\))1664 2858 y Fm(\002)1699 2925 y Fr(\031)1746 2937 y Fo(\030)1782 2925 y Fw(\()p Fr(@)1858 2937 y Fo(\030)1895 2925 y Fr(v)1935 2937 y Fo(\030)1972 2925 y Fw(\))f(+)g Fr(@)2149 2937 y Fo(\030)2186 2925 y Fr(\031)2233 2937 y Fo(\030)2270 2925 y Fw(\()p Fr(v)2342 2937 y Fo(\030)2378 2925 y Fw(\))2410 2858 y Fm(\003)2464 2925 y Fw(+)g Fr(@)2591 2937 y Fo(\030)2627 2925 y Fr(\031)2674 2937 y Fo(\030)2711 2925 y Fw(\()p Fr(L)2800 2937 y Fo(\030)2836 2925 y Fr(')2890 2937 y Fo(\030)2927 2925 y Fw(\))2959 2858 y Fm(\011)3007 2925 y Fr(v)3047 2937 y Fo(\030)3084 2925 y Fr(:)456 3071 y Fw(Observing)26 b(no)n(w)h(that,)h(b)n(y)g(\(2.29\))o(,)g Fr(\031)1653 3083 y Fo(\030)1690 3071 y Fw(\()p Fr(@)1766 3083 y Fo(\030)1802 3071 y Fr(v)1842 3083 y Fo(\030)1879 3071 y Fw(\))19 b(+)f Fr(@)2057 3083 y Fo(\030)2093 3071 y Fr(\031)2140 3083 y Fo(\030)2177 3071 y Fw(\()p Fr(v)2249 3083 y Fo(\030)2286 3071 y Fw(\))23 b(=)g(0)k(w)n(e)g(get)748 3218 y Fr(L)805 3230 y Fo(\030)841 3218 y Fr(@)885 3230 y Fo(\030)921 3218 y Fr(')975 3230 y Fo(\030)1035 3218 y Fw(=)c Fr(T)1172 3230 y Fo(\030)1221 3218 y Fw([)q Fr(@)1289 3230 y Fo(\030)1325 3218 y Fr(G)1390 3230 y Fo(\030)1445 3218 y Fq(\000)18 b Fr(@)1572 3230 y Fo(\030)1608 3218 y Fr(p)1650 3230 y Fo(\030)1687 3218 y Fr(J)26 b Fq(\003)18 b Fr(')1873 3230 y Fo(\030)1928 3218 y Fq(\000)g Fr(\031)2058 3230 y Fo(\030)2095 3218 y Fw(\()p Fr(G)2192 3230 y Fo(\030)2229 3218 y Fw(\))p Fr(@)2305 3230 y Fo(\030)2341 3218 y Fr(v)2381 3230 y Fo(\030)2418 3218 y Fw(\)])h(+)f Fr(\025)2623 3230 y Fo(\030)2660 3218 y Fr(H)7 b Fw(\()p Fr(\030)t Fw(\))p Fr(v)2880 3230 y Fo(\030)2917 3218 y Fr(;)292 b Fw(\(4.10\))456 3364 y(with)685 3528 y Fr(H)7 b Fw(\()p Fr(\030)t Fw(\))83 b(=)g Fq(\000)1192 3472 y Fw(1)p 1171 3509 85 4 v 1171 3585 a Fr(\025)1219 3597 y Fo(\030)1279 3528 y Fw([)p Fr(\031)1349 3540 y Fo(\030)1386 3528 y Fw(\()p Fr(@)1462 3540 y Fo(\030)1498 3528 y Fr(p)1540 3540 y Fo(\030)1577 3528 y Fr(J)26 b Fq(\003)18 b Fr(')1763 3540 y Fo(\030)1800 3528 y Fw(\))g(+)g Fr(@)1977 3540 y Fo(\030)2014 3528 y Fr(\031)2061 3540 y Fo(\030)2097 3528 y Fw(\()p Fr(L)2186 3540 y Fo(\030)2223 3528 y Fr(')2277 3540 y Fo(\030)2313 3528 y Fw(\)])948 3787 y(=)83 b Fq(\000)1192 3731 y Fw(1)p 1171 3768 V 1171 3844 a Fr(\025)1219 3856 y Fo(\030)1279 3674 y Fm(Z)1362 3695 y Fp(1)1325 3863 y Fs(0)1432 3787 y Fr(dx)1551 3645 y Fm(")1609 3731 y Fr(@)1653 3743 y Fo(\030)1689 3731 y Fr(p)1731 3743 y Fo(\030)p 1609 3768 159 4 v 1649 3844 a Fr(p)1691 3856 y Fo(\030)1778 3787 y Fr(v)1818 3799 y Fo(\030)1854 3787 y Fr(J)27 b Fq(\003)18 b Fr(')2041 3799 y Fo(\030)2096 3787 y Fw(+)2179 3645 y Fm( )2254 3731 y Fr(@)2298 3743 y Fo(\030)2335 3731 y Fr(v)2375 3743 y Fo(\030)p 2254 3768 158 4 v 2294 3844 a Fr(p)2336 3856 y Fo(\030)2440 3787 y Fq(\000)2533 3731 y Fr(@)2577 3743 y Fo(\030)2613 3731 y Fr(p)2655 3743 y Fo(\030)p 2533 3768 159 4 v 2573 3844 a Fr(p)2615 3816 y Fs(2)2615 3869 y Fo(\030)2701 3787 y Fr(v)2741 3799 y Fo(\030)2778 3645 y Fm(!)2857 3787 y Fr(L)2914 3799 y Fo(\030)2950 3787 y Fr(')3004 3799 y Fo(\030)3041 3645 y Fm(#)3103 3787 y Fw(\()p Fr(x)p Fw(\))948 4070 y(=)83 b Fq(\000)1192 4013 y Fw(1)p 1171 4050 85 4 v 1171 4127 a Fr(\025)1219 4139 y Fo(\030)1279 3957 y Fm(Z)1362 3977 y Fp(1)1325 4145 y Fs(0)1432 4070 y Fr(dx)1551 3928 y Fm( )1616 4070 y Fr(@)1660 4082 y Fo(\030)1697 4070 y Fr(v)1737 4082 y Fo(\030)1773 4070 y Fr(J)27 b Fq(\003)18 b Fr(')1960 4082 y Fo(\030)2015 4070 y Fw(+)2108 4013 y Fr(@)2152 4025 y Fo(\030)2188 4013 y Fr(p)2230 4025 y Fo(\030)p 2108 4050 159 4 v 2148 4127 a Fr(p)2190 4098 y Fs(2)2190 4152 y Fo(\030)2276 4070 y Fr(v)2316 4082 y Fo(\030)2353 4070 y Fr(')2407 4082 y Fo(\030)2462 4070 y Fq(\000)2555 4013 y Fr(@)2599 4025 y Fo(\030)2636 4013 y Fr(v)2676 4025 y Fo(\030)p 2555 4050 158 4 v 2595 4127 a Fr(p)2637 4139 y Fo(\030)2722 4070 y Fr(')2776 4082 y Fo(\030)2813 3928 y Fm(!)2892 4070 y Fw(\()p Fr(x)p Fw(\))948 4352 y(=)83 b Fq(\000)1192 4296 y Fw(1)p 1171 4333 85 4 v 1171 4409 a Fr(\025)1219 4421 y Fo(\030)1279 4239 y Fm(Z)1362 4259 y Fp(1)1325 4427 y Fs(0)1432 4352 y Fr(dx)14 b(')1590 4364 y Fo(\030)1627 4352 y Fw(\()p Fr(x)p Fw(\))1752 4210 y Fm( )1819 4352 y Fr(J)27 b Fq(\003)18 b Fr(@)1996 4364 y Fo(\030)2032 4352 y Fr(v)2072 4364 y Fo(\030)2127 4352 y Fw(+)2220 4296 y Fr(@)2264 4308 y Fo(\030)2300 4296 y Fr(p)2342 4308 y Fo(\030)p 2220 4333 159 4 v 2260 4409 a Fr(p)2302 4380 y Fs(2)2302 4434 y Fo(\030)2388 4352 y Fr(v)2428 4364 y Fo(\030)2483 4352 y Fq(\000)2576 4296 y Fr(@)2620 4308 y Fo(\030)2657 4296 y Fr(v)2697 4308 y Fo(\030)p 2576 4333 158 4 v 2616 4409 a Fr(p)2658 4421 y Fo(\030)2743 4210 y Fm(!)2823 4352 y Fw(\()p Fr(x)p Fw(\))p Fr(;)275 b Fw(\(4.11\))456 4568 y(where,)23 b(in)h(the)g(last)f(line,)i(w)n(e)e (used)g(that,)i(since)e Fr(@)2023 4580 y Fo(\030)2060 4568 y Fr(v)2100 4580 y Fo(\030)2136 4568 y Fw(,)i Fr(J)8 b Fw(,)24 b(and)f Fr(')2496 4580 y Fo(\030)2556 4568 y Fw(are)g(symmetric)g(functions,)456 4668 y(then:)1021 4692 y Fm(Z)1104 4713 y Fp(1)1067 4881 y Fs(0)1174 4805 y Fr(dx)14 b(@)1322 4817 y Fo(\030)1359 4805 y Fr(v)1399 4817 y Fo(\030)1436 4805 y Fw(\()p Fr(x)p Fw(\))p Fr(J)27 b Fq(\003)18 b Fr(')1734 4817 y Fo(\030)1771 4805 y Fw(\()p Fr(x)p Fw(\))24 b(=)1994 4692 y Fm(Z)2077 4713 y Fp(1)2040 4881 y Fs(0)2147 4805 y Fr(dx)14 b(')2305 4817 y Fo(\030)2342 4805 y Fw(\()p Fr(x)p Fw(\))p Fr(J)28 b Fq(\003)18 b Fr(@)2631 4817 y Fo(\030)2667 4805 y Fr(v)2707 4817 y Fo(\030)2744 4805 y Fw(\()p Fr(x)p Fw(\))p Fr(:)456 4976 y Fw(No)n(w,)27 b(b)n(y)g(di\013eren)n(tiating)g Fr(L)1375 4988 y Fo(\030)1411 4976 y Fr(v)1451 4988 y Fo(\030)1511 4976 y Fw(=)22 b Fr(\025)1646 4988 y Fo(\030)1683 4976 y Fr(v)1723 4988 y Fo(\030)1760 4976 y Fw(,)1050 5168 y Fr(J)27 b Fq(\003)18 b Fr(@)1227 5180 y Fo(\030)1263 5168 y Fr(v)1303 5180 y Fo(\030)1363 5168 y Fw(=)1460 5112 y Fr(d\025)1551 5124 y Fo(\030)p 1460 5149 128 4 v 1483 5225 a Fr(d\030)1609 5112 y(v)1649 5124 y Fo(\030)p 1608 5149 79 4 v 1608 5225 a Fr(p)1650 5237 y Fo(\030)1715 5168 y Fw(+)g Fr(\025)1846 5180 y Fo(\030)1892 5112 y Fr(@)1936 5124 y Fo(\030)1973 5112 y Fr(v)2013 5124 y Fo(\030)p 1892 5149 158 4 v 1932 5225 a Fr(p)1974 5237 y Fo(\030)2078 5168 y Fq(\000)2171 5112 y Fr(@)2215 5124 y Fo(\030)2251 5112 y Fr(p)2293 5124 y Fo(\030)p 2171 5149 159 4 v 2211 5225 a Fr(p)2253 5237 y Fo(\030)2339 5168 y Fr(J)27 b Fq(\003)18 b Fr(v)2512 5180 y Fo(\030)2567 5168 y Fw(+)2660 5112 y Fr(@)2704 5124 y Fo(\030)2740 5112 y Fr(v)2780 5124 y Fo(\030)p 2660 5149 158 4 v 2699 5225 a Fr(p)2741 5237 y Fo(\030)2827 5168 y Fr(;)p eop %%Page: 18 18 18 17 bop 456 251 a Fs(18)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fw(so)27 b(that)806 654 y Fr(H)7 b Fw(\()p Fr(\030)t Fw(\))24 b(=)e Fq(\000)1194 598 y Fw(1)p 1172 635 85 4 v 1172 711 a Fr(\025)1220 723 y Fo(\030)1281 541 y Fm(Z)1364 562 y Fp(1)1327 730 y Fs(0)1434 654 y Fr(dx)14 b(')1592 666 y Fo(\030)1629 654 y Fw(\()p Fr(x)p Fw(\))1754 512 y Fm( )1831 598 y Fr(d\025)1922 610 y Fo(\030)p 1831 635 128 4 v 1853 711 a Fr(d\030)1979 598 y(v)2019 610 y Fo(\030)p 1978 635 79 4 v 1978 711 a Fr(p)2020 723 y Fo(\030)2085 654 y Fw(+)k Fr(\025)2216 666 y Fo(\030)2263 598 y Fr(@)2307 610 y Fo(\030)2343 598 y Fr(v)2383 610 y Fo(\030)p 2263 635 158 4 v 2302 711 a Fr(p)2344 723 y Fo(\030)2448 654 y Fq(\000)2541 598 y Fr(@)2585 610 y Fo(\030)2622 598 y Fr(p)2664 610 y Fo(\030)p 2541 635 159 4 v 2581 711 a Fr(p)2623 682 y Fs(2)2623 736 y Fo(\030)2710 654 y Fr(L)2767 666 y Fo(\030)2803 654 y Fr(v)2843 666 y Fo(\030)2879 512 y Fm(!)2959 654 y Fw(\()p Fr(x)p Fw(\))p Fr(:)456 874 y Fw(Using)28 b(again)g Fr(L)970 886 y Fo(\030)1006 874 y Fr(v)1046 886 y Fo(\030)1107 874 y Fw(=)d Fr(\025)1245 886 y Fo(\030)1282 874 y Fr(v)1322 886 y Fo(\030)1387 874 y Fw(and)k(observing)e(the)i(\014rst)g(term)f(on)h(the)g(r.h.s.)g (is)f(prop)r(ortional)456 974 y(to)f Fr(\031)604 986 y Fo(\030)641 974 y Fw(\()p Fr(')727 986 y Fo(\030)764 974 y Fw(\))c(=)g(0,)k(w)n(e)g(\014nally)h(obtain:)1121 1202 y Fr(H)7 b Fw(\()p Fr(\030)t Fw(\))24 b(=)e Fq(\000)1491 1089 y Fm(Z)1574 1109 y Fp(1)1537 1277 y Fs(0)1644 1202 y Fr(dx)14 b(')1802 1214 y Fo(\030)1839 1202 y Fw(\()p Fr(x)p Fw(\))1964 1060 y Fm( )2041 1145 y Fr(@)2085 1157 y Fo(\030)2121 1145 y Fr(v)2161 1157 y Fo(\030)p 2041 1182 158 4 v 2080 1258 a Fr(p)2122 1270 y Fo(\030)2226 1202 y Fq(\000)2319 1145 y Fr(@)2363 1157 y Fo(\030)2400 1145 y Fr(p)2442 1157 y Fo(\030)p 2319 1182 159 4 v 2359 1258 a Fr(p)2401 1230 y Fs(2)2401 1284 y Fo(\030)2488 1202 y Fr(v)2528 1214 y Fo(\030)2564 1060 y Fm(!)2644 1202 y Fw(\()p Fr(x)p Fw(\))p Fr(:)454 b Fw(\(4.12\))456 1426 y(Recalling)27 b(no)n(w)f Fr(L)1046 1438 y Fo(\030)1105 1426 y Fw(=)d Fr(D)r(f)9 b Fq(j)1337 1438 y Fo(n)1378 1447 y Fi(\030)1442 1426 y Fw(and)28 b(\(3.14\))o(,)f(w)n(e)h(ha)n(v)n (e:)1277 1577 y Fr(@)1321 1589 y Fo(\030)1357 1577 y Fr(G)1422 1589 y Fo(\030)1482 1577 y Fw(=)23 b Fq(\000)p Fr(L)1692 1589 y Fo(\030)1727 1577 y Fr(@)1771 1589 y Fo(\030)1808 1577 y Fr(n)1858 1589 y Fo(\030)1912 1577 y Fw(+)18 b Fr(K)2066 1589 y Fo(\030)2120 1577 y Fw(+)g Fr(Z)2260 1589 y Fo(\030)2296 1577 y Fr(J)27 b Fq(\003)18 b Fr(@)2473 1589 y Fo(\030)2509 1577 y Fr(')2563 1589 y Fo(\030)2600 1577 y Fr(;)609 b Fw(\(4.13\))456 1727 y(where)580 1915 y Fr(K)651 1927 y Fo(\030)710 1915 y Fw(=)22 b Fq(\000)p Fr(\014)913 1881 y Fs(3)950 1915 y Fw(\()p Fr(J)27 b Fq(\003)18 b Fr(')1169 1927 y Fo(\030)1206 1915 y Fw(\))1238 1881 y Fs(2)1289 1802 y Fm(Z)1372 1823 y Fs(1)1335 1991 y(0)1409 1915 y Fr(ds)c(s)p Fw(\(1)19 b Fq(\000)f Fr(s)p Fw(\))c(tanh)1971 1878 y Fp(000)2032 1915 y Fq(f)p Fr(\014)t Fw([)p Fr(J)27 b Fq(\003)18 b Fw(\()p Fr(n)2363 1927 y Fo(\030)2418 1915 y Fw(+)g Fr(s')2594 1927 y Fo(\030)2630 1915 y Fw(\))h(+)f Fr(h)p Fw(])p Fq(g)p Fr(J)26 b Fq(\003)18 b Fr(@)3053 1927 y Fo(\030)3089 1915 y Fr(n)3139 1927 y Fo(\030)3175 1915 y Fr(;)34 b Fw(\(4.14\))580 2154 y Fr(Z)637 2166 y Fo(\030)696 2154 y Fw(=)22 b Fq(\000)p Fw(2)p Fr(\014)941 2120 y Fs(2)978 2154 y Fr(J)27 b Fq(\003)18 b Fr(')1165 2166 y Fo(\030)1215 2041 y Fm(Z)1298 2061 y Fs(1)1261 2230 y(0)1335 2154 y Fr(ds)c Fw(\(1)19 b Fq(\000)f Fr(s)p Fw(\))c(tanh)1858 2117 y Fp(00)1900 2154 y Fq(f)p Fr(\014)t Fw([)p Fr(J)26 b Fq(\003)18 b Fw(\()p Fr(n)2230 2166 y Fo(\030)2285 2154 y Fw(+)g Fr(s')2461 2166 y Fo(\030)2498 2154 y Fw(\))h(+)f Fr(h)p Fw(])p Fq(g)779 2393 y(\000)g Fr(\014)913 2358 y Fs(3)950 2393 y Fw(\()p Fr(J)27 b Fq(\003)18 b Fr(')1169 2405 y Fo(\030)1206 2393 y Fw(\))1238 2358 y Fs(2)1289 2280 y Fm(Z)1372 2300 y Fs(1)1335 2468 y(0)1409 2393 y Fr(ds)c(s)p Fw(\(1)19 b Fq(\000)f Fr(s)p Fw(\))c(tanh)1971 2356 y Fp(000)2032 2393 y Fq(f)p Fr(\014)t Fw([)p Fr(J)27 b Fq(\003)18 b Fw(\()p Fr(n)2363 2405 y Fo(\030)2418 2393 y Fw(+)g Fr(s')2594 2405 y Fo(\030)2630 2393 y Fw(\))h(+)f Fr(h)p Fw(])p Fq(g)p Fr(:)332 b Fw(\(4.15\))456 2589 y(Observing)30 b(that)i Fr(T)1088 2601 y Fo(\030)1124 2589 y Fr(L)1181 2601 y Fo(\030)1217 2589 y Fr(@)1261 2601 y Fo(\030)1297 2589 y Fr(n)1347 2601 y Fo(\030)1414 2589 y Fw(=)e Fr(L)1566 2601 y Fo(\030)1602 2589 y Fr(T)1651 2601 y Fo(\030)1687 2589 y Fr(@)1731 2601 y Fo(\030)1767 2589 y Fr(n)1817 2601 y Fo(\030)1853 2589 y Fw(,)j(from)f(\(4.10\))f (and)h(\(4.13\))f(w)n(e)h(obtain)g Fr(@)3198 2601 y Fo(\030)3234 2589 y Fr(')3288 2601 y Fo(\030)3357 2589 y Fw(b)n(y)456 2689 y(solving)26 b(the)i(linear)f(equation:)1691 2804 y Fr( )f Fw(=)d Fr(U)1916 2816 y Fo(\030)1952 2804 y Fr( )e Fw(+)d Fr(g)2150 2816 y Fo(\030)2186 2804 y Fr(;)456 2937 y Fw(where)698 3087 y Fr(U)755 3099 y Fo(\030)791 3087 y Fr( )952 3040 y(:)931 3087 y Fw(=)83 b Fr(L)1136 3051 y Fp(\000)p Fs(1)1136 3112 y Fo(\030)1224 3087 y Fr(T)1273 3099 y Fo(\030)1309 3087 y Fr(Z)1366 3099 y Fo(\030)1402 3087 y Fr(J)27 b Fq(\003)18 b Fr( )s(;)772 3229 y(g)812 3241 y Fo(\030)952 3182 y Fr(:)931 3229 y Fw(=)83 b Fr(L)1136 3194 y Fp(\000)p Fs(1)1136 3254 y Fo(\030)1224 3229 y Fr(T)1273 3241 y Fo(\030)1323 3229 y Fw([)p Fr(K)1417 3241 y Fo(\030)1471 3229 y Fq(\000)18 b Fr(@)1598 3241 y Fo(\030)1635 3229 y Fr(p)1677 3241 y Fo(\030)1713 3229 y Fr(J)26 b Fq(\003)18 b Fr(')1899 3241 y Fo(\030)1955 3229 y Fq(\000)g Fr(\031)2085 3241 y Fo(\030)2121 3229 y Fw(\()p Fr(G)2218 3241 y Fo(\030)2255 3229 y Fw(\))p Fr(@)2331 3241 y Fo(\030)2368 3229 y Fr(v)2408 3241 y Fo(\030)2444 3229 y Fw(])h(+)f Fr(H)7 b Fw(\()p Fr(\030)t Fw(\))p Fr(v)2789 3241 y Fo(\030)2844 3229 y Fq(\000)18 b Fr(T)2976 3241 y Fo(\030)3012 3229 y Fr(@)3056 3241 y Fo(\030)3093 3229 y Fr(n)3143 3241 y Fo(\030)3179 3229 y Fr(:)456 3386 y Fw(F)-7 b(rom)24 b(\(2.34\))g(and)g(\(4.15\))g Fq(k)p Fr(U)1402 3398 y Fo(\030)1437 3386 y Fq(k)1479 3398 y Fp(1)1572 3386 y Fq(\024)f Fr(C)6 b Fq(k)p Fr(')1821 3398 y Fo(\030)1857 3386 y Fq(k)1899 3398 y Fp(1)1969 3386 y Fw(,)26 b(hence,)f(b)n(y)f(using)g(the)h(\014rst)g(b)r(ound)g (in)g(\(4.9\))o(,)456 3486 y(there)i(is)h Fr(`)787 3498 y Fs(7)847 3486 y Fq(\025)c Fr(`)971 3498 y Fs(6)1035 3486 y Fw(suc)n(h)k(that)g Fq(k)p Fr(U)1502 3498 y Fo(\030)1538 3486 y Fq(k)1580 3498 y Fp(1)1673 3486 y Fq(\024)23 b Fw(1)p Fr(=)p Fw(2)k(for)g Fr(\030)h Fq(2)23 b Fw(\000)2235 3498 y Fo(`;\024)2326 3486 y Fw(.)38 b(Then)28 b(for)f(suc)n(h)h(v)-5 b(alues)27 b(of)h Fr(\030)k Fw(the)456 3586 y(equation)d(has)g(a)h (unique)g(solution)f Fr(@)1651 3598 y Fo(\030)1688 3586 y Fr(')1742 3598 y Fo(\030)1805 3586 y Fw(=)e(\(1)19 b Fq(\000)h Fr(U)2132 3598 y Fo(\030)2168 3586 y Fw(\))2200 3556 y Fp(\000)p Fs(1)2289 3586 y Fr(g)2329 3598 y Fo(\030)2365 3586 y Fw(.)44 b(The)30 b(con)n(tin)n(uit)n(y)g(of)f Fr(\030)i Fq(7!)c Fr(@)3317 3598 y Fo(\030)3354 3586 y Fr(')3408 3598 y Fo(\030)456 3685 y Fw(follo)n(ws)f(from)h(that)h(of) g Fr(\030)f Fq(7!)c Fr(U)1425 3697 y Fo(\030)1489 3685 y Fw(and)k Fr(\030)h Fq(7!)23 b Fr(g)1860 3697 y Fo(\030)1896 3685 y Fw(,)k(whic)n(h)h(can)f(b)r(e)h(easily)f(pro)n(v)n(ed.)555 3785 y(W)-7 b(e)31 b(are)e(left)h(with)h(the)f(second)g(b)r(ound)g(in)h (\(4.9\))o(.)45 b(Since)30 b Fq(k)p Fr(@)2498 3797 y Fo(\030)2534 3785 y Fr(')2588 3797 y Fo(\030)2624 3785 y Fq(k)2666 3797 y Fp(1)2763 3785 y Fq(\024)d Fw(2)p Fq(k)p Fr(g)2979 3797 y Fo(\030)3014 3785 y Fq(k)3056 3797 y Fp(1)3156 3785 y Fw(w)n(e)j(ha)n(v)n(e)456 3885 y(only)h(to)h(pro)n(v)n(e)e(an)i(analogous)e(estimate)h(for)h(the)g (kno)n(wn)f(term)h Fr(g)2609 3897 y Fo(\030)2645 3885 y Fw(.)50 b(F)-7 b(rom)32 b(\(2.34\))o(,)h(\(4.14\))o(,)456 3984 y(and)27 b(\(2.33\))o(,)553 4135 y Fq(k)p Fr(g)635 4147 y Fo(\030)671 4135 y Fq(k)713 4147 y Fp(1)805 4135 y Fq(\024)c Fr(C)972 4068 y Fm(\000)1010 4135 y Fq(k)p Fr(')1106 4147 y Fo(\030)1142 4135 y Fq(k)1184 4101 y Fs(2)1184 4156 y Fp(1)1273 4135 y Fw(+)18 b Fq(k)p Fr(')1452 4147 y Fo(\030)1488 4135 y Fq(k)1530 4147 y Fp(1)1618 4135 y Fw(+)g Fq(j)p Fr(\031)1771 4147 y Fo(\030)1808 4135 y Fw(\()p Fr(G)1905 4147 y Fo(\030)1942 4135 y Fw(\))p Fq(j)h Fw(+)f Fq(j)p Fr(H)7 b Fw(\()p Fr(\030)t Fw(\))p Fq(j)19 b Fw(+)f Fq(k)p Fr(@)2513 4147 y Fo(\030)2549 4135 y Fr(n)2599 4147 y Fo(\030)2653 4135 y Fq(\000)g Fr(\031)2783 4147 y Fo(\030)2820 4135 y Fw(\()p Fr(@)2896 4147 y Fo(\030)2933 4135 y Fr(n)2983 4147 y Fo(\030)3019 4135 y Fw(\))p Fr(v)3091 4147 y Fo(\030)3128 4135 y Fw(\))p Fq(k)3202 4147 y Fp(1)3272 4068 y Fm(\001)3324 4135 y Fr(;)456 4285 y Fw(where)31 b(w)n(e)h(ha)n(v)n(e)f(used)h Fq(k)p Fr(@)1303 4297 y Fo(\030)1339 4285 y Fr(p)1381 4297 y Fo(\030)1417 4285 y Fq(k)1459 4297 y Fp(1)1559 4285 y Fq(\024)e Fw(const)p Fq(k)p Fr(@)1930 4297 y Fo(\030)1966 4285 y Fr(n)2016 4297 y Fo(\030)2052 4285 y Fq(k)2094 4297 y Fp(1)2196 4285 y Fw(and)i(\(2.19\))o(,)h(and)f Fq(k)p Fr(@)2882 4297 y Fo(\030)2918 4285 y Fr(v)2958 4297 y Fo(\030)2994 4285 y Fq(k)3036 4297 y Fp(1)3137 4285 y Fq(\024)e Fw(const)o(,)456 4385 y(whic)n(h)22 b(follo)n(ws)g(b)n(y)h(\(2.26\))f(and)h(\(2.36\))o(.)35 b(Recalling)22 b Fr(G)2150 4397 y Fo(\030)2210 4385 y Fw(=)h Fq(\000)p Fw(\()p Fr(f)9 b Fw(\()p Fr(n)2527 4397 y Fo(\030)2562 4385 y Fw(\))g(+)g Fr(R)2740 4397 y Fo(\030)2776 4385 y Fw([)p Fr(')2853 4397 y Fo(\030)2890 4385 y Fw(]\),)24 b(\(2.7\))o(,)g(\(2.20\))o(,)456 4485 y(\(2.21\))o(,)k(and)f(\(2.31\))g (w)n(e)g(ha)n(v)n(e)1376 4639 y Fq(j)p Fr(\031)1446 4651 y Fo(\030)1483 4639 y Fw(\()p Fr(G)1580 4651 y Fo(\030)1617 4639 y Fw(\))p Fq(j)c(\024)g Fr(C)1862 4571 y Fm(\000)1900 4639 y Fr(e)1939 4604 y Fp(\000)p Fs(2)p Fo(\013\030)2122 4639 y Fw(+)18 b Fq(k)p Fr(')2301 4651 y Fo(\030)2337 4639 y Fq(k)2379 4651 y Fp(1)2449 4571 y Fm(\001)2501 4639 y Fr(:)456 4789 y Fw(F)-7 b(rom)36 b(\(2.35\))o(,)i(\(2.36\),)g (and)e Fq(k)p Fr(@)1486 4801 y Fo(\030)1522 4789 y Fr(p)1564 4801 y Fo(\030)1600 4789 y Fq(k)1642 4801 y Fp(1)1750 4789 y Fq(\024)h Fw(const,)i(w)n(e)c(get)i(that)f(also)g Fq(j)p Fr(H)7 b Fw(\()p Fr(\030)t Fw(\))p Fq(j)38 b(\024)f Fr(C)6 b Fq(k)p Fr(')3273 4801 y Fo(\030)3309 4789 y Fq(k)3351 4801 y Fp(1)3421 4789 y Fw(.)456 4888 y(Then,)24 b(from)e(the)h(\014rst)g(b)r(ound)g(in)g(\(4.9\),)h(\(2.40\))o(,)g(and) f(the)g(previous)f(estimates,)h(w)n(e)g(conclude)456 4991 y(that)k Fq(k)p Fr(g)717 5003 y Fo(\030)753 4991 y Fq(k)795 5003 y Fp(1)888 4991 y Fq(\024)22 b Fr(C)6 b(e)1079 4961 y Fp(\000)12 b Fs(min)o Fp(f)p Fs(2)p Fo(\013)p Fs(;)p Fo(\016)1412 4969 y Fl(1)1444 4961 y Fp(g)p Fo(\030)1515 4991 y Fw(.)37 b(The)27 b(prop)r(osition)g(is)g(th)n(us)h(pro)n(v)n (ed.)665 b Fg(\003)555 5116 y Fw(T)-7 b(o)31 b(conclude)h(the)g(pro)r (of)f(of)g(Theorem)g(2.9)f(w)n(e)i(are)e(left)i(with)h(the)e(equation)g Fr(P)12 b Fw(\()p Fr(\030)t Fw(\))31 b(=)e(0,)456 5216 y(where)18 b Fr(P)12 b Fw(\()p Fr(\030)t Fw(\))23 b(=)g Fr(\031)1014 5228 y Fo(\030)1064 5216 y Fw(\()q Fr(f)9 b Fw(\()p Fr(n)1229 5228 y Fo(\030)1265 5216 y Fw(\))18 b(+)g Fr(R)1461 5228 y Fo(\030)1498 5216 y Fw([)p Fr(')1575 5228 y Fo(\030)1611 5216 y Fw(]\))q(.)34 b(W)-7 b(e)19 b(\014rst)f(observ)n(e)f(that)h(from)h(\(2.20\))o(,)h(\(2.21\))o(,)h (Lemma)p eop %%Page: 19 19 19 18 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(19)456 450 y Fw(2.4,)43 b(and)e(Prop)r(osition)f(4.2,)j (for)e(an)n(y)f Fr(\024)46 b(>)f Fw(0)c(there)f(are)h Fr(`)2449 462 y Fs(8)2531 450 y Fq(\025)k Fr(`)2676 462 y Fs(7)2754 450 y Fw(suc)n(h)c(that,)k(for)40 b(an)n(y)456 550 y Fr(h)22 b(<)h(e)653 520 y Fp(\000)p Fs(2)p Fo(\013`)809 528 y Fl(8)873 550 y Fw(and)k Fr(\030)h Fq(2)23 b Fw(\000)1228 562 y Fo(`)1256 570 y Fl(8)1288 562 y Fo(;\024)1351 550 y Fw(,)1433 735 y Fq(j)p Fr(P)12 b Fw(\()p Fr(\030)t Fw(\))19 b Fq(\000)f Fr(V)h Fw(\()p Fr(\030)t Fw(\))p Fq(j)24 b(\024)f Fr(C)6 b(e)2137 701 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)2372 709 y Fl(2)2404 701 y Fs(\))p Fo(\030)3232 735 y Fw(\(4.16\))456 910 y(\(recall)27 b Fr(\016)747 922 y Fs(2)807 910 y Fw(=)22 b(min)q Fq(f)p Fr(\016)1112 922 y Fs(0)1148 910 y Fw(;)14 b Fr(\016)1222 922 y Fs(1)1259 910 y Fw(;)g(2)p Fr(\013)p Fq(g)p Fw(\).)37 b(W)-7 b(e)28 b(then)g(ha)n(v)n(e:)1197 1092 y Fr(P)12 b Fw(\()p Fr(`)p Fw(\))83 b Fq(\024)f Fr(V)19 b Fw(\()p Fr(`)p Fw(\))g(+)f Fr(C)6 b(e)1963 1058 y Fp(\000)p Fs(2\()p Fo(\013)p Fs(+)p Fo(\016)2198 1066 y Fl(2)2230 1058 y Fs(\))p Fo(`)1444 1221 y Fw(=)82 b Fq(\000)1670 1154 y Fm(\000)1708 1221 y Fr(\026K)24 b Fq(\000)18 b Fr(C)6 b(e)2040 1187 y Fp(\000)p Fo(\016)2122 1195 y Fl(2)2154 1187 y Fo(`)2186 1154 y Fm(\001)2238 1221 y Fr(e)2277 1187 y Fp(\000)p Fs(2)p Fo(\013`)2455 1221 y Fw(+)18 b(2)p Fr(m)2653 1233 y Fo(\014)2697 1221 y Fr(\026h;)579 1370 y(P)658 1303 y Fm(\000)696 1370 y Fw(\(2)p Fr(\013)p Fw(\))855 1336 y Fp(\000)p Fs(1)944 1370 y Fq(j)c Fw(log)g Fr(h)p Fq(j)19 b Fw(+)f Fr(\024)1323 1303 y Fm(\001)1444 1370 y Fq(\025)82 b Fr(V)1672 1303 y Fm(\000)1710 1370 y Fw(\(2)p Fr(\013)p Fw(\))1869 1336 y Fp(\000)p Fs(1)1959 1370 y Fq(j)14 b Fw(log)g Fr(h)p Fq(j)k Fw(+)g Fr(\024)2337 1303 y Fm(\001)2393 1370 y Fq(\000)g Fr(C)6 b(e)2580 1336 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)2815 1344 y Fl(2)2848 1336 y Fs(\))p Fo(\024)2917 1370 y Fr(h)2965 1336 y Fs(1+\(2)p Fo(\013)p Fs(\))3177 1311 y Fj(\000)p Fl(1)3254 1336 y Fo(\016)3284 1344 y Fl(2)1444 1528 y Fw(=)1591 1436 y Fm(h)1630 1528 y Fr(\026)p Fw(\(2)p Fr(m)1827 1540 y Fo(\014)1890 1528 y Fq(\000)18 b Fr(K)6 b(e)2089 1494 y Fp(\000)p Fs(2)p Fo(\013\024)2260 1528 y Fw(\))18 b Fq(\000)g Fr(C)6 b(e)2497 1494 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)2732 1502 y Fl(2)2765 1494 y Fs(\))p Fo(\024)2834 1528 y Fr(h)2882 1494 y Fs(\(2)p Fo(\013)p Fs(\))3010 1469 y Fj(\000)p Fl(1)3087 1494 y Fo(\016)3117 1502 y Fl(2)3154 1436 y Fm(i)3207 1528 y Fr(h:)456 1733 y Fw(Let)27 b Fr(\024)652 1745 y Fs(0)712 1733 y Fr(>)c Fw(0)k(b)r(e)h(suc)n(h)g(that)f(2)p Fr(m)1464 1745 y Fo(\014)1527 1733 y Fq(\000)18 b Fr(K)6 b(e)1726 1703 y Fp(\000)p Fs(2)p Fo(\013\024)1893 1711 y Fl(0)1952 1733 y Fq(\025)22 b Fr(m)2112 1745 y Fo(\014)2185 1733 y Fw(and,)27 b(for)g(an)n(y)g Fr(\024)c Fq(\025)g Fr(\024)2860 1745 y Fs(0)2897 1733 y Fw(,)28 b(let)f Fr(`)3102 1745 y Fs(9)3167 1733 y Fw(so)g(large)456 1836 y(that)32 b Fr(\026K)26 b Fq(\000)21 b Fr(C)6 b(e)977 1806 y Fp(\000)p Fo(\016)1059 1814 y Fl(2)1091 1806 y Fo(`)1119 1814 y Fl(9)1185 1836 y Fq(\025)30 b Fr(\026K)q(=)p Fw(2.)48 b(Then,)33 b(for)e(an)n(y)g Fr(\024)f Fq(\025)f Fr(\024)2314 1848 y Fs(0)2383 1836 y Fw(and)i Fr(`)f Fq(\025)f Fr(`)2742 1848 y Fs(9)2811 1836 y Fw(there)i(is)h Fr(h)3163 1848 y Fo(\024;`)3285 1836 y Fw(suc)n(h)456 1936 y(that,)c(for)f(an)n(y)f Fr(h)d(<)g(h)1149 1948 y Fo(\024;`)1239 1936 y Fw(,)1212 2115 y Fr(P)12 b Fw(\()p Fr(`)p Fw(\))23 b Fr(<)g Fw(0)p Fr(;)179 b(P)1810 2048 y Fm(\000)1848 2115 y Fw(\(2)p Fr(\013)p Fw(\))2007 2080 y Fp(\000)p Fs(1)2097 2115 y Fq(j)14 b Fw(log)f Fr(h)p Fq(j)19 b Fw(+)f Fr(\024)2475 2048 y Fm(\001)2536 2115 y Fr(>)k Fw(0)p Fr(:)456 2290 y Fw(Since)i(the)h(function)g Fr(\030)i Fq(7!)c Fr(P)12 b Fw(\()p Fr(\030)t Fw(\))25 b(is)f(easily)f(seen)h(to)h(b)r(e)f(con)n (tin)n(uous)g(\(actually)f(di\013eren)n(tiable\))456 2389 y(it)28 b(follo)n(ws)e(it)i(has)f(at)g(least)g(one)g(zero)g(in)h (\000)1814 2401 y Fo(`;\024)1932 2389 y Fw(for)f(an)n(y)f Fr(h)i Fw(small)f(enough.)36 b(In)28 b(order)e(to)h(pro)n(v)n(e)456 2489 y(uniqueness)40 b(it)g(is)g(su\016cien)n(t)h(to)f(sho)n(w)f(the)h (function)h(is)f(strictly)g(monotone.)74 b(Recalling)456 2589 y Fr(G)521 2601 y Fo(\030)601 2542 y Fr(:)580 2589 y Fw(=)23 b Fq(\000)p Fw(\()p Fr(f)9 b Fw(\()p Fr(n)897 2601 y Fo(\030)933 2589 y Fw(\))19 b(+)f Fr(R)1130 2601 y Fo(\030)1166 2589 y Fw([)p Fr(')1243 2601 y Fo(\030)1280 2589 y Fw(]\),)28 b(using)g(\(4.13\))o(,)g(and)f Fr(v)2071 2559 y Fp(\003)2068 2612 y Fo(\030)2110 2589 y Fr(L)2167 2601 y Fo(\030)2226 2589 y Fw(=)22 b Fr(\025)2361 2601 y Fo(\030)2398 2589 y Fr(v)2441 2559 y Fp(\003)2438 2612 y Fo(\030)2507 2589 y Fw(w)n(e)27 b(ha)n(v)n(e:)611 2774 y Fr(P)676 2740 y Fp(0)699 2774 y Fw(\()p Fr(\030)t Fw(\))d(=)f Fr(\025)963 2786 y Fo(\030)1000 2774 y Fr(\031)1047 2786 y Fo(\030)1083 2774 y Fw(\()p Fr(@)1159 2786 y Fo(\030)1196 2774 y Fr(n)1246 2786 y Fo(\030)1282 2774 y Fw(\))c Fq(\000)f Fr(\031)1463 2786 y Fo(\030)1500 2774 y Fw(\()p Fr(K)1603 2786 y Fo(\030)1657 2774 y Fw(+)g Fr(Z)1797 2786 y Fo(\030)1833 2774 y Fr(J)27 b Fq(\003)18 b Fr(@)2010 2786 y Fo(\030)2046 2774 y Fr(')2100 2786 y Fo(\030)2137 2774 y Fw(\))g(+)g Fr(@)2314 2786 y Fo(\030)2351 2774 y Fr(\031)2398 2786 y Fo(\030)2435 2774 y Fw(\()p Fr(f)9 b Fw(\()p Fr(n)2599 2786 y Fo(\030)2635 2774 y Fw(\)\))19 b(+)f Fr(@)2845 2786 y Fo(\030)2881 2774 y Fr(\031)2928 2786 y Fo(\030)2965 2774 y Fw(\()p Fr(R)3060 2786 y Fo(\030)3097 2774 y Fw([)p Fr(')3174 2786 y Fo(\030)3210 2774 y Fw(]\))p Fr(:)456 2949 y Fw(F)-7 b(rom)40 b(\(2.31\))o(,)45 b(\(2.38\))o(,)f(\(4.14\),)g (\(4.15\))o(,)h(and)40 b(Prop)r(osition)g(4.2,)j(it)f(follo)n(ws)e (that,)k(for)d(all)456 3048 y Fr(h)22 b(<)h(e)653 3018 y Fp(\000)p Fs(2)p Fo(\013`)809 3026 y Fl(9)873 3048 y Fw(and)k Fr(\030)h Fq(2)23 b Fw(\000)1228 3060 y Fo(`)1256 3068 y Fl(9)1288 3060 y Fo(;\024)1351 3048 y Fw(,)960 3234 y Fq(j)p Fr(\031)1030 3246 y Fo(\030)1067 3234 y Fw(\()p Fr(K)1170 3246 y Fo(\030)1225 3234 y Fw(+)18 b Fr(Z)1365 3246 y Fo(\030)1401 3234 y Fr(J)26 b Fq(\003)18 b Fr(@)1577 3246 y Fo(\030)1614 3234 y Fr(')1668 3246 y Fo(\030)1704 3234 y Fw(\))p Fq(j)h Fw(+)f Fq(j)p Fr(@)1928 3246 y Fo(\030)1965 3234 y Fr(\031)2012 3246 y Fo(\030)2048 3234 y Fw(\()p Fr(R)2143 3246 y Fo(\030)2180 3234 y Fw([)p Fr(')2257 3246 y Fo(\030)2294 3234 y Fw(]\))p Fq(j)23 b(\024)g Fr(C)6 b(e)2587 3200 y Fp(\000)p Fs(2\()p Fo(\013)p Fs(+)p Fo(\016)2822 3208 y Fl(2)2854 3200 y Fs(\))p Fo(\030)2917 3234 y Fr(:)456 3409 y Fw(On)28 b(the)h(other)g(hand,)g(from)g (\(2.20\))o(,)g(\(2.21\))o(,)g(and)g(using)h(\(2.42\))o(,)f(w)n(e)g(ha) n(v)n(e)f Fq(j)p Fr(@)2947 3421 y Fo(\030)2983 3409 y Fr(\031)3030 3421 y Fo(\030)3067 3409 y Fw(\()p Fr(f)9 b Fw(\()p Fr(n)3231 3421 y Fo(\030)3267 3409 y Fw(\)\))p Fq(j)26 b(\024)456 3515 y Fr(C)6 b(e)560 3485 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)795 3493 y Fl(2)827 3485 y Fs(\))p Fo(\030)889 3515 y Fw(.)37 b(F)-7 b(rom)28 b(\(2.25\))e(and)i(\(2.41\))f(w)n(e)g(conclude)g(that)1424 3746 y Fr(P)1489 3712 y Fp(0)1513 3746 y Fw(\()p Fr(\030)t Fw(\))c Fq(\025)1738 3690 y Fr(e)1777 3659 y Fp(\000)p Fs(2)p Fo(\013\030)p 1738 3727 204 4 v 1743 3803 a Fr(c)1779 3815 y Fs(1)1816 3751 y Fq(p)p 1886 3751 51 4 v 1886 3803 a Fr(\026)1965 3679 y Fm(\000)2003 3746 y Fw(1)18 b Fq(\000)g Fr(C)6 b(e)2250 3712 y Fp(\000)p Fo(\016)2332 3720 y Fl(2)2364 3712 y Fo(\030)2401 3679 y Fm(\001)2453 3746 y Fr(:)456 3972 y Fw(Then,)30 b(for)g(an)n(y)f Fr(\024)d Fq(\025)h Fr(\024)1201 3984 y Fs(0)1238 3972 y Fw(,)j(there)g(is)g Fr(`)1627 3984 y Fs(5)1691 3972 y Fq(\025)c Fr(`)1817 3984 y Fs(9)1884 3972 y Fw(suc)n(h)j(that,)i(for)e(eac)n(h)g Fr(h)e Fq(\024)f Fr(h)2811 3984 y Fo(\024;`)2898 3992 y Fl(5)2934 3972 y Fw(,)31 b(the)f(equation)456 4073 y Fr(P)12 b Fw(\()p Fr(\030)t Fw(\))23 b(=)g(0)k(has)g(a)g(unique)h (solution)f(on)g(\000)1773 4085 y Fo(`)1801 4093 y Fl(5)1833 4085 y Fo(;\024)1896 4073 y Fw(.)37 b(Theorem)26 b(2.9)h(th)n(us)g (follo)n(ws)g(with)3092 4051 y(\026)3084 4073 y Fr(\030)32 b Fw(equal)27 b(to)456 4173 y(suc)n(h)i(a)h(solution,)43 b(\026)-55 b Fr(')27 b Fw(=)f Fr(')1290 4178 y Fs(\026)1283 4193 y Fo(\030)1320 4173 y Fw(,)k(and)g Fr(h)1585 4185 y Fs(0)1649 4173 y Fw(=)c Fr(h)1788 4185 y Fo(\024;`)1875 4193 y Fl(5)1911 4173 y Fw(.)44 b(W)-7 b(e)30 b(ha)n(v)n(e)f(only)g(to) h(c)n(hec)n(k)f(the)h(limits)g(\(2.53\))o(.)456 4272 y(By)d(our)g(c)n(hoice)f(of)i Fr(`)1110 4284 y Fs(9)1175 4272 y Fw(and)f(since)g Fr(`)1574 4284 y Fs(5)1634 4272 y Fq(\025)c Fr(`)1757 4284 y Fs(9)1794 4272 y Fw(,)905 4485 y Fr(P)12 b Fw(\()p Fr(\030)t Fw(\))23 b Fq(\024)g(\000)1260 4429 y Fr(\026K)p 1260 4466 127 4 v 1302 4542 a Fw(2)1396 4485 y Fr(e)1435 4451 y Fp(\000)p Fs(2)p Fo(\013\030)1617 4485 y Fw(+)c(2)p Fr(m)1816 4497 y Fo(\014)1860 4485 y Fr(\026h)166 b Fq(8)14 b Fr(\030)26 b Fq(2)d Fw(\000)2377 4497 y Fo(`)2405 4505 y Fl(5)2437 4497 y Fo(;\024)2583 4485 y Fq(8)14 b Fr(h)22 b Fq(\024)g Fr(h)2849 4497 y Fo(\024;`)2936 4505 y Fl(5)2972 4485 y Fr(;)456 4695 y Fw(whic)n(h)27 b(implies)983 4673 y(\026)975 4695 y Fr(\030)g Fq(!)c Fw(+)p Fq(1)28 b Fw(as)e Fr(h)d Fq(#)g Fw(0.)36 b(On)28 b(the)g(other)f(hand,)g(b)n(y)i(\(4.16\))o(,)1166 4812 y Fm(\014)1166 4862 y(\014)1193 4882 y Fr(e)1232 4848 y Fs(2)p Fo(\013)1315 4833 y Fs(\026)1308 4848 y Fo(\030)1344 4882 y Fr(V)19 b Fw(\()1452 4861 y(\026)1443 4882 y Fr(\030)5 b Fw(\))1516 4812 y Fm(\014)1516 4862 y(\014)1567 4882 y Fw(=)1654 4812 y Fm(\014)1654 4862 y(\014)1682 4882 y Fw(2)p Fr(m)1797 4894 y Fo(\014)1841 4882 y Fr(\026he)1978 4848 y Fs(2)p Fo(\013)2061 4833 y Fs(\026)2054 4848 y Fo(\030)2109 4882 y Fq(\000)18 b Fr(\026K)2319 4812 y Fm(\014)2319 4862 y(\014)2369 4882 y Fq(\024)23 b Fr(C)6 b(e)2561 4848 y Fp(\000)p Fo(\016)2643 4856 y Fl(2)2682 4833 y Fs(\026)2675 4848 y Fo(\030)2711 4882 y Fr(:)456 5057 y Fw(Letting)32 b Fr(h)e Fq(#)h Fw(0)g(in)i(the)f(ab)r(o)n(v)n(e)f(inequalit)n(y)h(w)n(e) g(th)n(us)g(obtain)g(the)g(\014rst)g(limit)h(in)f(\(2.53\);)i(the)456 5157 y(second)27 b(one)g(then)h(follo)n(ws)e(b)n(y)i(the)g(\014rst)f(b) r(ound)h(in)g(\(4.9\).)1066 b Fg(\003)p eop %%Page: 20 20 20 19 bop 456 251 a Fs(20)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)1235 450 y Fw(5.)41 b Fv(Sp)-6 b(a)g(tial)32 b(pr)n(oper)-6 b(ties)33 b(of)f(the)g(bump)555 600 y Fw(In)26 b(this)g(section)g(w)n(e)f(pro)n(v)n(e)f(Prop)r(osition)g (2.10.)35 b(The)26 b(critical)f(droplet)g(is)h(uniquely)g(deter-)456 699 y(mined)i(b)n(y)f(its)h(restriction)e(to)i(the)g(semi-space)e Fn(R)2043 711 y Fs(+)2104 699 y Fw(,)i(whic)n(h)g(solv)n(es)1146 843 y Fr(q)s Fw(\()p Fr(x)p Fw(\))c(=)f(tanh)14 b Fq(f)o Fr(\014)k Fw([\()p Fr(J)1796 855 y Fs(+)1852 843 y Fr(q)s Fw(\)\()p Fr(x)p Fw(\))i(+)e Fr(h)p Fw(])p Fq(g)13 b Fr(;)180 b(x)23 b Fq(2)h Fn(R)2670 855 y Fs(+)2731 843 y Fr(;)520 b Fw(\(5.1\))456 986 y(where,)27 b(for)g(an)n(y)g Fr(u)22 b Fq(2)i Fr(C)6 b Fw(\()p Fn(R)1303 998 y Fs(+)1364 986 y Fw(\),)591 1187 y(\()p Fr(J)669 1199 y Fp(\006)725 1187 y Fr(u)p Fw(\)\()p Fr(x)p Fw(\))961 1140 y Fr(:)940 1187 y Fw(=)1028 1074 y Fm(Z)1111 1095 y Fs(+)p Fp(1)1074 1263 y Fs(0)1232 1187 y Fr(dy)17 b(J)1379 1199 y Fp(\006)1435 1187 y Fw(\()p Fr(x;)d(y)s Fw(\))p Fr(u)p Fw(\()p Fr(y)s Fw(\))p Fr(;)180 b(J)2032 1199 y Fp(\006)2088 1187 y Fw(\()p Fr(x;)14 b(y)s Fw(\))2325 1140 y Fr(:)2304 1187 y Fw(=)23 b Fr(J)8 b Fw(\()p Fr(x)19 b Fq(\000)f Fr(y)s Fw(\))g Fq(\006)g Fr(J)8 b Fw(\()p Fr(x)19 b Fw(+)f Fr(y)s Fw(\))p Fr(:)136 b Fw(\(5.2\))456 1377 y(Observ)n(e)38 b(that,)45 b(since)40 b Fr(J)48 b Fw(is)41 b(non)f(increasing)f(on)h Fn(R)2171 1389 y Fs(+)2233 1377 y Fw(,)j Fr(J)2345 1389 y Fp(\006)2401 1377 y Fw(\()p Fr(x;)14 b(y)s Fw(\))46 b Fq(\025)e Fw(0)c(for)g(all)g Fr(x;)14 b(y)48 b Fq(\025)c Fw(0;)456 1477 y(moreo)n(v)n(er,)27 b(since)j(sup)p Fq(f)p Fr(x)c Fq(2)h Fn(R)33 b Fw(:)26 b Fr(J)8 b Fw(\()p Fr(x)p Fw(\))28 b Fr(>)e Fw(0)p Fq(g)g Fw(=)g(1,)k(if)g Fr(x)d(>)f Fw(1)k(then)g Fr(J)2635 1489 y Fp(\006)2691 1477 y Fw(\()p Fr(x;)14 b(y)s Fw(\))27 b(=)f Fr(J)8 b Fw(\()p Fr(x)21 b Fq(\000)e Fr(y)s Fw(\))30 b(for)456 1576 y(all)d Fr(y)f Fq(\025)c Fw(0.)555 1676 y(T)-7 b(o)34 b(pro)n(v)n(e)f(the)h(prop)r (osition)f(w)n(e)h(will)h(exploit)f(the)g(fact)g(that)h(the)f(critical) g(droplet)g(is)g(a)456 1775 y(solution)e(of)40 b(\(5.1\))32 b(whic)n(h)h(is)g(close,)h(for)e Fr(h)h Fw(small,)i(to)d(a)h(suitable)g (re\015ected)g(instan)n(ton,)h(see)456 1875 y(\(2.12\))o(.)555 1975 y(Let)51 b(\026)-58 b Fr(m)784 1987 y Fo(\030)816 1971 y Fj(\003)890 1975 y Fw(b)r(e)35 b(as)f(in)h(Theorem)f(2.1.)57 b(Since)35 b Fr(\014)t Fw(\(1)23 b Fq(\000)g Fr(m)2301 1945 y Fs(2)2301 1998 y Fo(\014)2346 1975 y Fw(\))35 b Fr(<)f Fw(1,)i(from)f(\(1.10\))f(and)g(\(2.13\))456 2080 y(there)27 b(are)g Fr(\022)e Fq(2)e Fw(\(0)p Fr(;)14 b Fw(1\),)28 b Fr(h)1233 2092 y Fs(1)1293 2080 y Fq(2)23 b Fw(\(0)p Fr(;)14 b(h)1530 2092 y Fs(0)1567 2080 y Fw(],)28 b(and)f(a)g(p)r(ositiv)n(e)h(in)n(teger)e Fr(`)2488 2050 y Fp(\003)2549 2080 y Fq(2)d Fw(\(1)p Fr(;)14 b(\030)2778 2050 y Fp(\003)2835 2080 y Fq(\000)k Fw(1\))27 b(suc)n(h)h(that)887 2225 y Fr(\014)952 2157 y Fm(\000)990 2225 y Fw(1)18 b Fq(\000)34 b Fw(\026)-58 b Fr(m)1206 2237 y Fo(\030)1238 2220 y Fj(\003)1277 2225 y Fw(\()p Fr(x)p Fw(\))1388 2190 y Fs(2)1426 2157 y Fm(\001)1487 2225 y Fq(\024)23 b Fr(\022)168 b Fq(8)14 b(j)p Fr(x)k Fq(\000)g Fr(\030)2054 2190 y Fp(\003)2093 2225 y Fq(j)23 b(\025)f Fr(`)2261 2190 y Fp(\003)2318 2225 y Fq(\000)c Fw(1)p Fr(;)96 b Fq(8)14 b Fr(h)22 b Fq(2)h Fw(\(0)p Fr(;)14 b(h)2930 2237 y Fs(1)2967 2225 y Fw(])p Fr(:)261 b Fw(\(5.3\))456 2368 y(On)27 b(the)h(other)f(hand,)h(recalling)e(the)i(de\014nition)g (\(2.57\))o(,)g(\(2.12\))f(implies)1443 2512 y(lim)1448 2567 y Fo(h)p Fp(#)p Fs(0)1573 2442 y Fm(\015)1573 2492 y(\015)1619 2512 y Fr(p)18 b Fq(\000)g Fr(\014)t Fw(\(1)h Fq(\000)33 b Fw(\026)-57 b Fr(m)2062 2478 y Fs(2)2062 2533 y Fo(\030)2094 2516 y Fj(\003)2133 2512 y Fw(\))2165 2442 y Fm(\015)2165 2492 y(\015)2211 2548 y Fp(1)2304 2512 y Fw(=)23 b(0)p Fr(:)817 b Fw(\(5.4\))456 2701 y(F)-7 b(rom)30 b(\(5.3\))h(and)g(\(5.4\))g(it)g(follo)n(ws)f(there)h(are)f Fr(\016)i Fq(2)d Fw(\()p Fr(\022)r(;)14 b Fw(1\))31 b(and)g Fr(h)2547 2713 y Fs(2)2613 2701 y Fq(2)e Fw(\(0)p Fr(;)14 b(h)2856 2713 y Fs(1)2893 2701 y Fw(])31 b(suc)n(h)g(that,)h(for)456 2800 y Fr(`)491 2770 y Fp(\003)551 2800 y Fq(2)24 b Fw(\(1)p Fr(;)14 b(\030)781 2770 y Fp(\003)837 2800 y Fq(\000)k Fw(1\))28 b(as)f(b)r(efore,)1099 2943 y Fr(p)p Fw(\()p Fr(x)p Fw(\))d Fq(\024)f Fr(\016)169 b Fq(8)14 b(j)p Fr(x)k Fq(\000)g Fr(\030)1842 2909 y Fp(\003)1880 2943 y Fq(j)23 b(\025)g Fr(`)2049 2909 y Fp(\003)2105 2943 y Fq(\000)18 b Fw(1)p Fr(;)97 b Fq(8)14 b Fr(h)21 b Fq(2)j Fw(\(0)p Fr(;)14 b(h)2718 2955 y Fs(2)2755 2943 y Fw(])p Fr(:)473 b Fw(\(5.5\))456 3098 y Fz(Lemma)21 b(5.1.)33 b Fk(L)l(et)22 b Fr(h)h Fq(2)g Fw(\(0)p Fr(;)14 b(h)1413 3110 y Fs(2)1450 3098 y Fw(])p Fk(,)25 b Fr(h)1571 3110 y Fs(2)1631 3098 y Fk(and)e Fr(`)1820 3068 y Fp(\003)1881 3098 y Fk(b)l(e)g(as)g(in)29 b Fw(\(5.5\))p Fk(.)36 b(Then,)26 b(for)d(e)l(ach)h Fr(k)i Fq(2)d Fw([0)p Fr(;)14 b(\030)3239 3068 y Fp(\003)3281 3098 y Fq(\000)s Fr(`)3384 3068 y Fp(\003)3421 3098 y Fw(])456 3198 y Fk(and)30 b Fr(s)23 b Fq(\025)f Fr(\030)806 3167 y Fp(\003)863 3198 y Fw(+)c Fr(`)981 3167 y Fp(\003)1019 3198 y Fk(,)30 b(we)g(have:)1033 3399 y Fr(q)1073 3364 y Fp(0)1097 3399 y Fw(\()p Fr(x)p Fw(\))84 b(=)1439 3286 y Fm(Z)1522 3306 y Fo(k)q Fs(+1)1486 3474 y Fo(k)1647 3399 y Fr(dy)17 b(H)1817 3411 y Fo(k)1858 3399 y Fw(\()p Fr(x;)d(y)s Fw(\))p Fr(q)2090 3364 y Fp(0)2114 3399 y Fw(\()p Fr(y)s Fw(\))170 b Fq(8)14 b Fr(x)22 b Fq(2)i Fw([0)p Fr(;)14 b(k)s Fw(\))p Fr(;)470 b Fw(\(5.6\))1033 3625 y Fr(q)1073 3591 y Fp(0)1097 3625 y Fw(\()p Fr(x)p Fw(\))84 b(=)1439 3512 y Fm(Z)1522 3532 y Fo(s)1486 3701 y(s)p Fp(\000)p Fs(1)1606 3625 y Fr(dy)17 b(K)1778 3637 y Fo(s)1813 3625 y Fw(\()p Fr(x;)d(y)s Fw(\))p Fr(q)2045 3591 y Fp(0)2069 3625 y Fw(\()p Fr(y)s Fw(\))170 b Fq(8)14 b Fr(x)22 b Fq(2)i Fw(\()p Fr(s;)14 b Fw(+)p Fq(1)p Fw(\))p Fr(;)407 b Fw(\(5.7\))456 3822 y Fk(wher)l(e)30 b Fr(H)759 3834 y Fo(k)800 3822 y Fw(\()p Fr(x;)14 b(y)s Fw(\))p Fk(,)32 b Fr(x)24 b Fq(2)g Fw(\(0)p Fr(;)14 b(k)s Fw(\))p Fk(,)30 b(and)h Fr(K)1676 3834 y Fo(s)1711 3822 y Fw(\()p Fr(x;)14 b(y)s Fw(\))p Fk(,)31 b Fr(x)24 b(>)g(s)p Fk(,)30 b(ar)l(e)h(non)f(ne)l(gative)g(c)l(ontinuous)g(func-)456 3921 y(tions)f(of)i Fr(y)s Fk(,)f(strictly)g(p)l(ositive)h(for)g(some)f Fr(y)c Fq(2)d Fw([)p Fr(k)s(;)14 b(k)21 b Fw(+)d(1])p Fk(,)30 b Fr(y)c Fq(2)d Fw([)p Fr(s)18 b Fq(\000)g Fw(1)p Fr(;)c(s)p Fw(])29 b Fk(r)l(esp)l(e)l(ctively.)456 4098 y Fz(Pro)s(of.)78 b Fw(W)-7 b(e)39 b(start)e(with)i(the)f(pro)r(of)g (of)45 b(\(5.6\))o(.)69 b(W)-7 b(e)39 b(di\013eren)n(tiate)f(\(5.1\))f (at)i Fr(x)i Fq(2)g Fw([0)p Fr(;)14 b(k)s Fw(\),)456 4197 y(obtaining)27 b(\(recall)g(\(5.2\))o(\))844 4403 y Fr(q)884 4369 y Fp(0)908 4403 y Fw(\()p Fr(x)p Fw(\))d(=)e Fr(p)p Fw(\()p Fr(x)p Fw(\))1297 4290 y Fm(Z)1381 4311 y Fo(k)1344 4479 y Fs(0)1422 4403 y Fr(dy)16 b(J)1568 4415 y Fp(\000)1625 4403 y Fw(\()p Fr(x;)e(y)s Fw(\))p Fr(q)1857 4369 y Fp(0)1880 4403 y Fw(\()p Fr(y)s Fw(\))19 b(+)f Fr(p)p Fw(\()p Fr(x)p Fw(\))2257 4290 y Fm(Z)2341 4311 y Fo(k)q Fs(+1)2304 4479 y Fo(k)2466 4403 y Fr(dy)e(J)2612 4415 y Fp(\000)2669 4403 y Fw(\()p Fr(x;)e(y)s Fw(\))p Fr(q)2901 4369 y Fp(0)2925 4403 y Fw(\()p Fr(y)s Fw(\))p Fr(:)456 4588 y Fw(After)28 b Fr(N)36 b Fw(iteration)27 b(w)n(e)g(get)926 4789 y Fr(q)966 4755 y Fp(0)989 4789 y Fw(\()p Fr(x)p Fw(\))d(=)1212 4676 y Fm(Z)1295 4697 y Fo(k)q Fs(+1)1258 4865 y Fo(k)1420 4789 y Fr(dy)16 b(H)1596 4746 y Fs(\()p Fo(N)6 b Fs(\))1589 4814 y Fo(k)1711 4789 y Fw(\()p Fr(x;)14 b(y)s Fw(\))p Fr(q)1943 4755 y Fp(0)1967 4789 y Fw(\()p Fr(y)s Fw(\))19 b(+)2177 4676 y Fm(Z)2260 4697 y Fo(k)2223 4865 y Fs(0)2300 4789 y Fr(dy)e(D)2472 4746 y Fs(\()p Fo(N)6 b Fs(\))2470 4814 y Fo(k)2587 4789 y Fw(\()p Fr(x;)14 b(y)s Fw(\))p Fr(q)2819 4755 y Fp(0)2843 4789 y Fw(\()p Fr(y)s Fw(\))p Fr(;)300 b Fw(\(5.8\))456 4975 y(where)884 5144 y Fr(H)960 5100 y Fs(\()p Fo(N)6 b Fs(\))953 5169 y Fo(k)1075 5144 y Fw(\()p Fr(x;)14 b(y)s Fw(\))1311 5096 y Fr(:)1291 5144 y Fw(=)1411 5040 y Fo(N)1381 5065 y Fm(X)1378 5240 y Fo(n)p Fs(=1)1517 5144 y Fr(D)1588 5100 y Fs(\()p Fo(n)p Fs(\))1586 5169 y Fo(k)1685 5144 y Fw(\()p Fr(x;)g(y)s Fw(\))p Fr(;)181 b(D)2152 5100 y Fs(\(1\))2150 5169 y Fo(k)2241 5144 y Fw(\()p Fr(x;)14 b(y)s Fw(\))2477 5096 y Fr(:)2457 5144 y Fw(=)22 b Fr(p)p Fw(\()p Fr(x)p Fw(\))p Fr(J)2743 5156 y Fp(\000)2800 5144 y Fw(\()p Fr(x;)14 b(y)s Fw(\))p Fr(;)p eop %%Page: 21 21 21 20 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(21)456 450 y Fw(and,)27 b(for)g Fr(n)c(>)g Fw(1,)k(setting)h Fr(x)23 b Fw(=)g Fr(y)1492 462 y Fs(0)1556 450 y Fw(and)28 b Fr(y)d Fw(=)e Fr(y)1913 462 y Fo(n)1958 450 y Fw(,)896 657 y Fr(D)967 614 y Fs(\()p Fo(n)p Fs(\))965 682 y Fo(k)1064 657 y Fw(\()p Fr(y)1137 669 y Fs(0)1174 657 y Fr(;)14 b(y)1252 669 y Fo(n)1297 657 y Fw(\))23 b(=)1440 544 y Fm(Z)1523 565 y Fo(k)1486 733 y Fs(0)1564 657 y Fr(dy)1648 669 y Fs(1)1699 657 y Fq(\001)14 b(\001)g(\001)1809 544 y Fm(Z)1892 565 y Fo(k)1855 733 y Fs(0)1933 657 y Fr(dy)2017 669 y Fo(n)p Fp(\000)p Fs(1)2208 553 y Fo(n)2176 578 y Fm(Y)2175 755 y Fo(i)p Fs(=1)2296 657 y Fr(p)p Fw(\()p Fr(y)2411 669 y Fo(i)p Fp(\000)p Fs(1)2524 657 y Fw(\))p Fr(J)2602 669 y Fp(\000)2658 657 y Fw(\()p Fr(y)2731 669 y Fo(i)p Fp(\000)p Fs(1)2844 657 y Fr(;)g(y)2922 669 y Fo(i)2949 657 y Fw(\))p Fr(:)456 875 y Fw(The)27 b(assumptions)g(on)g Fr(J)36 b Fw(imply)950 1020 y(0)22 b Fq(\024)h Fr(J)1148 1032 y Fp(\000)1204 1020 y Fw(\()p Fr(x;)14 b(y)s Fw(\))24 b Fq(\024)e Fr(J)8 b Fw(\()p Fr(x)20 b Fq(\000)e Fr(y)s Fw(\))p Fr(;)97 b(J)1985 1032 y Fp(\000)2041 1020 y Fw(\(0)p Fr(;)14 b(y)s Fw(\))22 b Fq(\021)h Fw(0)166 b Fq(8)14 b Fr(x;)g(y)25 b Fq(2)e Fn(R)2889 1032 y Fs(+)3274 1020 y Fw(\(5.9\))456 1165 y(and)1046 1275 y(sup)p Fq(fj)p Fr(y)e Fq(\000)d Fr(x)p Fq(j)23 b(2)h Fn(R)1607 1287 y Fs(+)1705 1275 y Fw(:)37 b Fr(J)1811 1287 y Fp(\000)1867 1275 y Fw(\()p Fr(x;)14 b(y)s Fw(\))23 b Fr(>)g Fw(0)p Fq(g)f Fr(>)h Fw(0)165 b Fq(8)14 b Fr(x)23 b(>)f Fw(0)p Fr(:)378 b Fw(\(5.10\))456 1403 y(F)-7 b(rom)27 b(\(2.57\))o(,)h(\(5.5\))o(,)g(and)f(\(5.9\))h(w)n (e)f(get)1323 1564 y(0)22 b Fq(\024)h Fr(D)1546 1521 y Fs(\()p Fo(n)p Fs(\))1544 1589 y Fo(k)1643 1564 y Fw(\()p Fr(y)1716 1576 y Fs(0)1753 1564 y Fr(;)14 b(y)1831 1576 y Fo(n)1876 1564 y Fw(\))24 b Fq(\024)e Fr(\016)2059 1530 y Fo(n)p Fp(\000)p Fs(1)2189 1564 y Fr(J)2243 1530 y Fo(n)2289 1564 y Fw(\()p Fr(y)2362 1576 y Fs(0)2399 1564 y Fr(;)14 b(y)2477 1576 y Fo(n)2522 1564 y Fw(\))p Fr(:)655 b Fw(\(5.11\))456 1709 y(Since)25 b Fr(J)724 1679 y Fo(n)769 1709 y Fw(\()p Fr(y)842 1721 y Fs(0)879 1709 y Fr(;)14 b(y)957 1721 y Fo(n)1002 1709 y Fw(\))25 b(is)g(a)f(probabilit)n(y)g(densit)n(y)h(and)g Fq(k)p Fr(q)2153 1679 y Fp(0)2176 1709 y Fq(k)2218 1721 y Fp(1)2311 1709 y Fr(<)d Fq(1)p Fw(,)k(the)f(second)f(in)n(tegral)g(in)h(the)456 1809 y(r.h.s.)i(of)34 b(\(5.8\))27 b(v)-5 b(anishes)27 b(as)g Fr(N)32 b Fq(!)23 b Fw(+)p Fq(1)k Fw(and)h(w)n(e)f(obtain)g (\(5.6\))h(with)1482 2021 y Fr(H)1551 2033 y Fo(k)1592 2021 y Fw(\()p Fr(x;)14 b(y)s Fw(\))24 b(=)1925 1917 y Fp(1)1898 1942 y Fm(X)1895 2118 y Fo(n)p Fs(=1)2034 2021 y Fr(D)2105 1977 y Fs(\()p Fo(n)p Fs(\))2103 2046 y Fo(k)2202 2021 y Fw(\()p Fr(x;)14 b(y)s Fw(\))p Fr(;)815 b Fw(\(5.12\))456 2242 y(and)33 b(the)i(series)d(con)n(v)n(erges)g(exp) r(onen)n(tially)h(fast.)56 b(Clearly)33 b Fr(H)2473 2254 y Fo(k)2513 2242 y Fw(\()p Fr(x;)14 b Fq(\001)p Fw(\))35 b(is)f(non)g(negativ)n(e)f(and)456 2341 y(con)n(tin)n(uous.)i(Moreo)n (v)n(er,)25 b(from)j(\(5.10\))o(,)g(it)g(is)f(strictly)g(p)r(ositiv)n (e)h(for)f(some)g Fr(y)e Fq(2)f Fw([)p Fr(k)s(;)14 b(k)21 b Fw(+)d(1].)555 2441 y(The)28 b(case)f Fr(x)c(>)g(s)k Fw(can)h(b)r(e)g(treated)f(in)h(the)g(same)f(manner,)g(getting)926 2648 y Fr(K)997 2660 y Fo(s)1032 2648 y Fw(\()p Fr(x;)14 b(y)s Fw(\))23 b(=)1364 2544 y Fp(1)1338 2569 y Fm(X)1335 2745 y Fo(n)p Fs(=1)1474 2648 y Fr(R)1538 2614 y Fs(\()p Fo(n)p Fs(\))1537 2669 y Fo(s)1635 2648 y Fw(\()p Fr(x;)14 b(y)s Fw(\))p Fr(;)180 b(R)2094 2614 y Fs(\(1\))2093 2669 y Fo(s)2183 2648 y Fw(\()p Fr(x;)14 b(y)s Fw(\))2419 2601 y Fr(:)2399 2648 y Fw(=)22 b Fr(p)p Fw(\()p Fr(x)p Fw(\))p Fr(J)8 b Fw(\()p Fr(x)20 b Fq(\000)e Fr(y)s Fw(\))p Fr(;)258 b Fw(\(5.13\))456 2865 y(where,)27 b(for)g Fr(n)c(>)f Fw(1,)28 b(setting)f Fr(x)d Fw(=)e Fr(y)1570 2877 y Fs(0)1635 2865 y Fw(and)27 b Fr(y)f Fw(=)d Fr(y)1992 2877 y Fo(n)2037 2865 y Fw(,)705 3072 y Fr(R)769 3038 y Fs(\()p Fo(n)p Fs(\))768 3092 y Fo(s)865 3072 y Fw(\()p Fr(y)938 3084 y Fs(0)976 3072 y Fr(;)14 b(y)1054 3084 y Fo(n)1098 3072 y Fw(\))24 b(=)1241 2959 y Fm(Z)1324 2979 y Fs(+)p Fp(1)1287 3148 y Fo(s)1446 3072 y Fr(dy)1530 3084 y Fs(1)1581 3072 y Fq(\001)14 b(\001)g(\001)1691 2959 y Fm(Z)1774 2979 y Fs(+)p Fp(1)1737 3148 y Fo(s)1896 3072 y Fr(dy)1980 3084 y Fo(n)p Fp(\000)p Fs(1)2171 2968 y Fo(n)2138 2993 y Fm(Y)2137 3170 y Fo(i)p Fs(=1)2259 3072 y Fr(p)p Fw(\()p Fr(y)2374 3084 y Fo(i)p Fp(\000)p Fs(1)2486 3072 y Fw(\))p Fr(J)8 b Fw(\()p Fr(y)2645 3084 y Fo(i)p Fp(\000)p Fs(1)2777 3072 y Fq(\000)18 b Fr(y)2901 3084 y Fo(i)2928 3072 y Fw(\))p Fr(:)249 b Fw(\(5.14\))456 3294 y(In)27 b(\(5.13\))g(w)n(e)g (used)h(that)g Fr(J)1337 3306 y Fp(\000)1393 3294 y Fw(\()p Fr(u;)14 b(v)s Fw(\))23 b(=)g Fr(J)8 b Fw(\()p Fr(u)18 b Fq(\000)g Fr(v)s Fw(\))28 b(for)f Fr(u)c(>)g(s)g(>)f Fw(1)27 b(and)h Fr(v)e Fq(\025)d Fw(0.)461 b Fg(\003)456 3417 y Fz(Pro)s(of)34 b(of)g(the)g(monotonicit)m(y)e(prop)s(ert)m(y)-8 b(.)43 b Fw(W)-7 b(e)30 b(\014rst)g(pro)n(v)n(e)e(that)h(there)h(is)f Fr(h)3080 3429 y Fs(3)3144 3417 y Fq(2)e Fw(\(0)p Fr(;)14 b(h)3385 3429 y Fs(2)3421 3417 y Fw(])456 3516 y(suc)n(h)27 b(that)1159 3626 y Fr(q)1199 3592 y Fp(0)1223 3626 y Fw(\()p Fr(x)p Fw(\))d Fr(<)e Fw(0)166 b Fq(8)14 b(j)p Fr(x)k Fq(\000)g Fr(\030)1925 3592 y Fp(\003)1963 3626 y Fq(j)23 b(\024)g Fr(`)2132 3592 y Fp(\003)2170 3626 y Fr(;)97 b Fq(8)14 b Fr(h)21 b Fq(2)j Fw(\(0)p Fr(;)14 b(h)2658 3638 y Fs(3)2695 3626 y Fw(])p Fr(:)491 b Fw(\(5.15\))456 3754 y(T)-7 b(o)21 b(pro)n(v)n(e)f(\(5.15\))g(w)n(e)h(di\013eren)n (tiate)h(\(5.1\))f(for)g Fq(j)p Fr(x)6 b Fq(\000)g Fr(\030)2095 3724 y Fp(\003)2134 3754 y Fq(j)23 b(\024)g Fr(`)2303 3724 y Fp(\003)2340 3754 y Fw(.)35 b(Recalling)21 b Fr(J)2799 3766 y Fs(+)2854 3754 y Fw(\()p Fr(x;)14 b(y)s Fw(\))24 b(=)f Fr(J)8 b Fw(\()p Fr(x)e Fq(\000)g Fr(y)s Fw(\))456 3854 y(when)27 b Fr(x)19 b Fw(+)f Fr(y)26 b(>)d Fw(1,)k(w)n(e)g(get,) 599 3999 y Fr(q)639 3965 y Fp(0)662 3999 y Fw(\()p Fr(x)p Fw(\))d(=)f Fr(p)p Fw(\()p Fr(x)p Fw(\)\()p Fr(J)k Fq(\003)18 b Fr(q)s Fw(\))1275 3965 y Fp(0)1299 3999 y Fw(\()p Fr(x)p Fw(\))24 b(=)f Fr(p)p Fw(\()p Fr(x)p Fw(\)\()p Fr(J)1761 3965 y Fp(0)1803 3999 y Fq(\003)18 b Fw(\()p Fr(q)k Fq(\000)34 b Fw(\026)-58 b Fr(m)2110 4011 y Fo(\030)2142 3995 y Fj(\003)2181 3999 y Fw(\)\)\()p Fr(x)p Fw(\))20 b(+)e Fr(p)p Fw(\()p Fr(x)p Fw(\)\()p Fr(J)28 b Fq(\003)33 b Fw(\026)-57 b Fr(m)2851 3965 y Fp(0)2851 4019 y Fo(\030)2883 4003 y Fj(\003)2922 3999 y Fw(\)\()p Fr(x)p Fw(\))p Fr(:)144 b Fw(\(5.16\))456 4147 y(Since)43 b(\026)-58 b Fr(m)745 4159 y Fo(\030)777 4143 y Fj(\003)844 4147 y Fw(is)27 b(strictly)h(decreasing)e(on)h Fn(R)1783 4159 y Fs(+)1844 4147 y Fw(,)h(from)f(\(2.12\))g(w)n(e)g(get)h(\(5.15\))o(.)555 4246 y(F)-7 b(rom)32 b(\(5.15\))g(and)g(Lemma)g(5.1)g(it)h(follo)n(ws)e Fr(q)2034 4216 y Fp(0)2058 4246 y Fw(\()p Fr(x)p Fw(\))h Fr(<)e Fw(0)i(for)g(all)h Fr(x)e(>)g Fw(0)h(and)g Fr(h)f Fq(2)g Fw(\(0)p Fr(;)14 b(h)3361 4258 y Fs(3)3398 4246 y Fw(],)456 4346 y(th)n(us)27 b(getting)h(the)g(monotonicit)n(y)e(prop) r(ert)n(y)h(of)g(the)h(bump.)1008 b Fg(\003)555 4468 y Fw(W)-7 b(e)33 b(will)f(pro)n(v)n(e)e(Prop)r(osition)h(2.10)f(with)j Fr(h)1965 4438 y Fp(\003)2033 4468 y Fw(=)d Fr(h)2176 4480 y Fs(3)2214 4468 y Fw(.)50 b(W)-7 b(e)32 b(are)f(th)n(us)h(left)h (with)g(the)f(pro)r(of)456 4568 y(of)39 b(\(2.55\))o(.)54 b(W)-7 b(e)34 b(follo)n(w)e(the)i(same)e(strategy)g(used)i(in)f([6,)h (Sect.)g(3].)53 b(In)34 b(fact)f(large)f(part)h(of)456 4668 y(that)27 b(strategy)e(can)i(b)r(e)g(adapted)g(to)g(our)f(con)n (text)g(without)h(mo)r(di\014cation.)37 b(W)-7 b(e)27 b(\014rst)g(need)g(a)456 4767 y(w)n(eak)n(er)e(result.)456 4923 y Fz(Lemma)k(5.2.)40 b Fk(Ther)l(e)31 b(ar)l(e)f Fr(\021)c(>)d Fw(0)29 b Fk(and)h Fr(c)23 b(>)g Fw(0)29 b Fk(such)h(that)1064 5070 y Fq(j)p Fr(q)1127 5036 y Fp(0)1151 5070 y Fw(\()p Fr(x)p Fw(\))p Fq(j)24 b(\024)f Fr(ce)1472 5036 y Fp(\000)p Fo(\021)r Fs(\()p Fo(x)p Fp(\000)p Fo(\030)1706 5044 y Fi(q)1738 5036 y Fs(\))1938 5070 y Fq(8)14 b Fr(x)22 b Fq(2)i Fn(R)2201 5082 y Fs(+)2262 5070 y Fr(;)99 b Fq(8)14 b Fr(h)21 b Fq(2)j Fw(\(0)p Fr(;)14 b(h)2752 5036 y Fp(\003)2790 5070 y Fw(])p Fr(;)396 b Fw(\(5.17\))456 5216 y Fk(wher)l(e)30 b Fr(\030)726 5228 y Fo(q)786 5216 y Fw(=)23 b Fr(\030)910 5228 y Fo(q)947 5216 y Fw(\()p Fr(h)p Fw(\))30 b Fk(is)g(the)g(\(unique\))f(p)l (ositive)i(zer)l(o)f(of)h(the)f(bump.)p eop %%Page: 22 22 22 21 bop 456 251 a Fs(22)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 454 y Fz(Pro)s(of.)42 b Fw(By)27 b(the)h(de\014nition)g (\(5.14\),)f Fr(R)1726 411 y Fs(\()p Fo(n)p Fs(\))1725 464 y Fo(s)1823 454 y Fw(\()p Fr(x;)14 b(y)s Fw(\))24 b(=)f(0)k(if)h Fr(x)c(>)f(n)18 b Fw(+)g Fr(s)28 b Fw(and)f Fr(y)f Fq(2)e Fw([)p Fr(s)18 b Fq(\000)g Fw(1)p Fr(;)c(s)p Fw(],)28 b(and)456 553 y(it)f(satis\014es)g(a)f(b)r(ound)i(analogous)d (to)i(\(5.11\))o(.)37 b(Then,)27 b(from)g(\(5.7\),)g(for)g(an)n(y)f Fr(x)d(>)g(s)g Fq(\025)g Fr(\030)3211 523 y Fp(\003)3266 553 y Fw(+)18 b Fr(`)3384 523 y Fp(\003)3421 553 y Fw(,)456 653 y(w)n(e)27 b(ha)n(v)n(e)955 798 y Fq(j)p Fr(q)1018 764 y Fp(0)1041 798 y Fw(\()p Fr(x)p Fw(\))p Fq(j)d(\024)f Fr(\014)t Fq(k)p Fr(q)1420 764 y Fp(0)1443 798 y Fq(k)1485 810 y Fp(1)1616 719 y Fm(X)1569 897 y Fo(n)p Fp(\025)p Fo(x)p Fp(\000)p Fo(s)1796 798 y Fr(\016)1836 764 y Fo(n)p Fp(\000)p Fs(1)1990 798 y Fq(\024)f Fr(\016)2117 764 y Fp(\000)p Fs(1)2206 798 y Fr(\014)t Fq(k)p Fr(q)2339 764 y Fp(0)2363 798 y Fq(k)2405 810 y Fp(1)2475 798 y Fr(e)2514 764 y Fp(\000)p Fs(\()p Fo(x)p Fp(\000)p Fo(s)p Fs(\))p Fp(j)10 b Fs(log)i Fo(\016)r Fp(j)2922 798 y Fr(:)287 b Fw(\(5.18\))456 1049 y(Let)24 b Fr(\030)637 1061 y Fo(q)699 1049 y Fw(b)r(e)g(the)h(\(unique\))g(zero)f(of)g Fr(q)s Fw(\()p Fr(x)p Fw(\))h(in)g Fn(R)1867 1061 y Fs(+)1928 1049 y Fw(.)36 b(By)25 b(\(2.12\))40 b(\026)-58 b Fr(m)2425 1061 y Fo(\030)2457 1045 y Fj(\003)2496 1049 y Fw(\()p Fr(\030)2564 1061 y Fo(q)2601 1049 y Fw(\))24 b(=)38 b(\026)-58 b Fr(m)p Fw(\()p Fr(\030)2889 1019 y Fp(\003)2940 1049 y Fq(\000)12 b Fr(\030)3053 1061 y Fo(q)3089 1049 y Fw(\))25 b(v)-5 b(anishes)456 1149 y(as)27 b Fr(h)22 b Fq(#)h Fw(0,)k(hence)1536 1294 y(lim)1541 1348 y Fo(h)p Fp(#)p Fs(0)1665 1294 y Fw([)p Fr(\030)1724 1306 y Fo(q)1761 1294 y Fw(\()p Fr(h)p Fw(\))19 b Fq(\000)f Fr(\030)2015 1260 y Fp(\003)2053 1294 y Fw(\()p Fr(h)p Fw(\)])24 b(=)e(0)p Fr(:)868 b Fw(\(5.19\))456 1501 y(In)27 b(particular)g(\(5.19\))g (implies)g(there)h(is)f Fr(`)c(<)f Fq(1)28 b Fw(suc)n(h)f(that)1359 1680 y Fr(\030)1395 1692 y Fo(q)1451 1680 y Fw(+)18 b Fr(`)23 b Fq(\025)f Fr(\030)1719 1646 y Fp(\003)1776 1680 y Fw(+)c Fr(`)1894 1646 y Fp(\003)2098 1680 y Fq(8)c Fr(h)22 b Fq(2)h Fw([0)p Fr(;)14 b(h)2457 1646 y Fp(\003)2494 1680 y Fw(])p Fr(:)692 b Fw(\(5.20\))456 1860 y(and)27 b(\(5.17\))g(follo)n(ws)f(from)i(\(5.18\))f(with)h Fr(s)23 b Fw(=)f Fr(\030)1940 1872 y Fo(q)1996 1860 y Fw(+)c Fr(`)p Fw(.)1243 b Fg(\003)555 2000 y Fw(F)-7 b(rom)27 b(\(5.7\),)h(w)n(e)f(ha)n(v)n(e,)f(for)i(eac)n(h)e Fr(s)d Fq(\025)g Fr(`)p Fw(,)939 2226 y Fr(q)979 2192 y Fp(0)1002 2226 y Fw(\()p Fr(x)c Fw(+)f Fr(\030)1219 2238 y Fo(q)1256 2226 y Fw(\))24 b(=)1399 2113 y Fm(Z)1482 2134 y Fo(s)1445 2302 y(s)p Fp(\000)p Fs(1)1566 2226 y Fr(dy)17 b(G)1732 2238 y Fo(s)1767 2226 y Fw(\()p Fr(x;)d(y)s Fw(\))p Fr(q)1999 2192 y Fp(0)2023 2226 y Fw(\()p Fr(y)22 b Fw(+)c Fr(\030)2237 2238 y Fo(q)2274 2226 y Fw(\))p Fr(;)180 b Fq(8)14 b Fr(h)22 b Fq(2)h Fw(\(0)p Fr(;)14 b(h)2877 2192 y Fp(\003)2915 2226 y Fw(])p Fr(;)271 b Fw(\(5.21\))456 2461 y(where,)27 b(setting)g Fr(p)1033 2473 y Fo(\030)1063 2481 y Fi(q)1100 2461 y Fw(\()p Fr(x)p Fw(\))1256 2414 y Fr(:)1236 2461 y Fw(=)22 b Fr(p)p Fw(\()p Fr(x)d Fw(+)f Fr(\030)1582 2473 y Fo(q)1619 2461 y Fw(\),)668 2716 y Fr(G)733 2728 y Fo(s)769 2716 y Fw(\()p Fr(x;)c(y)s Fw(\))1005 2669 y Fr(:)984 2716 y Fw(=)1101 2612 y Fp(1)1074 2637 y Fm(X)1072 2813 y Fo(n)p Fs(=1)1211 2603 y Fm(Z)1294 2623 y Fs(+)p Fp(1)1257 2791 y Fo(s)1415 2716 y Fr(dy)1499 2728 y Fs(1)1550 2716 y Fq(\001)g(\001)g(\001)1661 2603 y Fm(Z)1744 2623 y Fs(+)p Fp(1)1707 2791 y Fo(s)1865 2716 y Fr(dy)1949 2728 y Fo(n)p Fp(\000)p Fs(1)2140 2612 y Fo(n)2108 2637 y Fm(Y)2107 2814 y Fo(i)p Fs(=1)2228 2716 y Fr(p)2270 2728 y Fo(\030)2300 2736 y Fi(q)2337 2716 y Fw(\()p Fr(y)2410 2728 y Fo(i)p Fp(\000)p Fs(1)2523 2716 y Fw(\))p Fr(J)8 b Fw(\()p Fr(y)2682 2728 y Fo(i)p Fp(\000)p Fs(1)2813 2716 y Fq(\000)18 b Fr(y)2937 2728 y Fo(i)2965 2716 y Fw(\))p Fr(:)212 b Fw(\(5.22\))456 2972 y(W)-7 b(e)28 b(observ)n(e)d(that)j Fr(p)1116 2984 y Fo(\030)1146 2992 y Fi(q)1183 2972 y Fw(\()p Fr(x)p Fw(\))h(is)f(a)f(strictly)g (decreasing)f(function)i(of)g Fr(x)g Fw(for)f Fr(x)d(>)e Fw(0,)1070 3185 y Fr(p)1112 3197 y Fo(\030)1142 3205 y Fi(q)1179 3185 y Fw(\()p Fr(x)p Fw(\))i Fr(>)33 b Fw(inf)1402 3236 y Fo(x>)p Fs(0)1538 3185 y Fr(p)1580 3197 y Fo(\030)1610 3205 y Fi(q)1647 3185 y Fw(\()p Fr(x)p Fw(\))24 b(=)f Fr(p)1912 3197 y Fp(1)2005 3185 y Fw(=)g Fr(\014)2158 3093 y Fm(\020)2207 3185 y Fw(1)18 b Fq(\000)g Fw(\()p Fr(m)2455 3150 y Fp(\000)2455 3210 y Fo(\014)s(;h)2559 3185 y Fw(\))2591 3151 y Fs(2)2628 3093 y Fm(\021)2701 3185 y Fr(<)23 b Fw(1)401 b(\(5.23\))456 3400 y(and,)27 b(b)n(y)g(Lemma)h(5.2,)f(there)g(is)g Fr(c)1540 3370 y Fp(0)1587 3400 y Fr(>)22 b Fw(0)28 b(suc)n(h)f(that)1240 3580 y Fr(p)1282 3592 y Fo(\030)1312 3600 y Fi(q)1350 3580 y Fw(\()p Fr(x)p Fw(\))d Fq(\024)e Fr(p)1614 3592 y Fp(1)1703 3580 y Fw(+)c Fr(c)1822 3545 y Fp(0)1845 3580 y Fr(e)1884 3545 y Fp(\000)p Fo(\021)r(x)2014 3580 y Fr(;)180 b Fq(8)14 b Fr(h)22 b Fq(2)h Fw([0)p Fr(;)14 b(h)2576 3545 y Fp(\003)2613 3580 y Fw(])p Fr(:)573 b Fw(\(5.24\))555 3759 y(In)29 b([6)o(,)g(Thm.)f(3.1])g(the)g (asymptotics)g(of)44 b(\026)-58 b Fr(m)1932 3729 y Fp(0)1955 3759 y Fw(\()p Fr(x)p Fw(\))30 b(follo)n(ws)d(from)h(an)g(analogous)e (\(to)i(\(5.21\))o(\))456 3859 y(expression)35 b(for)52 b(\026)-58 b Fr(m)1074 3829 y Fp(0)1098 3859 y Fw(\()p Fr(x)p Fw(\),)40 b(where)c Fr(p)1563 3871 y Fo(\030)1593 3879 y Fi(q)1630 3859 y Fw(\()p Fr(x)p Fw(\))i(is)f(replaced)f(b)n(y)h Fr(p)2389 3871 y Fs(\026)-46 b Fo(m)2439 3859 y Fw(\()p Fr(x)p Fw(\))2610 3812 y Fr(:)2589 3859 y Fw(=)38 b Fr(\014)t Fw(\(1)25 b Fq(\000)40 b Fw(\026)-58 b Fr(m)p Fw(\()p Fr(x)p Fw(\))3115 3829 y Fs(2)3154 3859 y Fw(\))37 b(in)g(the)456 3958 y(de\014nition)k(of)h Fr(G)1012 3970 y Fo(s)1048 3958 y Fw(\()p Fr(x;)14 b(y)s Fw(\).)79 b(The)41 b(pro)r(of)g(do)r(es)g (not)h(dep)r(end)g(on)f(the)h(sp)r(eci\014c)g(form)f(of)g(the)456 4058 y(function)24 b Fr(p)832 4070 y Fs(\026)-46 b Fo(m)882 4058 y Fw(\()p Fr(x)p Fw(\),)26 b(but)f(only)f(on)g(the)h(monotonicit)n (y)e(prop)r(ert)n(y)g(and)h(the)h(analogous)d(of)30 b(\(5.23\))456 4158 y(and)k(\(5.24\))o(.)57 b(Then)35 b(a)f(result)g(as)g([6,)i(Thm.)f (3.1])e(holds)h(in)h(our)f(case.)56 b(W)-7 b(e)35 b(conclude)f(that)456 4257 y(there)27 b(exist)g Fr(M)32 b(>)23 b Fw(0)k(and)g Fr(\016)f Fq(2)e Fw(\(0)p Fr(;)14 b(\015)5 b Fw(\),)27 b Fr(\015)32 b Fw(as)27 b(in)h(\(2.54\))o(,)g(suc)n(h)f(that)683 4441 y(lim)630 4491 y Fo(x)p Fp(!)p Fs(+)p Fp(1)865 4441 y Fr(e)904 4407 y Fo(\015)t(x)984 4441 y Fr(q)1024 4407 y Fp(0)1047 4441 y Fw(\()p Fr(x)19 b Fw(+)f Fr(\030)1264 4453 y Fo(q)1301 4441 y Fw(\))24 b(=)e Fq(\000)p Fr(M)t(;)233 b Fw(lim)1797 4491 y Fo(x)p Fp(!)p Fs(+)p Fp(1)2032 4441 y Fr(e)2071 4407 y Fo(\016)r(x)2158 4441 y Fw(\()p Fr(e)2229 4407 y Fo(\015)t(x)2309 4441 y Fr(q)2349 4407 y Fp(0)2373 4441 y Fw(\()p Fr(x)19 b Fw(+)f Fr(\030)2590 4453 y Fo(q)2627 4441 y Fw(\))h(+)f Fr(M)9 b Fw(\))23 b(=)f(0)p Fr(:)174 b Fw(\(5.25\))456 4657 y(As)40 b(in)i([6)o(])f(the)g(constan)n(t)f Fr(M)50 b Fw(is)40 b(non)h(zero)f(b)r(ecause)g(of)h(the)g(monotonicit)n (y)f(prop)r(ert)n(y)g(of)456 4756 y Fr(q)496 4726 y Fp(0)519 4756 y Fw(\()p Fr(x)p Fw(\).)49 b(Moreo)n(v)n(er,)30 b(since)h(0)e Fr(<)g(p)1509 4768 y Fp(1)1608 4756 y Fr(<)g Fw(1)i(and)g(\(5.24\))f(holds)h(uniformly)h(in)f Fr(h)p Fw(,)h(the)g(constan)n(t)456 4856 y Fr(M)f Fw(=)23 b Fr(M)9 b Fw(\()p Fr(h)p Fw(\))28 b(app)r(earing)e(in)i(\(5.25\))f (remains)f(b)r(ounded)i(a)n(w)n(a)n(y)e(from)h(0)g(as)g Fr(h)c Fq(#)g Fw(0.)555 4956 y(Analogously)k(w)n(e)h(obtain)f(\(2.55\)) h(\(with)h Fr(A)24 b Fw(=)g Fr(M)9 b(\015)2184 4925 y Fp(\000)p Fs(1)2272 4956 y Fw(\))28 b(from)g(\(5.25\))g(b)n(y)g (arguing)e(exactly)456 5055 y(as)f(in)h(the)g(pro)r(ofs)f(of)h([6)o(,)g (Thms.)37 b(3.2,)25 b(3.3])g(where)g(\(2.9\))h(follo)n(ws)f(as)g(a)g (corollary)e(of)j([6,)g(Thm.)456 5155 y(3.1].)36 b(W)-7 b(e)28 b(omit)f(the)h(details.)1997 b Fg(\003)p eop %%Page: 23 23 23 22 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(23)1337 450 y Fw(6.)41 b Fv(The)32 b(inv)-7 b(ariant)30 b(manif)n(old)i Fq(W)555 600 y Fw(In)g(this)g(section)f(w)n (e)g(pro)n(v)n(e)f(Theorem)h(2.12,)g(i.e.)h(the)g(existence)f(of)h(a)f (one)g(dimensional,)456 699 y(in)n(v)-5 b(arian)n(t,)40 b(expanding)f(manifold)f Fq(W)46 b Fw(in)39 b Fr(C)1907 669 y Fs(sym)2028 699 y Fw(\()p Fn(R)p Fw(;)14 b([)p Fq(\000)p Fw(1)p Fr(;)g Fw(1]\))44 b(consisting)38 b(of)h(t)n(w)n(o)f (branc)n(hes)456 799 y(that)27 b(originate)g(from)g(the)h(bump)g Fr(q)s Fw(.)555 898 y(F)-7 b(or)28 b Fr(h)d Fq(2)g Fw(\(0)p Fr(;)14 b(h)1017 868 y Fp(\003)1054 898 y Fw(])29 b(\()p Fr(h)1186 868 y Fp(\003)1253 898 y Fw(as)f(in)g(Prop)r(osition)f (2.10\))h(let)h Fr(L)p Fw(,)f Fr(\025)p Fw(,)h Fr(v)j Fw(b)r(e)d(as)f(in)h(Prop)r(osition)e(2.11.)456 998 y(W)-7 b(e)27 b(next)g(deriv)n(e)e(some)i(prop)r(erties)e(of)i(the)g(ev)n (olution)f Fr(S)2270 1010 y Fo(t)2299 998 y Fw(\()p Fr(q)20 b Fw(+)d Fr(u)2518 1010 y Fs(0)2554 998 y Fw(\))28 b(starting)d(from)i (an)f(initial)456 1098 y(datum)35 b Fr(q)26 b Fw(+)d Fr(u)925 1110 y Fs(0)996 1098 y Fw(with)35 b Fr(u)1240 1110 y Fs(0)1312 1098 y Fw(small.)58 b(W)-7 b(e)35 b(set)f Fr(u)1917 1110 y Fo(t)2002 1051 y Fr(:)1981 1098 y Fw(=)g Fr(S)2131 1110 y Fo(t)2160 1098 y Fw(\()p Fr(q)27 b Fw(+)c Fr(u)2392 1110 y Fs(0)2428 1098 y Fw(\))h Fq(\000)f Fr(q)s Fw(.)58 b(Since)35 b Fr(S)2968 1110 y Fo(t)2997 1098 y Fw(\()p Fr(q)s Fw(\))g(=)g Fr(q)j Fw(and)456 1197 y Fr(f)9 b Fw(\()p Fr(q)s Fw(\))23 b(=)f(0)28 b(w)n(e)f(ha)n(v)n(e)1264 1291 y Fr(du)1355 1303 y Fo(t)p 1264 1328 120 4 v 1288 1404 a Fr(dt)1417 1347 y Fw(=)c Fr(Lu)1610 1359 y Fo(t)1657 1347 y Fw(+)18 b([)p Fr(f)9 b Fw(\()p Fr(q)21 b Fw(+)d Fr(u)2034 1359 y Fo(t)2063 1347 y Fw(\))h Fq(\000)f Fr(f)9 b Fw(\()p Fr(q)s Fw(\))18 b Fq(\000)g Fr(Lu)2557 1359 y Fo(t)2586 1347 y Fw(])c Fr(;)628 b Fw(\(6.1\))456 1493 y(whic)n(h)27 b(implies)1012 1688 y Fr(u)1060 1700 y Fo(t)1112 1688 y Fw(=)22 b Fr(e)1238 1653 y Fo(Lt)1313 1688 y Fr(u)1361 1700 y Fs(0)1416 1688 y Fw(+)1499 1575 y Fm(Z)1582 1595 y Fo(t)1545 1763 y Fs(0)1611 1688 y Fr(ds)14 b(e)1746 1653 y Fo(L)p Fs(\()p Fo(t)p Fp(\000)p Fo(s)p Fs(\))1970 1688 y Fw([)p Fr(f)9 b Fw(\()p Fr(q)21 b Fw(+)d Fr(u)2264 1700 y Fo(s)2299 1688 y Fw(\))h Fq(\000)f Fr(f)9 b Fw(\()p Fr(q)s Fw(\))19 b Fq(\000)f Fr(Lu)2794 1700 y Fo(s)2828 1688 y Fw(])c Fr(:)386 b Fw(\(6.2\))456 1875 y(Then)27 b(b)n(y)i(\(2.7\))e(and)h(\(2.64\))e(\(with)j Fr(\020)g Fw(=)23 b(0\))1215 2003 y Fm(\015)1215 2053 y(\015)1261 2074 y Fr(u)1309 2086 y Fo(t)1356 2074 y Fq(\000)18 b Fr(e)1478 2039 y Fo(Lt)1553 2074 y Fr(u)1601 2086 y Fs(0)1637 2003 y Fm(\015)1637 2053 y(\015)1684 2107 y Fp(1)1777 2074 y Fq(\024)23 b Fr(C)1924 2086 y Fs(2)1975 1961 y Fm(Z)2058 1981 y Fo(t)2021 2149 y Fs(0)2087 2074 y Fr(ds)14 b(e)2222 2039 y Fo(\025)p 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Fs(0)1225 2886 y Fw(\))p Fk(,)1351 2977 y Fm(\015)1351 3027 y(\015)1397 3048 y Fr(u)1445 3060 y Fo(t)1492 3048 y Fq(\000)18 b Fr(e)1614 3013 y Fo(Lt)1689 3048 y Fr(u)1737 3060 y Fs(0)1774 2977 y Fm(\015)1774 3027 y(\015)1820 3081 y Fp(1)1913 3048 y Fq(\024)23 b Fr(N)2090 2980 y Fm(\000)2128 3048 y Fr(e)2167 3013 y Fo(\025t)2236 3048 y Fq(k)p Fr(u)2326 3060 y Fs(0)2362 3048 y Fq(k)2404 3060 y Fp(1)2474 2980 y Fm(\001)2512 2994 y Fs(2)3274 3048 y Fw(\(6.6\))456 3189 y Fk(and)1443 3295 y Fq(k)p Fr(u)1533 3307 y Fo(t)1562 3295 y Fq(k)1603 3320 y Fp(1)1696 3295 y Fq(\024)g Fw(\(1)18 b(+)g Fr(C)2018 3307 y Fs(1)2056 3295 y Fw(\))p Fr(e)2127 3260 y Fo(\025t)2196 3295 y Fq(k)p Fr(u)2286 3307 y Fs(0)2322 3295 y Fq(k)2364 3307 y Fp(1)2434 3295 y Fr(:)817 b Fw(\(6.7\))456 3418 y Fk(with)30 b Fr(C)695 3430 y Fs(1)762 3418 y Fk(as)g(in)37 b Fw(\(2.64\))o Fk(.)456 3592 y Fz(Pro)s(of.)j Fw(The)24 b(lemma)f(will)h(follo)n(w)f (with)h Fr(N)1878 3545 y(:)1857 3592 y Fw(=)f(4)p Fr(C)2046 3604 y Fs(2)2083 3592 y Fr(\025)2131 3562 y Fp(\000)p Fs(1)2221 3592 y Fw(.)35 b(W)-7 b(e)24 b(pro)n(v)n(e)e(\(6.6\))h(b)n(y) h(con)n(tradiction.)456 3692 y(Fix)32 b Fr(\034)39 b(<)30 b(\033)s Fw(\()p Fr(u)909 3704 y Fs(0)946 3692 y Fw(\))j(and)e (de\014ne)h Fr(\032)1463 3704 y Fo(\034)1555 3645 y Fr(:)1535 3692 y Fw(=)d Fr(e)1668 3662 y Fo(\025\034)1749 3692 y Fq(k)p Fr(u)1839 3704 y Fs(0)1875 3692 y Fq(k)1917 3704 y Fp(1)1987 3692 y Fw(.)50 b(Let)32 b Fr(T)41 b Fq(\024)29 b Fr(\034)42 b Fw(b)r(e)32 b(the)g(\014rst)g(time)g(when)g (the)456 3791 y(inequalit)n(y)i(\(6.6\))e(b)r(ecomes)h(an)g(equalit)n (y)-7 b(.)53 b(Then,)35 b(b)n(y)f(\(6.3\))f(with)h Fr(t)e Fw(=)g Fr(T)45 b Fw(\(and)33 b(supp)r(osing)456 3891 y(without)28 b(loss)e(of)i(generalit)n(y)e(that)i Fq(k)p Fr(u)1670 3903 y Fs(0)1706 3891 y Fq(k)1748 3903 y Fp(1)1841 3891 y Fr(>)23 b Fw(0\),)528 4094 y Fr(N)618 4026 y Fm(\000)656 4094 y Fr(e)695 4059 y Fo(\025T)786 4094 y Fq(k)p Fr(u)876 4106 y Fs(0)913 4094 y Fq(k)955 4106 y Fp(1)1025 4026 y Fm(\001)1063 4040 y Fs(2)1183 4094 y Fq(\024)82 b Fr(C)1389 4106 y Fs(2)1441 3981 y Fm(Z)1524 4001 y Fo(T)1487 4169 y Fs(0)1576 4094 y Fr(ds)14 b(e)1711 4059 y Fo(\025)p Fs(\()p Fo(T)9 b Fp(\000)p Fo(s)p Fs(\))1951 4001 y Fm(h)1991 4094 y Fr(e)2030 4059 y Fo(\025s)2104 4094 y Fq(k)p Fr(u)2194 4106 y Fs(0)2230 4094 y Fq(k)2272 4106 y Fp(1)2361 4094 y Fw(+)18 b Fr(N)2533 4026 y Fm(\000)2571 4094 y Fr(e)2610 4059 y Fo(\025s)2685 4094 y Fq(k)p Fr(u)2775 4106 y Fs(0)2811 4094 y Fq(k)2853 4106 y Fp(1)2923 4026 y Fm(\001)2961 4040 y Fs(2)2998 4001 y Fm(i)3038 4019 y Fs(2)1183 4335 y Fq(\024)82 b Fr(C)1389 4347 y Fs(2)1441 4335 y Fr(e)1480 4300 y Fo(\025T)1571 4335 y Fq(k)p Fr(u)1661 4347 y Fs(0)1698 4335 y Fq(k)1740 4347 y Fp(1)1823 4222 y Fm(Z)1906 4242 y Fo(T)1869 4410 y Fs(0)1959 4335 y Fr(ds)14 b(e)2094 4300 y Fo(\025s)2168 4335 y Fq(k)p Fr(u)2258 4347 y Fs(0)2294 4335 y Fq(k)2336 4347 y Fp(1)2420 4335 y Fw(\(1)k(+)g Fr(N)9 b(\032)2714 4347 y Fo(\034)2755 4335 y Fw(\))2788 4293 y Fs(2)1183 4535 y Fq(\024)82 b Fr(C)1389 4547 y Fs(2)1441 4535 y Fw(\(1)18 b(+)g Fr(N)9 b(\032)1735 4547 y Fo(\034)1776 4535 y Fw(\))1809 4493 y Fs(2)1860 4535 y Fr(\025)1908 4501 y Fp(\000)p Fs(1)2011 4468 y Fm(\000)2049 4535 y Fr(e)2088 4501 y Fo(\025T)2179 4535 y Fq(k)p Fr(u)2269 4547 y Fs(0)2306 4535 y Fq(k)2348 4547 y Fp(1)2418 4468 y Fm(\001)2456 4481 y Fs(2)2516 4535 y Fr(<)23 b(N)2693 4468 y Fm(\000)2731 4535 y Fr(e)2770 4501 y Fo(\025T)2862 4535 y Fq(k)p Fr(u)2952 4547 y Fs(0)2988 4535 y Fq(k)3030 4547 y Fp(1)3100 4468 y Fm(\001)3138 4481 y Fs(2)3189 4535 y Fr(;)62 b Fw(\(6.8\))456 4676 y(where)20 b(in)h(the)g(last)g (inequalit)n(y)f(w)n(e)g(used)h Fr(N)9 b(\032)1861 4688 y Fo(\034)1925 4676 y Fr(<)23 b Fw(1.)34 b(W)-7 b(e)21 b(ha)n(v)n(e)f(th)n(us)h(reac)n(hed)e(a)i(con)n(tradiction)456 4776 y(and)27 b(\(6.6\))g(is)h(pro)n(v)n(ed)e(for)h(all)g Fr(t)c Fq(\024)g Fr(\034)9 b Fw(.)38 b(Hence,)27 b(b)n(y)i(\(2.64\))o (,)949 4938 y Fq(k)p Fr(u)1039 4950 y Fo(t)1067 4938 y Fq(k)1109 4950 y Fp(1)1262 4938 y Fq(\024)82 b Fr(C)1468 4950 y Fs(1)1506 4938 y Fr(e)1545 4903 y Fo(\025t)1613 4938 y Fq(k)p Fr(u)1703 4950 y Fs(0)1739 4938 y Fq(k)1781 4950 y Fp(1)1870 4938 y Fw(+)18 b Fr(N)2042 4870 y Fm(\000)2080 4938 y Fr(e)2119 4903 y Fo(\025t)2188 4938 y Fq(k)p Fr(u)2278 4950 y Fs(0)2314 4938 y Fq(k)2356 4950 y Fp(1)2426 4870 y Fm(\001)2464 4884 y Fs(2)1262 5075 y Fq(\024)82 b Fw(\()q Fr(C)1501 5087 y Fs(1)1557 5075 y Fw(+)18 b Fr(N)9 b(\032)1759 5087 y Fo(\034)1800 5075 y Fw(\))14 b Fr(e)1885 5040 y Fo(\025t)1953 5075 y Fq(k)p Fr(u)2043 5087 y Fs(0)2079 5075 y Fq(k)2121 5087 y Fp(1)2214 5075 y Fq(\024)23 b Fw(\(1)18 b(+)g Fr(C)2536 5087 y Fs(1)2574 5075 y Fw(\))p Fr(e)2645 5040 y Fo(\025t)2713 5075 y Fq(k)p Fr(u)2803 5087 y Fs(0)2840 5075 y Fq(k)2882 5087 y Fp(1)3274 5075 y Fw(\(6.9\))456 5216 y(for)27 b(all)g Fr(t)c Fq(\024)g Fr(\034)9 b Fw(,)28 b(and)g(Lemma)f(6.1)g(is)g(pro)n(v)n(ed.)1503 b Fg(\003)p eop %%Page: 24 24 24 23 bop 456 251 a Fs(24)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)555 450 y Fw(W)e(e)25 b(use)g(in)f(the)h(sequel)f(the)h(follo)n (wing)f(notation.)35 b(F)-7 b(or)24 b Fr(v)s Fw(,)i Fr(N)33 b Fw(as)24 b(in)h(Prop)r(osition)d(2.11)i(and)456 550 y(Lemma)j(6.1)g(w)n(e)g(denote)g(b)n(y)h Fr(\032)f Fw(an)n(y)g(p)r (ositiv)n(e)g(n)n(um)n(b)r(er)h(suc)n(h)f(that)1716 688 y Fr(N)9 b(\032)p Fq(k)p Fr(v)s Fq(k)1962 700 y Fp(1)2055 688 y Fr(<)22 b Fw(1)1048 b(\(6.10\))456 826 y(and)27 b(de\014ne)1445 929 y Fr( )1499 941 y Fp(\006)p Fo(")1631 882 y Fr(:)1610 929 y Fw(=)c Fr(q)e Fq(\006)d Fr("v)s(;)180 b(")23 b Fq(2)g Fw([0)p Fr(;)14 b(\032)p Fw(])p Fr(;)777 b Fw(\(6.11\))1241 1092 y Fr(\034)9 b Fw(\()p Fr(\032;)14 b(")p Fw(\))1513 1045 y Fr(:)1492 1092 y Fw(=)1593 1036 y(1)p 1590 1073 49 4 v 1590 1149 a Fr(\025)1662 1092 y Fw(log)1793 1036 y Fr(\032)p 1793 1073 43 4 v 1795 1149 a(")1846 1092 y(;)180 b Fw(i.e.)51 b Fr(e)2245 1058 y Fo(\025\034)7 b Fs(\()p Fo(\032;")p Fs(\))2486 1092 y Fw(=)2584 1036 y Fr(\032)p 2584 1073 V 2586 1149 a(")2636 1092 y(:)573 b Fw(\(6.12\))456 1241 y(W)-7 b(e)40 b(observ)n(e)e(that)i Fq(\006)p Fr("v)s Fw(,)i Fr(")h Fq(2)g Fw([0)p Fr(;)14 b(\032)p Fw(])39 b(satisfy)h(the)g(h)n(yp)r(othesis)f(of)g(Lemma)h(6.1) f(and)g(that)456 1340 y Fr(\034)9 b Fw(\()p Fr(\032;)14 b(")p Fw(\))23 b Fr(<)g(\033)s Fw(\()p Fq(\006)p Fr("v)s Fw(\),)28 b Fr(\033)s Fw(\()p Fq(\001)p Fw(\))h(as)e(in)h(\(6.5\).)36 b(Hence,)28 b(for)f(an)n(y)g Fr(t)c Fq(\024)g Fr(\034)9 b Fw(\()p Fr(\032;)14 b(")p Fw(\),)1136 1429 y Fm(\015)1136 1479 y(\015)1183 1499 y Fr(S)1234 1511 y Fo(t)1277 1499 y Fw(\()p Fr( )1363 1511 y Fp(\006)p Fo(")1450 1499 y Fw(\))19 b Fq(\000)1584 1432 y Fm(\000)1622 1499 y Fr(q)j Fq(\006)c Fr(e)1803 1465 y Fo(\025t)1871 1499 y Fr("v)1953 1432 y Fm(\001)1991 1429 y(\015)1991 1479 y(\015)2037 1533 y Fp(1)2130 1499 y Fq(\024)23 b Fr(N)2308 1432 y Fm(\000)2346 1499 y Fr(e)2385 1465 y Fo(\025t)2453 1499 y Fr(")p Fq(k)p Fr(v)s Fq(k)2619 1511 y Fp(1)2688 1432 y Fm(\001)2726 1445 y Fs(2)3232 1499 y Fw(\(6.13\))456 1637 y(and)1262 1741 y Fq(k)p Fr(S)1355 1753 y Fo(t)1398 1741 y Fw(\()p Fr( )1484 1753 y Fp(\006)p Fo(")1572 1741 y Fw(\))18 b Fq(\000)g Fr(q)s Fq(k)1787 1766 y Fp(1)1880 1741 y Fq(\024)23 b Fw(\(1)18 b(+)g Fr(C)2202 1753 y Fs(1)2240 1741 y Fw(\))p Fr(e)2311 1706 y Fo(\025t)2379 1741 y Fr(")p Fq(k)p Fr(v)s Fq(k)2545 1753 y Fp(1)2615 1741 y Fr(:)594 b Fw(\(6.14\))456 1892 y Fz(Theorem)36 b(6.2.)44 b Fk(F)-6 b(or)34 b(any)i Fr(h)31 b Fq(2)i Fw(\(0)p Fr(;)14 b(h)1711 1862 y Fp(\003)1749 1892 y Fw(])34 b Fk(\()p Fr(h)1888 1862 y Fp(\003)1961 1892 y Fk(as)h(in)g(Pr)l(op)l(osition)h(2.10\),)i(ther)l(e)d(ar)l(e)g Fr(\032)d(>)g Fw(0)456 1992 y Fk(and)e Fr(w)678 1962 y Fp(\006)676 2012 y Fo(s)758 1992 y Fq(2)23 b Fr(C)901 1952 y Fs(sym)895 2014 y(0)1021 1992 y Fw(\()p Fn(R)p Fw(\))q Fk(,)36 b Fr(s)23 b Fq(\024)f Fw(0)p Fk(,)30 b(such)g(that,)g(for)h(any)f Fr(s)23 b Fq(\024)g Fw(0)p Fk(,)1352 2131 y Fw(lim)1361 2185 y Fo(")p Fp(#)p Fs(0)1482 2131 y Fq(k)p Fr(S)1575 2146 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)1835 2131 y Fw(\()p Fr( )1921 2143 y Fp(\006)p Fo(")2009 2131 y Fw(\))19 b Fq(\000)f Fr(w)2204 2097 y Fp(\006)2202 2151 y Fo(s)2260 2131 y Fq(k)2302 2143 y Fp(1)2395 2131 y Fw(=)23 b(0)p Fr(:)684 b Fw(\(6.15\))456 2308 y Fk(Mor)l(e)l(over)967 2446 y Fw(lim)917 2496 y Fo(s)p Fp(!\0001)1147 2446 y Fq(k)p Fr(w)1250 2412 y Fp(\006)1248 2467 y Fo(s)1324 2446 y Fq(\000)18 b Fr(q)s Fq(k)1489 2458 y Fp(1)1582 2446 y Fw(=)23 b(0;)183 b Fr(S)1969 2458 y Fo(t)1998 2446 y Fw(\()p Fr(w)2091 2412 y Fp(\006)2089 2467 y Fo(s)2148 2446 y Fw(\))24 b(=)e Fr(w)2352 2411 y Fp(\006)2350 2467 y Fo(s)p Fs(+)p Fo(t)2547 2446 y Fw(if)42 b Fr(s)18 b Fw(+)g Fr(t)23 b Fq(\024)g Fw(0)p Fr(:)249 b Fw(\(6.16\))555 2634 y(A)32 b(uniformit)n(y)e(in)i Fr(s)c Fq(\024)h Fw(0)h(of)h(the)h(limit)f(\(6.15\))g(is)g(pro)n(v)n (ed)e(in)i(Prop)r(osition)f(6.6)g(b)r(elo)n(w)h(to)456 2733 y(whic)n(h)c(w)n(e)g(refer)g(for)g(a)g(precise)g(statemen)n(t.)456 2852 y Fz(Pro)s(of)48 b(of)h(Theorem)e(6.2.)79 b Fw(W)-7 b(e)43 b(will)f(next)h(pro)n(v)n(e)d(that)j(if)f Fr(\032)g Fw(is)h(small)e(enough)h(then)456 2952 y Fq(f)p Fr(S)549 2967 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\))727 2952 y Fw(\()p Fr( )813 2964 y Fp(\006)p Fo(")901 2952 y Fw(\))38 b(:)f Fr(")h Fq(2)g Fw(\(0)p Fr(;)14 b(\032)p Fw(])p Fq(g)36 b Fw(is)g(a)g(Cauc)n(h)n(y)f(sequence)h(as)g Fr(")h Fq(#)g Fw(0.)63 b(Without)37 b(loss)f(of)g(gen-)456 3051 y(eralit)n(y)26 b(w)n(e)h(restrict)g(to)h(the)g(case)e(with)j(the) f(plus)f(sign.)37 b(Then)27 b(w)n(e)h(need)f(to)h(estimate)1177 3189 y Fr(S)1228 3204 y Fo(\034)7 b Fs(\()p Fo(\032;")1377 3188 y Fj(0)1399 3204 y Fs(\))1443 3189 y Fw(\()q Fr( )1530 3201 y Fo(")1561 3185 y Fj(0)1588 3189 y Fw(\))18 b Fq(\000)g Fr(S)1772 3204 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\))1965 3189 y Fw(\()p Fr( )2051 3201 y Fo(")2087 3189 y Fw(\))14 b Fr(;)180 b Fw(0)23 b Fr(<)f(")2527 3155 y Fp(0)2573 3189 y Fr(<)h(":)509 b Fw(\(6.17\))456 3327 y(Observing)26 b(that)1577 3438 y Fr( )1631 3450 y Fo(")1689 3438 y Fw(=)d Fr(q)f Fw(+)c Fr(e)1958 3404 y Fo(\025\034)7 b Fs(\()p Fo(";")2143 3378 y Fj(0)2165 3404 y Fs(\))2195 3438 y Fr(")2234 3404 y Fp(0)2257 3438 y Fr(v)s(;)456 3558 y Fw(b)n(y)28 b(\(6.13\))o(,)1312 3592 y Fm(\015)1312 3642 y(\015)1358 3663 y Fr(S)1409 3678 y Fo(\034)7 b Fs(\()p Fo(";")1555 3661 y Fj(0)1578 3678 y Fs(\))1621 3663 y Fw(\()q Fr( )1708 3675 y Fo(")1739 3659 y Fj(0)1766 3663 y Fw(\))19 b Fq(\000)f Fr( )1954 3675 y Fo(")1989 3592 y Fm(\015)1989 3642 y(\015)2035 3696 y Fp(1)2129 3663 y Fq(\024)k Fr(N)9 b Fq(k)p Fr(v)s Fq(k)2419 3628 y Fs(2)2419 3683 y Fp(1)2489 3663 y Fr(")2528 3628 y Fs(2)2565 3663 y Fr(:)644 b Fw(\(6.18\))555 3788 y(W)-7 b(e)24 b(th)n(us)g(need)g(to)f(compare)f Fr(S)1539 3800 y Fo(t)1582 3788 y Fw(\()q Fr( )1669 3800 y Fo(")1704 3788 y Fw(\))i(and)g Fr(S)1969 3800 y Fo(t)1998 3788 y Fw(\()16 b(~)-58 b Fr(m)p Fw(\),)25 b Fr(t)e Fq(\024)g Fr(\034)9 b Fw(\()p Fr(\032;)14 b(")p Fw(\),)25 b(for)e(all)g (functions)40 b(~)-58 b Fr(m)24 b Fw(suc)n(h)456 3888 y(that)1500 3991 y Fq(k)16 b Fw(~)-58 b Fr(m)18 b Fq(\000)g Fr( )1770 4003 y Fo(")1806 3991 y Fq(k)1848 4003 y Fp(1)1941 3991 y Fq(\024)k Fr(N)9 b Fq(k)p Fr(v)s Fq(k)2231 3957 y Fs(2)2231 4011 y Fp(1)2301 3991 y Fr(")2340 3957 y Fs(2)2377 3991 y Fr(:)832 b Fw(\(6.19\))456 4111 y(Let)1550 4215 y(\001)1619 4227 y Fo(t)1693 4168 y Fr(:)1672 4215 y Fw(=)22 b Fr(S)1810 4227 y Fo(t)1853 4215 y Fw(\()q Fr( )1940 4227 y Fo(")1975 4215 y Fw(\))d Fq(\000)f Fr(S)2160 4227 y Fo(t)2189 4215 y Fw(\()e(~)-58 b Fr(m)p Fw(\))p Fr(:)883 b Fw(\(6.20\))456 4335 y(By)28 b(\(2.7\))f(w)n(e)h(ha)n(v)n(e) 1031 4460 y Fr(d)p Fw(\001)1143 4472 y Fo(t)p 1031 4497 142 4 v 1066 4573 a Fr(dt)1206 4517 y Fw(=)23 b Fr(L)1351 4532 y Fo(S)1392 4540 y Fi(t)1419 4532 y Fs(\()p Fo( )1489 4540 y Fi(")1521 4532 y Fs(\))1551 4517 y Fw(\001)1620 4529 y Fo(t)1668 4517 y Fw(+)18 b Fr(R)1815 4473 y Fs(\(1\))1814 4537 y Fo(t)1903 4517 y Fr(;)180 b Fq(k)p Fr(R)2212 4473 y Fs(\(1\))2211 4537 y Fo(t)2301 4517 y Fq(k)2343 4529 y Fp(1)2436 4517 y Fq(\024)22 b Fr(k)2566 4529 y Fs(2)2604 4517 y Fq(k)p Fw(\001)2715 4529 y Fo(t)2744 4517 y Fq(k)2786 4482 y Fs(2)2786 4537 y Fp(1)2855 4517 y Fr(:)354 b Fw(\(6.21\))456 4683 y(Since)34 b Fq(k)p Fr(L)778 4695 y Fo(m)p Fs(+)p Fo(u)930 4683 y Fw(\001)24 b Fq(\000)e Fr(L)1167 4695 y Fo(m)1230 4683 y Fw(\001)p Fq(k)1341 4695 y Fp(1)1445 4683 y Fq(\024)34 b Fr(c)1580 4652 y Fp(0)1603 4683 y Fq(k)p Fr(u)p Fq(k)1735 4695 y Fp(1)1804 4683 y Fq(k)p Fw(\001)p Fq(k)1957 4695 y Fp(1)2061 4683 y Fw(with)h Fr(c)2293 4652 y Fp(0)2350 4683 y Fw(a)f(suitable)h(constan)n(t,)g(b)n (y)g(\(6.14\))456 4782 y(there)27 b(is)g Fr(C)810 4794 y Fs(3)876 4782 y Fw(so)g(that)591 4928 y Fr(R)655 4885 y Fs(\(2\))654 4949 y Fo(t)788 4881 y Fr(:)767 4928 y Fw(=)c Fr(L)912 4943 y Fo(S)953 4951 y Fi(t)980 4943 y Fs(\()p Fo( )1050 4951 y Fi(")1082 4943 y Fs(\))1112 4928 y Fw(\001)1181 4940 y Fo(t)1229 4928 y Fq(\000)18 b Fr(L)p Fw(\001)1438 4940 y Fo(t)1467 4928 y Fr(;)180 b Fq(k)p Fr(R)1776 4885 y Fs(\(2\))1775 4949 y Fo(t)1864 4928 y Fq(k)1906 4940 y Fp(1)1999 4928 y Fq(\024)23 b Fr(C)2146 4940 y Fs(3)2183 4928 y Fr(\032)p Fq(k)p Fw(\001)2337 4940 y Fo(t)2366 4928 y Fq(k)2408 4940 y Fp(1)2644 4928 y Fq(8)14 b Fr(t)22 b Fq(\024)h Fr(\034)9 b Fw(\()p Fr(\032;)14 b(")p Fw(\))p Fr(:)136 b Fw(\(6.22\))456 5066 y(Th)n(us)1492 5136 y Fr(d)p Fw(\001)1604 5148 y Fo(t)p 1492 5173 V 1526 5249 a Fr(dt)1667 5192 y Fw(=)22 b Fr(L)p Fw(\001)1880 5204 y Fo(t)1928 5192 y Fw(+)c Fr(R)2075 5149 y Fs(\(2\))2074 5212 y Fo(t)2182 5192 y Fw(+)g Fr(R)2329 5149 y Fs(\(1\))2328 5212 y Fo(t)3232 5192 y Fw(\(6.23\))p eop %%Page: 25 25 25 24 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(25)456 450 y Fw(and)1211 589 y(\001)1280 601 y Fo(t)1333 589 y Fw(=)22 b Fr(e)1459 555 y Fo(Lt)1534 589 y Fw(\001)1603 601 y Fs(0)1659 589 y Fw(+)1742 476 y Fm(Z)1825 497 y Fo(t)1788 665 y Fs(0)1868 589 y Fr(ds)14 b(e)2003 555 y Fo(L)p Fs(\()p Fo(t)p Fp(\000)p Fo(s)p Fs(\))2212 589 y Fw([)p Fr(R)2299 555 y Fs(\(2\))2298 610 y Fo(s)2407 589 y Fw(+)k Fr(R)2554 555 y Fs(\(1\))2553 610 y Fo(s)2643 589 y Fw(])p Fr(:)543 b Fw(\(6.24\))456 755 y(Then)27 b(b)n(y)i(\(2.64\))e(and)g(the)h(b)r(ounds)g(in)g (\(6.21\))o({\(6.22\))o(,)f(for)g(an)n(y)g Fr(t)c Fq(\024)g Fr(\034)9 b Fw(\()p Fr(\032;)14 b(")p Fw(\),)822 950 y Fq(k)p Fw(\001)933 962 y Fo(t)962 950 y Fq(k)1004 962 y Fp(1)1097 950 y Fq(\024)23 b Fr(C)1244 962 y Fs(1)1281 950 y Fr(e)1320 915 y Fo(\025t)1388 950 y Fw(\001)1457 962 y Fs(0)1513 950 y Fw(+)18 b Fr(C)1655 962 y Fs(1)1707 837 y Fm(Z)1790 857 y Fo(t)1753 1025 y Fs(0)1833 950 y Fr(ds)c(e)1968 915 y Fo(\025)p Fs(\()p Fo(t)p Fp(\000)p Fo(s)p Fs(\))2171 950 y Fw([)p Fr(C)2253 962 y Fs(3)2291 950 y Fr(\032)p Fq(k)p Fw(\001)2445 962 y Fo(s)2480 950 y Fq(k)2522 962 y Fp(1)2610 950 y Fw(+)k Fr(k)2736 962 y Fs(2)2774 950 y Fq(k)p Fw(\001)2885 962 y Fo(s)2920 950 y Fq(k)2962 915 y Fs(2)2962 970 y Fp(1)3032 950 y Fw(])p Fr(:)456 1129 y Fw(Calling)1647 1231 y Fr(\025)1695 1196 y Fp(\003)1777 1184 y Fr(:)1756 1231 y Fw(=)23 b Fr(\025)c Fw(+)f Fr(C)2053 1243 y Fs(2)2091 1231 y Fr(C)2150 1243 y Fs(3)2187 1231 y Fr(\032;)979 b Fw(\(6.25\))456 1350 y(w)n(e)27 b(ha)n(v)n(e,)f(b)n(y)i(iteration)f(and)g(recalling)g (\(6.4\),)1107 1545 y Fq(k)p Fw(\001)1218 1557 y Fo(t)1247 1545 y Fq(k)1289 1557 y Fp(1)1382 1545 y Fq(\024)c Fr(C)1529 1557 y Fs(1)1566 1545 y Fr(e)1605 1511 y Fo(\025)1644 1486 y Fj(\003)1679 1511 y Fo(t)1708 1545 y Fw(\001)1777 1557 y Fs(0)1833 1545 y Fw(+)18 b Fr(C)1975 1557 y Fs(2)2026 1432 y Fm(Z)2109 1452 y Fo(t)2073 1620 y Fs(0)2139 1545 y Fr(ds)c(e)2274 1511 y Fo(\025)2313 1486 y Fj(\003)2347 1511 y Fs(\()p Fo(t)p Fp(\000)p Fo(s)p Fs(\))2512 1545 y Fq(k)p Fw(\001)2623 1557 y Fo(s)2658 1545 y Fq(k)2700 1511 y Fs(2)2700 1565 y Fp(1)2770 1545 y Fr(:)439 b Fw(\(6.26\))456 1741 y(Setting)28 b Fr(W)820 1753 y Fo(t)893 1694 y Fr(:)872 1741 y Fw(=)23 b Fr(e)999 1710 y Fp(\000)p Fo(\025)1090 1685 y Fj(\003)1125 1710 y Fo(t)1154 1741 y Fq(k)p Fw(\001)1265 1753 y Fo(t)1294 1741 y Fq(k)1336 1753 y Fp(1)1433 1741 y Fw(and)28 b(using)g(\(6.19\))o(,)g(from)f(\(6.26\))g(w)n(e)g(get,)h (for)f(all)g Fr(t)c Fq(\024)g Fr(\034)9 b Fw(\()p Fr(\032;)14 b(")p Fw(\),)1333 1935 y Fr(W)1411 1947 y Fo(t)1464 1935 y Fq(\024)23 b Fr(C)1611 1947 y Fs(1)1648 1935 y Fr(N)9 b Fq(k)p Fr(v)s Fq(k)1851 1901 y Fs(2)1851 1956 y Fp(1)1920 1935 y Fr(")1959 1901 y Fs(2)2015 1935 y Fw(+)18 b Fr(C)2157 1947 y Fs(2)2208 1822 y Fm(Z)2291 1843 y Fo(t)2254 2011 y Fs(0)2321 1935 y Fr(ds)c(W)2507 1901 y Fs(2)2495 1956 y Fo(s)2544 1935 y Fr(;)665 b Fw(\(6.27\))456 2114 y(whic)n(h)27 b(implies:)1052 2313 y Fr(W)1130 2325 y Fo(t)1182 2313 y Fq(\024)c Fr(c")1345 2279 y Fs(2)1425 2209 y Fp(1)1398 2234 y Fm(X)1396 2410 y Fo(n)p Fs(=0)1535 2246 y Fm(\000)1573 2313 y Fr(c")1648 2279 y Fs(2)1685 2313 y Fr(t)1715 2246 y Fm(\001)1753 2262 y Fo(n)1812 2313 y Fr(;)180 b(c)2094 2266 y(:)2074 2313 y Fw(=)22 b Fr(C)2220 2325 y Fs(1)2258 2313 y Fw(\(1)c Fq(_)h Fr(C)2483 2325 y Fs(2)2521 2313 y Fw(\))p Fr(N)9 b Fq(k)p Fr(v)s Fq(k)2756 2279 y Fs(2)2756 2333 y Fp(1)2825 2313 y Fr(:)384 b Fw(\(6.28\))456 2526 y(Since)31 b Fr("\034)9 b Fw(\()p Fr(\032;)14 b(")p Fw(\))29 b Fq(!)g Fw(0)h(as)g Fr(")f Fq(#)f Fw(0,)j(w)n(e)g(can)f(c)n(ho)r(ose)g Fr(")2082 2538 y Fs(1)2148 2526 y Fq(2)f Fw(\(0)p Fr(;)14 b(\032)p Fw(])30 b(so)h(that)g(the)g(series)f(con)n(v)n(erges)456 2626 y(and)d Fr(W)695 2638 y Fo(t)748 2626 y Fq(\024)22 b Fw(2)p Fr(c")952 2596 y Fs(2)1016 2626 y Fw(for)28 b(all)f Fr(")c Fq(2)g Fw(\(0)p Fr(;)14 b(")1549 2638 y Fs(1)1586 2626 y Fw(])27 b(and)h Fr(t)23 b Fq(\024)g Fr(\034)9 b Fw(\()p Fr(\032;)14 b(")p Fw(\).)37 b(Cho)r(osing)27 b Fr(\032)g Fw(small)h(enough)f(so)g(that)1233 2811 y Fr(C)1292 2823 y Fs(2)1329 2811 y Fr(C)1388 2823 y Fs(3)1426 2811 y Fr(\032)c Fq(\024)1590 2754 y Fr(\025)p 1590 2792 49 4 v 1593 2868 a Fw(2)1782 2811 y(i.e.)110 b Fr(e)2037 2776 y Fo(\025)2076 2751 y Fj(\003)2111 2776 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\))2313 2811 y Fq(\024)2401 2718 y Fm(\020)2460 2754 y Fr(\032)p 2460 2792 43 4 v 2462 2868 a(")2513 2718 y Fm(\021)2563 2736 y Fs(3)p Fo(=)p Fs(2)3232 2811 y Fw(\(6.29\))456 2975 y(and)27 b(recalling)h(\(6.12\))o(,)f(\(6.25\),)g(and)h(the)g(de\014nition)g(of) f Fr(W)2322 2987 y Fo(t)2352 2975 y Fw(,)h(w)n(e)f(get)1092 3112 y Fq(k)p Fw(\001)1203 3124 y Fo(t)1232 3112 y Fq(k)1274 3124 y Fp(1)1367 3112 y Fq(\024)c Fr(C)1514 3124 y Fs(4)1551 3049 y Fq(p)p 1621 3049 39 4 v 1621 3112 a Fr(")o(;)180 b Fq(8)14 b Fr(")22 b Fq(2)h Fw(\(0)p Fr(;)14 b(")2212 3124 y Fs(1)2249 3112 y Fw(])83 b Fq(8)14 b Fr(t)22 b Fq(\024)h Fr(\034)9 b Fw(\()p Fr(\032;)14 b(")p Fw(\))p Fr(;)425 b Fw(\(6.30\))456 3252 y(with)28 b Fr(C)704 3264 y Fs(4)785 3205 y Fr(:)764 3252 y Fw(=)23 b(2)p Fr(c\032)973 3222 y Fs(3)p Fo(=)p Fs(2)1077 3252 y Fw(.)555 3352 y(By)29 b(\(6.18\))e(and)g(\(6.30\))g(w)n(e)g(conclude)g(that)706 3418 y Fm(\015)706 3468 y(\015)752 3489 y Fr(S)803 3504 y Fo(\034)7 b Fs(\()p Fo(\032;")952 3487 y Fj(0)974 3504 y Fs(\))1018 3489 y Fw(\()p Fr( )1104 3501 y Fo(")1135 3485 y Fj(0)1162 3489 y Fw(\))19 b Fq(\000)f Fr(S)1347 3504 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\))1540 3489 y Fw(\()p Fr( )1626 3501 y Fo(")1661 3489 y Fw(\))1694 3418 y Fm(\015)1694 3468 y(\015)1740 3523 y Fp(1)1833 3489 y Fq(\024)23 b Fr(C)1980 3501 y Fs(4)2017 3425 y Fq(p)p 2087 3425 V 2087 3489 a Fr(")193 b Fw(if)42 b(0)23 b Fr(<)f(")2600 3455 y Fp(0)2646 3489 y Fr(<)h(")g Fq(\024)f Fr(")2922 3501 y Fs(1)2959 3489 y Fr(;)250 b Fw(\(6.31\))456 3631 y(whic)n(h)33 b(sho)n(ws)g Fq(f)p Fr(S)1037 3646 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\))1229 3631 y Fw(\()p Fr( )1315 3643 y Fo(")1351 3631 y Fw(\))p Fq(g)34 b Fw(is)f(a)h(Cauc)n (h)n(y)e(sequence)i(as)f Fr(")g Fq(#)g Fw(0)g(for)g(all)h Fr(\032)g Fw(small)f(enough.)456 3730 y(Analogously)26 b(w)n(e)h(argue)f(for)h(the)h(case)f(with)h(the)g(min)n(us)f(sign.)555 3830 y(The)g(same)g(argumen)n(t)f(sho)n(ws)g(that)h(also)f Fr(S)1938 3845 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)2213 3830 y Fw(\()p Fr( )2299 3842 y Fp(\006)p Fo(")2386 3830 y Fw(\))28 b(is,)f(for)f(eac)n(h)h Fr(s)c Fq(\024)f Fw(0,)27 b(a)g(Cauc)n(h)n(y)456 3930 y(sequence.)79 b(Then)42 b Fr(S)1157 3945 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)1431 3930 y Fw(\()q Fr( )1518 3942 y Fp(\006)p Fo(")1605 3930 y Fw(\))42 b(con)n(v)n(erges)e(in)i(sup)f(norm)h(as)f Fr(")46 b Fq(#)g Fw(0)c(to)g(a)f(function)456 4038 y Fr(w)517 4008 y Fp(\006)515 4059 y Fo(s)573 4038 y Fw(,)49 b(hence)c(\(6.15\))o(.)87 b(Moreo)n(v)n(er)42 b(if)j Fr(t)29 b Fw(+)h Fr(s)51 b Fq(\024)f Fw(0,)f Fr(t)i Fq(\025)f Fw(0,)f(then)c Fr(S)2730 4050 y Fo(t)2772 3971 y Fm(\000)2811 4038 y Fr(S)2862 4053 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)3137 4038 y Fw(\()p Fr( )3223 4050 y Fo(")3258 4038 y Fw(\))3291 3971 y Fm(\001)3380 4038 y Fw(=)456 4138 y Fr(S)507 4153 y Fo(\034)g Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)p Fs(+)p Fo(t)858 4138 y Fw(\()p Fr( )944 4150 y Fo(")979 4138 y Fw(\))q(.)46 b(By)32 b(\(2.3\))e(for)h(eac)n(h)f Fr(t)e Fq(\025)g Fw(0,)j Fr(S)2036 4150 y Fo(t)2065 4138 y Fw(\()p Fr(m)p Fw(\))h(dep)r(ends)f(con)n(tin)n(uously)e(on)i Fr(m)p Fw(,)g(th)n(us)456 4248 y Fr(S)507 4260 y Fo(t)550 4180 y Fm(\000)588 4248 y Fr(S)639 4263 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)914 4248 y Fw(\()p Fr( )1000 4260 y Fo(")1035 4248 y Fw(\))1068 4180 y Fm(\001)1131 4248 y Fq(!)26 b Fr(S)1291 4260 y Fo(t)1320 4248 y Fw(\()p Fr(w)1413 4218 y Fp(\006)1411 4268 y Fo(s)1470 4248 y Fw(\))j(as)g Fr(")c Fq(#)g Fw(0.)41 b(On)28 b(the)i(other)e(hand)h Fr(S)2634 4263 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)p Fs(+)p Fo(t)2985 4248 y Fw(\()q Fr( )3072 4260 y Fo(")3107 4248 y Fw(\))26 b Fq(!)f Fr(w)3334 4212 y Fp(\006)3332 4268 y Fo(s)p Fs(+)p Fo(t)456 4357 y Fw(as)f Fr(")f Fq(#)g Fw(0,)i(hence)g Fr(S)1051 4369 y Fo(t)1094 4357 y Fw(\()p Fr(w)1187 4327 y Fp(\006)1185 4378 y Fo(s)1244 4357 y Fw(\))e(=)g Fr(w)1448 4322 y Fp(\006)1446 4378 y Fo(s)p Fs(+)p Fo(t)1558 4357 y Fw(,)j(pro)n(ving)e(the)h(second)g (relation)f(in)i(\(6.16\))o(.)36 b(Finally)-7 b(,)26 b(from)456 4457 y(\(6.14\))o(,)770 4527 y Fm(\015)770 4577 y(\015)816 4598 y Fr(S)867 4613 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)1142 4598 y Fw(\()p Fr( )1228 4610 y Fp(\006)p Fo(")1316 4598 y Fw(\))19 b Fq(\000)f Fr(q)1490 4527 y Fm(\015)1490 4577 y(\015)1536 4631 y Fp(1)1629 4598 y Fq(\024)23 b Fr(C)1776 4610 y Fs(5)1827 4598 y Fr(e)1866 4563 y Fo(\025s)1941 4598 y Fr(;)180 b(C)2203 4610 y Fs(5)2284 4551 y Fr(:)2263 4598 y Fw(=)23 b(\(1)18 b(+)g Fr(C)2585 4610 y Fs(1)2623 4598 y Fw(\))p Fr(\032)p Fq(k)p Fr(v)s Fq(k)2825 4610 y Fp(1)2895 4598 y Fr(;)314 b Fw(\(6.32\))456 4735 y(from)27 b(whic)n(h,)g(letting)h Fr(")23 b Fq(#)g Fw(0,)1560 4766 y Fm(\015)1560 4816 y(\015)1606 4837 y Fr(w)1667 4802 y Fp(\006)1665 4857 y Fo(s)1743 4837 y Fq(\000)18 b Fr(q)1866 4766 y Fm(\015)1866 4816 y(\015)1912 4870 y Fp(1)2005 4837 y Fq(\024)23 b Fr(C)2152 4849 y Fs(5)2203 4837 y Fr(e)2242 4802 y Fo(\025s)2317 4837 y Fr(;)892 b Fw(\(6.33\))456 4960 y(pro)n(ving)26 b(the)i(\014rst)f(statemen)n(t)h(in)f(\(6.16\),)g(Theorem)g(6.2)g(is)g (pro)n(v)n(ed.)730 b Fg(\003)456 5079 y Fz(Pro)s(of)31 b(of)h(Theorem)e(2.12.)36 b Fw(The)27 b(manifold)1078 5216 y Fq(W)1211 5169 y Fr(:)1190 5216 y Fw(=)c Fq(W)1367 5181 y Fs(+)1440 5216 y Fq([)c(W)1603 5181 y Fp(\000)1659 5216 y Fr(;)97 b Fq(W)1868 5181 y Fp(\006)1967 5169 y Fr(:)1946 5216 y Fw(=)2034 5148 y Fm(\010)2083 5216 y Fr(S)2134 5228 y Fo(t)2163 5216 y Fw(\()p Fr(w)2256 5181 y Fp(\006)2254 5236 y Fo(s)2313 5216 y Fw(\))37 b(:)g Fr(s)23 b Fq(\024)f Fw(0)h Fq(\024)g Fr(t)2774 5148 y Fm(\011)3232 5216 y Fw(\(6.34\))p eop %%Page: 26 26 26 25 bop 456 251 a Fs(26)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fw(and)31 b(b)r(oth)g(its)h(branc)n(hes)e Fq(W)1377 420 y Fp(\006)1464 450 y Fw(are)g(in)n(v)-5 b(arian)n(t)30 b(under)h Fr(S)2248 462 y Fo(t)2308 450 y Fw(whic)n(h)h(is)f(in)n(v)n(ertible)f(on)h Fq(W)3215 420 y Fp(\006)3271 450 y Fw(.)48 b(By)456 550 y(\(6.16\))26 b Fq(W)784 520 y Fp(\006)868 550 y Fw(originate)g(at)i Fr(s)23 b Fw(=)f Fq(\0001)28 b Fw(from)f Fr(q)s Fw(.)555 649 y(Recalling)h(\(6.12\))o(,)g(from)f(\(6.13\))614 723 y Fm(\015)614 773 y(\015)660 794 y Fr(S)711 809 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)986 794 y Fw(\()q Fr( )1073 806 y Fp(\006)p Fo(")1160 794 y Fw(\))19 b Fq(\000)1294 727 y Fm(\000)1332 794 y Fr(q)j Fq(\006)c Fr(e)1513 760 y Fo(\025s)1587 794 y Fr(\032v)1673 727 y Fm(\001)1711 723 y(\015)1711 773 y(\015)1757 828 y Fp(1)1851 794 y Fq(\024)k Fr(C)1997 806 y Fs(6)2049 794 y Fr(e)2088 760 y Fs(2)p Fo(\025s)2195 794 y Fr(;)180 b(C)2457 806 y Fs(6)2539 747 y Fr(:)2518 794 y Fw(=)23 b Fr(N)f Fw(\()q Fr(\032)p Fq(k)p Fr(v)s Fq(k)2898 806 y Fp(1)2967 794 y Fw(\))2999 752 y Fs(2)3050 794 y Fr(;)159 b Fw(\(6.35\))456 930 y(from)27 b(whic)n(h,)g(letting)h Fr(")23 b Fq(#)g Fw(0,)1355 995 y Fm(\015)1355 1045 y(\015)1401 1066 y Fr(w)1462 1031 y Fp(\006)1460 1086 y Fo(s)1537 1066 y Fq(\000)1620 998 y Fm(\000)1658 1066 y Fr(q)f Fq(\006)c Fr(e)1839 1031 y Fo(\025s)1913 1066 y Fr(\032v)1999 998 y Fm(\001)2038 995 y(\015)2038 1045 y(\015)2084 1099 y Fp(1)2177 1066 y Fq(\024)23 b Fr(C)2324 1078 y Fs(6)2375 1066 y Fr(e)2414 1031 y Fs(2)p Fo(\025s)2522 1066 y Fr(:)687 b Fw(\(6.36\))555 1206 y(Next,)28 b(b)n(y)h(\(6.1\))e(and)g(recalling)g (that)h Fr(f)9 b Fw(\()p Fr(q)s Fw(\))23 b(=)g(0,)531 1337 y Fr(dw)635 1307 y Fp(\006)633 1358 y Fo(s)p 531 1374 161 4 v 571 1450 a Fr(ds)785 1393 y Fw(=)83 b Fr(L)p Fw(\()p Fr(w)1083 1359 y Fp(\006)1081 1414 y Fo(s)1158 1393 y Fq(\000)18 b Fr(q)s Fw(\))g(+)1414 1326 y Fm(\002)1449 1393 y Fr(f)9 b Fw(\()p Fr(w)1592 1359 y Fp(\006)1590 1414 y Fo(s)1649 1393 y Fw(\))18 b Fq(\000)g Fr(f)9 b Fw(\()p Fr(q)s Fw(\))19 b Fq(\000)f Fr(L)2109 1326 y Fm(\000)2146 1393 y Fr(w)2207 1359 y Fp(\006)2205 1414 y Fo(s)2283 1393 y Fq(\000)g Fr(q)2406 1326 y Fm(\001\003)2501 1393 y Fw(=)23 b Fr(L)2660 1326 y Fm(\002)2694 1393 y Fr(w)2755 1359 y Fp(\006)2753 1414 y Fo(s)2830 1393 y Fq(\000)2913 1326 y Fm(\000)2951 1393 y Fr(q)f Fq(\006)c Fr(e)3132 1359 y Fo(\025s)3206 1393 y Fr(\032v)3292 1326 y Fm(\001\003)933 1558 y Fq(\006)c Fr(\025e)1099 1524 y Fo(\025s)1173 1558 y Fr(\032v)22 b Fw(+)1361 1491 y Fm(\002)1395 1558 y Fr(f)9 b Fw(\()p Fr(w)1538 1524 y Fp(\006)1536 1578 y Fo(s)1595 1558 y Fw(\))19 b Fq(\000)f Fr(f)9 b Fw(\()p Fr(q)s Fw(\))18 b Fq(\000)g Fr(L)2055 1491 y Fm(\000)2093 1558 y Fr(w)2154 1524 y Fp(\006)2152 1578 y Fo(s)2229 1558 y Fq(\000)g Fr(q)2352 1491 y Fm(\001\003)2438 1558 y Fr(:)771 b Fw(\(6.37\))456 1694 y(Denoting)32 b(b)n(y)g Fq(k)p Fr(L)p Fq(k)1080 1706 y Fp(1)1181 1694 y Fw(the)g(norm)g(of)g(the)h(op)r(erator)d Fr(L)i Fw(\(whic)n(h)h(is)f (\014nite\),)i(b)n(y)f(\(2.7\),)g(\(6.33\))o(,)456 1793 y(and)27 b(\(6.36\))g(w)n(e)g(ha)n(v)n(e:)908 1861 y Fm(\015)908 1910 y(\015)908 1960 y(\015)908 2010 y(\015)964 1925 y Fr(dw)1068 1895 y Fp(\006)1066 1945 y Fo(s)p 964 1962 V 1003 2038 a Fr(ds)1153 1981 y Fq(\007)18 b Fr(\025e)1323 1947 y Fo(\025s)1398 1981 y Fr(\032v)1484 1861 y Fm(\015)1484 1910 y(\015)1484 1960 y(\015)1484 2010 y(\015)1530 2064 y Fp(1)1624 1981 y Fq(\024)k Fr(C)1770 1993 y Fs(7)1822 1981 y Fr(e)1861 1947 y Fs(2)p Fo(\025s)1968 1981 y Fr(;)180 b(C)2230 1993 y Fs(7)2312 1934 y Fr(:)2291 1981 y Fw(=)22 b Fq(k)p Fr(L)p Fq(k)2519 1993 y Fp(1)2588 1981 y Fr(C)2647 1993 y Fs(6)2703 1981 y Fw(+)c Fr(k)2829 1993 y Fs(2)2867 1981 y Fr(C)2932 1947 y Fs(2)2926 2002 y(5)2969 1981 y Fr(:)240 b Fw(\(6.38\))555 2171 y(Recalling)27 b(that)h Fr(v)s Fw(\()p Fr(x)p Fw(\))c Fq(\031)39 b Fw(~)-58 b Fr(m)1435 2141 y Fp(0)1458 2171 y Fw(\()p Fr(\030)1530 2141 y Fp(\003)1587 2171 y Fq(\000)18 b Fr(x)p Fw(\))29 b(in)e(the)h(sense)g(of)34 b(\(2.69\))o(,)27 b(w)n(e)h(set)1309 2346 y Fr(s)1348 2358 y Fs(0)1408 2346 y Fw(:)46 b Fr(\032e)1559 2312 y Fo(\025s)1629 2320 y Fl(0)1689 2346 y Fw(=)1826 2290 y(1)p 1787 2327 120 4 v 1787 2351 a Fq(p)p 1856 2351 51 4 v 52 x Fr(\026)1916 2346 y(;)180 b(m)2192 2312 y Fp(\006)2192 2366 y Fo(s)2292 2299 y Fr(:)2271 2346 y Fw(=)22 b Fr(w)2419 2310 y Fp(\006)2417 2367 y Fo(s)p Fs(+)p Fo(s)2530 2375 y Fl(0)2568 2346 y Fr(:)641 b Fw(\(6.39\))456 2538 y(Then)27 b(\(6.38\))g(implies)1354 2570 y Fm(\015)1354 2620 y(\015)1354 2670 y(\015)1354 2719 y(\015)1410 2634 y Fr(dm)1526 2604 y Fp(\006)1526 2655 y Fo(s)p 1410 2671 173 4 v 1455 2747 a Fr(ds)1611 2690 y Fq(\007)18 b Fr(\025e)1781 2656 y Fo(\025s)1905 2634 y Fw(1)p 1866 2671 120 4 v 1866 2696 a Fq(p)p 1935 2696 51 4 v 51 x Fr(\026)1995 2690 y(v)2038 2570 y Fm(\015)2038 2620 y(\015)2038 2670 y(\015)2038 2719 y(\015)2084 2779 y Fp(1)2178 2690 y Fq(\024)23 b Fr(C)2325 2702 y Fs(7)2376 2690 y Fr(e)2415 2656 y Fs(2)p Fo(\025s)2523 2690 y Fr(;)456 2869 y Fw(whic)n(h)k(giv)n (es)g(\(2.73\))o(.)555 2968 y(The)h(pro)r(ofs)f(of)g(the)h(monotonicit) n(y)f(prop)r(ert)n(y)g(of)g Fr(m)2223 2938 y Fp(\006)2223 2989 y Fo(s)2307 2968 y Fw(and)h(of)f(the)h(b)r(ound)g(\(2.74\))f(will) h(b)r(e)456 3068 y(giv)n(en)e(in)i(Prop)r(osition)e(6.3)h(b)r(elo)n(w,) g(Theorem)g(2.12)f(is)i(then)g(pro)n(v)n(ed.)706 b Fg(\003)456 3236 y Fz(Prop)s(osition)28 b(6.3.)38 b Fk(F)-6 b(or)29 b(any)f Fr(s)23 b Fq(2)h Fn(R)p Fk(,)35 b(the)28 b(symmetric)h (functions)f Fr(m)2695 3206 y Fp(\006)2695 3257 y Fo(s)2779 3236 y Fk(ar)l(e)h(non)f(incr)l(e)l(asing)456 3336 y(on)h Fn(R)628 3348 y Fs(+)719 3336 y Fk(and)39 b Fw(\(2.74\))29 b Fk(holds.)555 3486 y Fw(In)24 b(order)f(to)h(pro)n(v)n(e)e(Prop)r (osition)g(6.3)h(w)n(e)h(need)g(the)g(follo)n(wing)f(prop)r(erties)g (of)h(the)g(\015o)n(w)f Fr(S)3392 3498 y Fo(t)3421 3486 y Fw(.)456 3636 y Fz(Theorem)37 b(6.4.)43 b(\(The)c(Comparison)e (Theorem,)h Fw([8])p Fz(\))d Fk(L)l(et)g Fr(m;)29 b Fw(~)-57 b Fr(m)32 b Fq(2)i Fr(L)2949 3648 y Fp(1)3018 3636 y Fw(\()p Fn(R)q Fw(\))41 b Fk(b)l(e)35 b(such)456 3736 y(that)24 b Fr(m)p Fw(\()p Fr(x)p Fw(\))g Fq(\024)39 b Fw(~)-58 b Fr(m)p Fw(\()p Fr(x)p Fw(\))25 b Fk(for)h(al)t(l)g Fr(x)d Fq(2)g Fn(R)p Fk(.)43 b(Then)26 b Fr(S)1900 3748 y Fo(t)1929 3736 y Fw(\()p Fr(m)p Fw(\)\()p Fr(x)p Fw(\))e Fq(\024)f Fr(S)2340 3748 y Fo(t)2369 3736 y Fw(\()16 b(~)-58 b Fr(m)p Fw(\)\()p Fr(x)p Fw(\))26 b Fk(for)g(al)t(l)f Fw(\()p Fr(t;)14 b(x)p Fw(\))24 b Fq(2)g Fn(R)3219 3748 y Fs(+)3287 3736 y Fq(\002)7 b Fn(R)p Fk(.)456 3886 y Fz(Lemma)31 b(6.5.)41 b Fk(L)l(et)30 b Fr(m)c Fq(2)f Fr(L)1367 3856 y Fs(sym)1367 3907 y Fp(1)1487 3886 y Fw(\()p Fn(R)p Fw(\))37 b Fk(b)l(e)31 b(a)h(non)f(incr)l(e)l(asing)g (function)g(on)g Fn(R)2880 3898 y Fs(+)2942 3886 y Fk(.)42 b(Then)32 b Fr(S)3278 3898 y Fo(t)3307 3886 y Fw(\()p Fr(m)p Fw(\))456 3986 y Fk(has)e(the)g(same)g(monotonicity)h(pr)l(op)l (erty)g(for)f(al)t(l)h Fr(t)23 b Fq(2)h Fn(R)2219 3998 y Fs(+)2280 3986 y Fk(.)456 4154 y Fz(Pro)s(of.)41 b Fw(The)28 b(\015o)n(w)f(solution)g Fr(S)1465 4166 y Fo(t)1494 4154 y Fw(\()p Fr(m)p Fw(\))h(solv)n(es)f(the)h(in)n(tegral)e(equation) 942 4348 y Fr(S)993 4360 y Fo(t)1022 4348 y Fw(\()p Fr(m)p Fw(\))e(=)e Fr(e)1309 4314 y Fp(\000)p Fo(t)1390 4348 y Fr(m)c Fw(+)1564 4235 y Fm(Z)1647 4255 y Fo(t)1611 4423 y Fs(0)1677 4348 y Fr(ds)c(e)1812 4314 y Fp(\000)p Fs(\()p Fo(t)p Fp(\000)p Fo(s)p Fs(\))2041 4348 y Fw(tanh)g Fq(f)p Fr(\014)k Fw([)p Fr(J)26 b Fq(\003)18 b Fr(S)2534 4360 y Fo(s)2570 4348 y Fw(\()p Fr(m)p Fw(\))h(+)f Fr(h)p Fw(])p Fq(g)13 b Fr(:)456 4534 y Fw(Since)27 b Fr(J)36 b Fw(is)27 b(smo)r(oth,)h(the)g(function)g Fr(g)1662 4546 y Fo(t)1691 4534 y Fw(\()p Fr(x)p Fw(\))1847 4487 y Fr(:)1826 4534 y Fw(=)22 b Fr(S)1964 4546 y Fo(t)1994 4534 y Fw(\()p Fr(m)p Fw(\)\()p Fr(x)p Fw(\))d Fq(\000)f Fr(e)2383 4504 y Fp(\000)p Fo(t)2464 4534 y Fr(m)p Fw(\()p Fr(x)p Fw(\))29 b(is)e(di\013eren)n(tiable.)37 b(F)-7 b(ur-)456 4634 y(ther)23 b(its)g(spatial)g(deriv)-5 b(ativ)n(e)22 b Fr(g)1421 4604 y Fp(0)1418 4654 y Fo(t)1447 4634 y Fw(\()p Fr(x)p Fw(\))i(is)g(an)f(an)n(tisymmetric)f(function)i(whic)n (h)f(satis\014es,)h(for)e(an)n(y)456 4733 y Fr(x)h Fq(2)h Fn(R)659 4745 y Fs(+)720 4733 y Fw(,)487 4925 y Fr(g)530 4891 y Fp(0)527 4946 y Fo(t)556 4925 y Fw(\()p Fr(x)p Fw(\))84 b(=)899 4812 y Fm(Z)982 4833 y Fo(t)945 5001 y Fs(0)1011 4925 y Fr(ds)14 b(p)1149 4937 y Fo(s)1184 4925 y Fw(\()p Fr(x)p Fw(\))g(\()q Fr(J)1388 4937 y Fp(\000)1444 4925 y Fr(g)1487 4891 y Fp(0)1484 4946 y Fo(s)1519 4925 y Fw(\))g(\()p Fr(x)p Fw(\))20 b(+)e Fr(z)1818 4937 y Fo(t)1847 4925 y Fw(\()p Fr(x)p Fw(\))p Fr(;)1251 b Fw(\(6.40\))479 5163 y Fr(p)521 5175 y Fo(s)556 5163 y Fw(\()p Fr(x)p Fw(\))772 5116 y Fr(:)751 5163 y Fw(=)1432 5106 y Fr(\014)p 909 5143 1099 4 v 909 5228 a Fw(cosh)1066 5191 y Fs(2)1117 5228 y Fq(f)o Fr(\014)19 b Fw([\()p Fr(J)26 b Fq(\003)18 b Fr(S)1462 5240 y Fo(s)1498 5228 y Fw(\()p Fr(m)p Fw(\)\))c(\()p Fr(x)p Fw(\))20 b(+)e Fr(h)p Fw(])p Fq(g)2017 5163 y Fr(;)97 b(z)2176 5175 y Fo(t)2205 5163 y Fw(\()p Fr(x)p Fw(\))2361 5116 y Fr(:)2340 5163 y Fw(=)2428 5050 y Fm(Z)2511 5070 y Fo(t)2474 5238 y Fs(0)2540 5163 y Fr(ds)14 b(e)2675 5128 y Fp(\000)p Fo(t)2756 5163 y Fr(p)2798 5175 y Fo(s)2833 5163 y Fw(\()p Fr(x)p Fw(\)\()p Fr(J)3022 5175 y Fp(\000)3098 5163 y Fq(\003)k Fr(m)3231 5128 y Fp(0)3254 5163 y Fw(\)\()p Fr(x)p Fw(\))p Fr(:)p eop %%Page: 27 27 27 26 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(27)456 450 y Fw(T)-7 b(o)27 b(get)h(\(6.40\))f(w)n(e)g (used)h(that,)g(since)f Fr(g)1715 462 y Fo(s)1778 450 y Fw(is)h(di\013eren)n(tiable,)g(and)f(since)h(b)r(oth)g Fr(g)2988 420 y Fp(0)2985 471 y Fo(s)3048 450 y Fw(and)f Fr(m)3282 420 y Fp(0)3333 450 y Fw(are)456 550 y(o)r(dd)g(functions,) 526 675 y Fr(d)p 502 712 91 4 v 502 788 a(dx)616 731 y Fw(\()q Fr(J)f Fq(\003)18 b Fr(S)832 743 y Fo(s)867 731 y Fw(\()p Fr(m)p Fw(\)\))d(\()p Fr(x)p Fw(\))24 b(=)f(\()p Fr(J)j Fq(\003)18 b Fr(g)1481 697 y Fp(0)1478 751 y Fo(s)1514 731 y Fw(\))c(\()p Fr(x)p Fw(\))19 b(+)f Fr(e)1812 697 y Fp(\000)p Fo(s)1899 731 y Fw(\()p Fr(J)27 b Fq(\003)18 b Fr(m)2137 697 y Fp(0)2160 731 y Fw(\)\()p Fr(x)p Fw(\))24 b(=)f(\()p Fr(J)2493 743 y Fp(\000)2549 731 y Fr(g)2592 697 y Fp(0)2589 751 y Fo(s)2625 731 y Fw(\))14 b(\()p Fr(x)p Fw(\))19 b(+)f Fr(e)2923 697 y Fp(\000)p Fo(s)3010 731 y Fw(\()p Fr(J)3088 743 y Fp(\000)3145 731 y Fr(m)3218 697 y Fp(0)3241 731 y Fw(\)\()p Fr(x)p Fw(\))p Fr(;)456 901 y Fw(where)31 b(for)h(an)n(y)g(function)h Fr(u)f Fw(on)g Fn(R)1578 913 y Fs(+)1639 901 y Fw(,)h Fr(J)1741 913 y Fp(\000)1797 901 y Fr(u)p Fw(,)h(is)e(de\014ned)g(in)h(\(5.2\).) 51 b(By)32 b(iteration)f(of)39 b(\(6.40\))o(,)456 1001 y(calling)27 b(\()p Fr(t;)14 b(x)p Fw(\))23 b(=)g(\()p Fr(s)1079 1013 y Fs(0)1117 1001 y Fr(;)14 b(x)1201 1013 y Fs(0)1238 1001 y Fw(\),)28 b(w)n(e)f(get)672 1210 y Fr(g)715 1176 y Fp(0)712 1231 y Fo(t)741 1210 y Fw(\()p Fr(x)p Fw(\))84 b(=)1113 1106 y Fp(1)1086 1131 y Fm(X)1084 1307 y Fo(n)p Fs(=1)1223 1097 y Fm(Z)1306 1118 y Fo(s)1337 1126 y Fl(0)1269 1286 y Fs(0)1374 1210 y Fr(ds)1456 1222 y Fs(1)1507 1210 y Fr(:)14 b(:)g(:)1618 1097 y Fm(Z)1701 1118 y Fo(s)1732 1126 y Fi(n)p Fj(\000)p Fl(1)1664 1286 y Fs(0)1850 1210 y Fr(ds)1932 1222 y Fo(n)1084 1498 y Fq(\002)1176 1385 y Fm(Z)1259 1406 y Fs(+)p Fp(1)1222 1574 y Fs(0)1380 1498 y Fr(dx)1470 1510 y Fs(1)1522 1498 y Fr(:)g(:)g(:)1633 1385 y Fm(Z)1716 1406 y Fs(+)p Fp(1)1679 1574 y Fs(0)1837 1498 y Fr(dx)1927 1510 y Fo(n)1987 1394 y(n)p Fp(\000)p Fs(1)1997 1419 y Fm(Y)1989 1598 y Fo(k)q Fs(=1)2127 1498 y Fr(p)2169 1510 y Fo(s)2204 1498 y Fw(\()p Fr(x)2283 1510 y Fo(k)2325 1498 y Fw(\))p Fr(J)2403 1510 y Fp(\000)2459 1498 y Fw(\()p Fr(x)2538 1510 y Fo(k)2579 1498 y Fr(;)g(x)2663 1510 y Fo(k)q Fs(+1)2789 1498 y Fw(\))p Fr(z)2860 1510 y Fo(s)2891 1518 y Fi(n)2936 1498 y Fw(\()p Fr(x)3015 1510 y Fo(n)3061 1498 y Fw(\))p Fr(:)116 b Fw(\(6.41\))456 1720 y(F)-7 b(rom)34 b(the)h(fact)f(that)h Fr(J)1234 1732 y Fp(\000)1325 1720 y Fq(\025)f Fw(0)g(and)h Fr(m)1742 1690 y Fp(0)1765 1720 y Fw(\()p Fr(x)p Fw(\))h Fq(\024)e Fw(0)g(for)g Fr(x)h Fq(\025)g Fw(0)f(it)h(follo)n(ws)e(that)i Fr(z)3074 1732 y Fo(s)3144 1720 y Fw(is)f(a)h(non)456 1820 y(p)r(ositiv)n(e)24 b(function)i(on)e Fn(R)1249 1832 y Fs(+)1311 1820 y Fw(.)36 b(Since)25 b Fr(J)1630 1832 y Fp(\000)1686 1820 y Fw(\()p Fr(x;)14 b(y)s Fw(\))26 b(and)e Fr(p)2104 1832 y Fo(s)2140 1820 y Fw(\()p Fr(x)p Fw(\))i(are)e(non)h(negativ)n(e)e(for)i Fr(x;)14 b(y)26 b Fq(\025)d Fw(0,)i(w)n(e)456 1919 y(conclude)j(from)g(\(6.41\))f(that) i(also)e Fr(g)1626 1889 y Fp(0)1623 1940 y Fo(t)1681 1919 y Fw(is)h(a)g(non)g(p)r(ositiv)n(e)g(function)h(on)f Fn(R)2801 1931 y Fs(+)2863 1919 y Fw(.)39 b(Then)28 b Fr(S)3193 1931 y Fo(t)3223 1919 y Fw(\()p Fr(m)p Fw(\))h(is)456 2019 y(non)24 b(increasing)f(on)h Fn(R)1165 2031 y Fs(+)1251 2019 y Fw(b)r(ecause)g(sum)h(of)f(t)n(w)n(o)g(functions)h(with)g(this)g (prop)r(ert)n(y)-7 b(.)35 b(The)25 b(lemma)456 2119 y(is)i(pro)n(v)n (ed.)2574 b Fg(\003)456 2240 y Fz(Pro)s(of)38 b(of)h(Prop)s(osition)e (6.3.)53 b Fw(Since)34 b(the)g(di\013erence)f(b)r(et)n(w)n(een)h Fr(m)2717 2210 y Fp(\006)2717 2260 y Fo(s)2807 2240 y Fw(and)f Fr(w)3035 2210 y Fp(\006)3033 2260 y Fo(s)3125 2240 y Fw(is)h(only)f(a)456 2339 y(time)28 b(shift,)g(see)f(\(6.39\))o (,)h(it)g(is)g(enough)f(to)g(pro)n(v)n(e)f(the)i(prop)r(osition)f(for)g Fr(w)2798 2309 y Fp(\006)2796 2360 y Fo(s)2854 2339 y Fw(.)555 2439 y(W)-7 b(e)27 b(start)f(with)i(the)f(monotonicit)n(y)e (prop)r(ert)n(y)-7 b(.)36 b(W)-7 b(e)27 b(use)g(Theorem)e(6.2)h(and)h (Lemma)f(6.5.)456 2539 y(Th)n(us)i(the)i(\014rst)f(step)g(is)g(to)g (sho)n(w)g(that)g(for)g Fr(")g Fw(small)f(the)i(functions)f Fr( )2712 2551 y Fp(\006)p Fo(")2826 2539 y Fw(=)c Fr(q)d Fq(\006)d Fr("v)32 b Fw(are)d(non)456 2638 y(increasing)d(on)h Fn(R)1013 2650 y Fs(+)1074 2638 y Fw(.)37 b(T)-7 b(o)27 b(this)h(purp)r(ose)f(w)n(e)g(\014rst)h(notice)f(that,)h(b)n(y)g (de\014nition)f(\(see)h(\(2.59\))o(\),)972 2824 y Fr(v)1015 2790 y Fp(0)1038 2824 y Fw(\()p Fr(x)p Fw(\))c(=)f Fq(\000)1385 2768 y Fw(2)p Fr(\014)p 1336 2805 192 4 v 1336 2881 a Fw(1)18 b(+)g Fr(\025)1537 2824 y(q)s Fw(\()p Fr(x)p Fw(\))p Fr(q)1728 2790 y Fp(0)1752 2824 y Fw(\()p Fr(x)p Fw(\)\()p Fr(J)28 b Fq(\003)18 b Fr(v)s Fw(\)\()p Fr(x)p Fw(\))i(+)2403 2768 y Fr(p)p 2328 2805 V 2328 2881 a Fw(1)e(+)g Fr(\025)2530 2824 y Fw(\()p Fr(J)2616 2790 y Fp(0)2658 2824 y Fq(\003)g Fr(v)s Fw(\)\()p Fr(x)p Fw(\))p Fr(:)305 b Fw(\(6.42\))456 3001 y(By)28 b(\(2.63\))o(,)g(for)f (a)g(suitable)h(constan)n(t)f Fr(C)1750 3013 y Fs(8)1787 3001 y Fw(,)1240 3144 y(sup)1241 3211 y Fo(x>)p Fs(1)1379 3073 y Fm(\014)1379 3123 y(\014)1406 3144 y Fr(e)1445 3110 y Fo(\015)1480 3118 y Fi(v)1515 3110 y Fo(x)1557 3144 y Fr(v)1600 3110 y Fp(0)1624 3144 y Fw(\()p Fr(x)p Fw(\))1735 3073 y Fm(\014)1735 3123 y(\014)1787 3144 y Fq(\024)22 b Fr(C)1933 3156 y Fs(8)1985 3144 y Fw(sup)1986 3211 y Fo(x>)p Fs(1)2137 3144 y Fr(e)2176 3110 y Fo(\015)2211 3118 y Fi(v)2246 3110 y Fo(x)2288 3144 y Fr(v)s Fw(\()p Fr(x)p Fw(\))i Fr(<)f Fq(1)p Fr(:)572 b Fw(\(6.43\))456 3338 y(Then)27 b(from)h(\(2.55\))o(,)g(since)f Fr(\015)1378 3350 y Fo(v)1440 3338 y Fr(>)c(\015)5 b Fw(,)27 b(w)n(e)h(get)1645 3530 y(sup)1646 3597 y Fo(x>)p Fs(1)1784 3410 y Fm(\014)1784 3459 y(\014)1784 3509 y(\014)1784 3559 y(\014)1822 3474 y Fr(v)1865 3444 y Fp(0)1888 3474 y Fw(\()p Fr(x)p Fw(\))p 1822 3511 179 4 v 1824 3587 a Fr(q)1864 3563 y Fp(0)1887 3587 y Fw(\()p Fr(x)p Fw(\))2010 3410 y Fm(\014)2010 3459 y(\014)2010 3509 y(\014)2010 3559 y(\014)2061 3530 y Fr(<)23 b Fq(1)p Fr(:)977 b Fw(\(6.44\))456 3724 y(F)-7 b(or)28 b Fr(x)e Fq(2)g Fw([0)p Fr(;)14 b Fw(1],)29 b(since)g Fr(q)1224 3694 y Fp(0)1247 3724 y Fw(\(0\))d(=)f(0,)30 b(w)n(e)e(need)i(to)f(sho)n(w)f(that)h Fr(q)2414 3694 y Fp(00)2457 3724 y Fw(\(0\))d Fq(6)p Fw(=)f(0.)41 b(This)29 b(is)g(easily)g(seen)456 3823 y(b)n(y)e(noticing)g(that)h(since)f Fr(q)1312 3793 y Fp(0)1336 3823 y Fw(\(0\))c(=)g(0)k(and)g(b)r(oth)h Fr(J)2033 3793 y Fp(0)2084 3823 y Fw(and)g Fr(q)2286 3793 y Fp(0)2337 3823 y Fw(are)e(an)n(tisymmetric)h(functions,)1143 4025 y Fr(q)1183 3991 y Fp(00)1225 4025 y Fw(\(0\))c(=)g Fq(\000)p Fw(2)p Fr(\014)t Fw(\(1)17 b Fq(\000)i Fr(q)s Fw(\(0\))1921 3991 y Fs(2)1958 4025 y Fw(\))2004 3912 y Fm(Z)2087 3933 y Fs(1)2050 4101 y(0)2124 4025 y Fr(dy)e(J)2279 3991 y Fp(0)2302 4025 y Fw(\()p Fr(y)s Fw(\))p Fr(q)2450 3991 y Fp(0)2474 4025 y Fw(\()p Fr(y)s Fw(\))23 b Fr(<)g Fw(0)p Fr(:)474 b Fw(\(6.45\))456 4210 y(In)38 b(the)h(last)e (inequalit)n(y)h(w)n(e)g(used)g(that,)j(b)n(y)d(our)g(assumptions)f(on) h(the)h(function)f Fr(J)47 b Fw(and)456 4310 y(Prop)r(osition)25 b(2.10,)h Fr(J)1154 4279 y Fp(0)1178 4310 y Fw(\()p Fr(x)p Fw(\))p Fr(q)1329 4279 y Fp(0)1353 4310 y Fw(\()p Fr(x)p Fw(\))e Fr(>)f Fw(0)k(for)f Fr(x)e Fq(2)f Fw(\(0)p Fr(;)14 b Fw(1\).)37 b(F)-7 b(rom)27 b(\(6.44\))f(and)h(\(6.45\))g(w)n(e)g (then)h(get)1645 4502 y(sup)1622 4572 y Fo(x)p Fp(2)p Fh(R)1752 4580 y Fl(+)1807 4382 y Fm(\014)1807 4431 y(\014)1807 4481 y(\014)1807 4531 y(\014)1845 4446 y Fr(v)1888 4416 y Fp(0)1911 4446 y Fw(\()p Fr(x)p Fw(\))p 1845 4483 V 1847 4559 a Fr(q)1887 4535 y Fp(0)1910 4559 y Fw(\()p Fr(x)p Fw(\))2033 4382 y Fm(\014)2033 4431 y(\014)2033 4481 y(\014)2033 4531 y(\014)2084 4502 y Fr(<)23 b Fq(1)p Fr(:)954 b Fw(\(6.46\))456 4709 y(Lemma)38 b(6.5)h(and)g(\(6.46\))f (imply)i(that)f(for)f(an)n(y)h Fr(s)j Fq(\024)g Fw(0)d(there)f(is)h Fr(")2702 4721 y Fo(s)2780 4709 y Fq(2)k Fw(\(0)p Fr(;)14 b(\032)p Fw(])38 b(suc)n(h)h(that)456 4809 y Fq(f)p Fr(S)549 4824 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)823 4809 y Fw(\()p Fr( )909 4821 y Fp(\006)p Fo(")997 4809 y Fw(\))53 b(:)g Fr(")38 b Fq(2)i Fw(\(0)p Fr(;)14 b(")1480 4821 y Fo(s)1514 4809 y Fw(])p Fq(g)37 b Fw(is)g(a)g(sequence)f(of)h (non)g(increasing)f(functions)h(on)g Fn(R)3360 4821 y Fs(+)3421 4809 y Fw(.)456 4913 y(Hence)28 b(from)h(\(6.15\))e(the)i (same)f(prop)r(ert)n(y)g(holds)g(for)g Fr(w)2242 4883 y Fp(\006)2240 4933 y Fo(s)2299 4913 y Fw(,)h Fr(s)24 b Fq(\024)g Fw(0.)40 b(Then)28 b(the)h(monotonicit)n(y)456 5012 y(prop)r(ert)n(y)d(of)i Fr(w)951 4982 y Fp(\006)949 5033 y Fo(s)1035 5012 y Fw(for)f(all)g Fr(s)c Fq(2)h Fn(R)33 b Fw(follo)n(ws)27 b(from)g(Lemma)h(6.5.)555 5112 y(W)-7 b(e)21 b(are)e(left)i(with)g(the)g(b)r(ound)g(\(2.74\))o(.) 35 b(Since)20 b Fr(q)k Fw(solv)n(es)19 b(\(1.7\))h(and)g(it)h(is)f (strictly)g(decreasing)456 5212 y(on)33 b Fn(R)631 5182 y Fs(+)692 5212 y Fw(,)j(it)e(follo)n(ws)f(that)h Fr(m)1378 5177 y Fp(\000)1378 5237 y Fo(\014)s(;h)1514 5212 y Fr(<)f(q)s Fw(\()p Fr(x)p Fw(\))i Fr(<)e(m)1969 5177 y Fs(+)1969 5237 y Fo(\014)s(;h)2106 5212 y Fw(for)g(all)h Fr(x)f Fq(2)h Fn(R)p Fw(.)62 b(W)-7 b(e)34 b(also)e(recall)h(that)h Fr(q)p eop %%Page: 28 28 28 27 bop 456 251 a Fs(28)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fw(satis\014es)25 b(\(2.55\))o(.)36 b(Since)26 b Fr(v)k Fw(is)c(a)f(p)r(ositiv)n(e)h(function)g(whic)n(h)g (satis\014es)f(\(2.63\))g(with)i Fr(\015)3103 462 y Fo(v)3165 450 y Fr(>)c(\015)5 b Fw(,)26 b(w)n(e)456 550 y(conclude)h(that,)h(for) f(all)g Fr(")h Fw(small)f(enough,)1237 696 y Fr(m)1310 660 y Fp(\000)1310 721 y Fo(\014)s(;h)1437 696 y Fq(\024)22 b Fr( )1578 708 y Fp(\000)p Fo(")1666 696 y Fw(\()p Fr(x)p Fw(\))i Fr(<)e(q)s Fw(\()p Fr(x)p Fw(\))j Fr(<)d( )2205 708 y Fo(")2241 696 y Fw(\()p Fr(x)p Fw(\))i Fr(<)e(m)2536 660 y Fs(+)2536 721 y Fo(\014)s(;h)2640 696 y Fr(:)569 b Fw(\(6.47\))456 849 y(Then)27 b(\(2.74\))g(follo)n(ws)g(from)g (Theorem)g(6.2)g(and)g(the)h(Comparison)e(Theorem.)401 b Fg(\003)555 972 y Fw(W)-7 b(e)30 b(conclude)g(this)g(section)f(b)n(y) g(pro)n(ving)f(Prop)r(osition)g(6.6)h(b)r(elo)n(w,)h(whic)n(h)g(is)f(a) g(stronger)456 1071 y(v)n(ersion)d(of)i(Theorem)g(6.2,)g(since)g(w)n(e) f(sho)n(w)h(that)g(the)h(curv)n(es)e Fq(f)p Fr(w)2558 1041 y Fp(\006)2556 1092 y Fo(s)2614 1071 y Fq(g)h Fw(are)f(the)i (limits,)g(in)f(sup)456 1171 y(norm,)f(of)g(the)h(curv)n(es)f Fr(S)1239 1186 y Fo(\034)7 b Fs(\()p Fo(\032;\016)r Fs(\))1418 1171 y Fq(C)1462 1183 y Fo(\016)1526 1171 y Fw(where,)27 b(for)g(an)n(y)g Fr(\016)f(>)d Fw(0,)1531 1319 y Fq(C)1575 1331 y Fo(\016)1655 1272 y Fr(:)1634 1319 y Fw(=)f Fq(f)p Fr( )1817 1331 y Fo(")1889 1319 y Fw(:)37 b(0)23 b Fr(<)g(")f(<)h(\016) s Fq(g)13 b Fr(:)456 1475 y Fz(Prop)s(osition)30 b(6.6.)40 b Fk(L)l(et)29 b Fr(\016)d(>)d Fw(0)p Fk(,)30 b Fr(s)23 b Fq(\024)f Fw(0)p Fk(,)30 b(and)1737 1621 y Fr(\016)s Fw(\()p Fr(s)p Fw(\))1924 1574 y Fr(:)1903 1621 y Fw(=)23 b Fr(e)2030 1587 y Fo(\025s)2105 1621 y Fr(\016)o(:)1068 b Fw(\(6.48\))456 1767 y Fk(Then)1264 1878 y Fw(lim)1272 1932 y Fo(\016)r Fp(#)p Fs(0)1393 1878 y Fw(sup)1398 1948 y Fo(s)p Fp(\024)p Fs(0)1532 1807 y Fm(\015)1532 1857 y(\015)1578 1878 y Fr(S)1629 1893 y Fo(\034)7 b Fs(\()p Fo(\032;\016)r Fs(\))1823 1811 y Fm(\000)1861 1878 y Fr( )1915 1893 y Fp(\006)p Fo(\016)r Fs(\()p Fo(s)p Fs(\))2087 1811 y Fm(\001)2143 1878 y Fq(\000)18 b Fr(w)2287 1844 y Fp(\006)2285 1899 y Fo(s)2344 1807 y Fm(\015)2344 1857 y(\015)2390 1912 y Fp(1)2483 1878 y Fw(=)23 b(0)p Fr(:)596 b Fw(\(6.49\))456 2112 y Fz(Pro)s(of.)71 b Fw(Without)37 b(loss)e(of)h(generalit)n(y)f(w)n(e)h(restrict)f(to)h(the)h(case)e (with)i(the)f(plus)h(sign)e(in)456 2212 y(\(6.48\))28 b(and)i(\(6.49\))o(.)43 b(W)-7 b(e)30 b(need)g(to)f(sho)n(w)g(that)h (for)f(an)n(y)g Fr(\021)g(>)d Fw(0)j(there)h(is)f Fr(\016)2830 2224 y Fo(\021)2897 2212 y Fr(>)d Fw(0)j(so)g(that)h(for)456 2312 y(an)n(y)c Fr(\016)h(<)22 b(\016)800 2324 y Fo(\021)868 2312 y Fw(and)28 b Fr(s)23 b Fq(\024)f Fw(0)1423 2355 y Fm(\015)1423 2405 y(\015)1470 2426 y Fr(S)1521 2441 y Fo(\034)7 b Fs(\()p Fo(\032;\016)r Fs(\))1714 2358 y Fm(\000)1752 2426 y Fr( )1806 2441 y Fo(\016)r Fs(\()p Fo(s)p Fs(\))1926 2358 y Fm(\001)1982 2426 y Fq(\000)18 b Fr(w)2126 2391 y Fs(+)2124 2446 y Fo(s)2182 2355 y Fm(\015)2182 2405 y(\015)2228 2459 y Fp(1)2322 2426 y Fq(\024)k Fr(\021)s(:)756 b Fw(\(6.50\))456 2563 y(W)-7 b(e)28 b(appro)n(ximate)e Fr(w)1140 2533 y Fs(+)1138 2583 y Fo(s)1223 2563 y Fw(b)n(y)i Fr(S)1390 2575 y Fo(t)1419 2563 y Fw(\()p Fr( )1505 2575 y Fo(")1541 2563 y Fw(\))g(for)f (suitable)h(v)-5 b(alues)27 b(of)h Fr(")g Fw(and)f Fr(t)p Fw(:)38 b(giv)n(en)27 b Fr(s)c Fq(\024)g Fw(0)k(let)h Fr(")3294 2575 y Fs(0)3359 2563 y Fw(b)r(e)456 2662 y(suc)n(h)f(that)h (for)f Fr(")c Fq(2)g Fw(\(0)p Fr(;)14 b(")1240 2674 y Fs(0)1277 2662 y Fw(])1421 2759 y Fm(\015)1421 2809 y(\015)1467 2830 y Fr(S)1518 2845 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)1793 2830 y Fw(\()p Fr( )1879 2842 y Fo(")1915 2830 y Fw(\))18 b Fq(\000)g Fr(w)2109 2795 y Fs(+)2107 2850 y Fo(s)2165 2759 y Fm(\015)2165 2809 y(\015)2211 2863 y Fp(1)2305 2830 y Fq(\024)2402 2773 y Fr(\021)p 2402 2810 45 4 v 2403 2886 a Fw(2)2456 2830 y Fr(:)753 b Fw(\(6.51\))456 2999 y(F)-7 b(or)27 b Fr(\016)f(<)c(\032)28 b Fw(w)n(e)f(ha)n(v)n(e)1236 3145 y Fr(S)1287 3160 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)1562 3145 y Fw(\()q Fr( )1649 3157 y Fo(")1684 3145 y Fw(\))24 b(=)e Fr(S)1878 3160 y Fo(\034)7 b Fs(\()p Fo(\032;\016)r Fs(\))2072 3077 y Fm(\000)2110 3145 y Fr(S)2161 3160 y Fo(\034)g Fs(\()p Fo(\016)r Fs(\()p Fo(s)p Fs(\))p Fo(;")p Fs(\))2435 3145 y Fw(\()p Fr( )2521 3157 y Fo(")2556 3145 y Fw(\))2589 3077 y Fm(\001)2641 3145 y Fr(:)568 b Fw(\(6.52\))456 3291 y(By)28 b(\(6.13\))o(,)491 3391 y Fm(\015)491 3441 y(\015)537 3462 y Fr(S)588 3477 y Fo(\034)7 b Fs(\()p Fo(\016)r Fs(\()p Fo(s)p Fs(\))p Fo(;")p Fs(\))862 3462 y Fw(\()p Fr( )948 3474 y Fo(")983 3462 y Fw(\))19 b Fq(\000)f Fr( )1171 3477 y Fo(\016)r Fs(\()p Fo(s)p Fs(\))1291 3391 y Fm(\015)1291 3441 y(\015)1337 3496 y Fp(1)1430 3462 y Fw(=)1518 3366 y Fm(\015)1518 3416 y(\015)1518 3466 y(\015)1564 3462 y Fr(S)1615 3477 y Fo(\034)7 b Fs(\()p Fo(\016)r Fs(\()p Fo(s)p Fs(\))p Fo(;")p Fs(\))1889 3462 y Fw(\()p Fr( )1975 3474 y Fo(")2011 3462 y Fw(\))19 b Fq(\000)f Fr(q)j Fq(\000)d Fr(e)2325 3428 y Fo(\025\034)7 b Fs(\()p Fo(\016)r Fs(\()p Fo(s)p Fs(\))p Fo(;")p Fs(\))2624 3462 y Fr("v)2706 3366 y Fm(\015)2706 3416 y(\015)2706 3466 y(\015)2752 3520 y Fp(1)2846 3462 y Fw(=)22 b Fr(N)9 b Fq(k)p Fr(v)s Fq(k)3136 3428 y Fs(2)3136 3482 y Fp(1)3205 3462 y Fr(\016)s Fw(\()p Fr(s)p Fw(\))3348 3428 y Fs(2)3386 3462 y Fr(:)3232 3591 y Fw(\(6.53\))456 3690 y(W)-7 b(e)28 b(de\014ne)1280 3801 y Fr(D)1349 3813 y Fo(t)1422 3754 y Fr(:)1401 3801 y Fw(=)23 b Fr(S)1540 3813 y Fo(t)1583 3734 y Fm(\000)1621 3801 y Fr( )1675 3816 y Fo(\016)r Fs(\()p Fo(s)p Fs(\))1795 3734 y Fm(\001)1851 3801 y Fq(\000)18 b Fr(S)1985 3813 y Fo(t)2028 3734 y Fm(\000)2066 3801 y Fr(S)2117 3816 y Fo(\034)7 b Fs(\()p Fo(\016)r Fs(\()p Fo(s)p Fs(\))p Fo(;")p Fs(\))2391 3801 y Fw(\()p Fr( )2477 3813 y Fo(")2513 3801 y Fw(\))2545 3734 y Fm(\001)2597 3801 y Fr(;)612 b Fw(\(6.54\))456 3930 y(so)27 b(that)1051 3970 y Fm(\015)1051 4020 y(\015)1097 4041 y Fr(D)1166 4056 y Fo(\034)7 b Fs(\()p Fo(\032;\016)r Fs(\))1346 3970 y Fm(\015)1346 4020 y(\015)1392 4074 y Fp(1)1486 4041 y Fw(=)1573 3970 y Fm(\015)1573 4020 y(\015)1619 4041 y Fr(S)1670 4056 y Fo(\034)g Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)1945 4041 y Fw(\()q Fr( )2032 4053 y Fo(")2067 4041 y Fw(\))19 b Fq(\000)f Fr(S)2252 4056 y Fo(\034)7 b Fs(\()p Fo(\032;\016)r Fs(\))2445 3974 y Fm(\000)2484 4041 y Fr( )2538 4056 y Fo(\016)r Fs(\()p Fo(s)p Fs(\))2657 3974 y Fm(\001)2695 3970 y(\015)2695 4020 y(\015)2741 4074 y Fp(1)2826 4041 y Fr(:)383 b Fw(\(6.55\))456 4169 y(The)23 b(analysis)e(of)i Fr(D)1088 4181 y Fo(t)1140 4169 y Fw(is)g(iden)n(tical)f(to)h(that)g(of)g(\001)1979 4181 y Fo(t)2032 4169 y Fw(in)g(the)g(pro)r(of)f(of)h(Theorem)f(6.2.)35 b(In)23 b(fact,)h(b)n(y)456 4269 y(comparing)j(\(6.19\))o({\(6.20\))g (with)h(\(6.53\))o({\(6.54\))o(,)g(w)n(e)f(see)h(that)g Fr(D)2569 4281 y Fo(t)2626 4269 y Fw(satis\014es)f(the)h(conditions)456 4369 y(de\014ning)19 b(the)g(function)h(\001)1281 4381 y Fo(t)1330 4369 y Fw(when)f(the)g(parameter)f Fr(")h Fw(app)r(earing)f(in)h(\(6.19\))o({\(6.20\))f(is)h(replaced)456 4468 y(b)n(y)27 b Fr(\016)s Fw(\()p Fr(s)p Fw(\).)37 b(Then)28 b(the)g(b)r(ound)g(\(6.30\))f(applied)h(to)f Fr(D)2091 4480 y Fo(t)2148 4468 y Fw(b)r(ecomes)999 4625 y Fq(k)o Fr(D)1109 4637 y Fo(t)1138 4625 y Fq(k)1180 4650 y Fp(1)1273 4625 y Fq(\024)c Fr(C)1420 4637 y Fs(4)1457 4550 y Fm(p)p 1540 4550 144 4 v 75 x Fr(\016)s Fw(\()p Fr(s)p Fw(\))167 b Fq(8)14 b Fr(\016)25 b Fq(2)e Fw(\(0)p Fr(;)14 b(")2201 4637 y Fs(1)2238 4625 y Fw(])83 b Fq(8)14 b Fr(t)22 b Fq(\024)h Fr(\034)9 b Fw(\()p Fr(\032;)14 b(\016)s Fw(\()p Fr(s)p Fw(\)\))p Fr(;)456 4771 y Fw(whic)n(h)27 b(implies)1286 4818 y Fm(\015)1286 4868 y(\015)1332 4888 y Fr(D)1401 4903 y Fo(\034)7 b Fs(\()p Fo(\032;\016)r Fs(\))1581 4818 y Fm(\015)1581 4868 y(\015)1627 4922 y Fp(1)1720 4888 y Fq(\024)23 b Fr(C)1867 4900 y Fs(4)1904 4814 y Fq(p)p 1974 4814 41 4 v 1974 4888 a Fr(\016)169 b Fq(8)14 b Fr(\016)25 b Fq(2)e Fw(\(0)p Fr(;)14 b(")2531 4900 y Fs(1)2568 4888 y Fw(])p Fr(:)618 b Fw(\(6.56\))456 5022 y(W)-7 b(e)28 b(then)g(c)n(ho)r(ose)e Fr(\016)1088 5034 y Fo(\021)1151 5022 y Fq(2)e Fw(\(0)p Fr(;)14 b(")1380 5034 y Fs(1)1417 5022 y Fw(])27 b(so)g(small)g(that)1723 5192 y Fr(C)1782 5204 y Fs(4)1819 5121 y Fm(p)p 1902 5121 78 4 v 71 x Fr(\016)1939 5204 y Fo(\021)2003 5192 y Fq(\024)2100 5136 y Fr(\021)p 2100 5173 45 4 v 2101 5249 a Fw(2)2154 5192 y Fr(;)1055 b Fw(\(6.57\))p eop %%Page: 29 29 29 28 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(29)456 450 y Fw(and)27 b(w)n(e)g(ha)n(v)n(e)g(b)n(y)h (\(6.51\))f(and)g(\(6.55\){\(6.57\))f(that,)i(for)f(all)g Fr(\016)f(<)d(\016)2575 462 y Fo(\021)2615 450 y Fw(,)597 530 y Fm(\015)597 579 y(\015)644 600 y Fr(S)695 615 y Fo(\034)7 b Fs(\()p Fo(\032;\016)r Fs(\))888 533 y Fm(\000)926 600 y Fr( )980 615 y Fo(\016)r Fs(\()p Fo(s)p Fs(\))1100 533 y Fm(\001)1156 600 y Fq(\000)18 b Fr(w)1300 566 y Fs(+)1298 621 y Fo(s)1356 530 y Fm(\015)1356 579 y(\015)1402 634 y Fp(1)1542 600 y Fq(\024)1676 530 y Fm(\015)1676 579 y(\015)1722 600 y Fr(S)1773 615 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)2048 600 y Fw(\()p Fr( )2134 612 y Fo(")2170 600 y Fw(\))18 b Fq(\000)g Fr(w)2364 566 y Fs(+)2362 621 y Fo(s)2420 530 y Fm(\015)2420 579 y(\015)2466 634 y Fp(1)1648 738 y Fw(+)1740 667 y Fm(\015)1740 717 y(\015)1786 738 y Fr(S)1837 753 y Fo(\034)7 b Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)2112 738 y Fw(\()q Fr( )2199 750 y Fo(")2234 738 y Fw(\))19 b Fq(\000)f Fr(S)2419 753 y Fo(\034)7 b Fs(\()p Fo(\032;\016)r Fs(\))2613 670 y Fm(\000)2651 738 y Fr( )2705 753 y Fo(\016)r Fs(\()p Fo(s)p Fs(\))2824 670 y Fm(\001)2862 667 y(\015)2862 717 y(\015)2908 771 y Fp(1)3002 738 y Fq(\024)22 b Fr(\021)s(:)76 b Fw(\(6.58\))456 885 y(Prop)r(osition)26 b(6.6)g(is)i(th)n(us)f(pro)n (v)n(ed.)1811 b Fg(\003)1387 1114 y Fw(7.)41 b Fv(Global)30 b(str)n(ucture)i(of)g Fq(W)555 1263 y Fw(In)f(this)g(section)f(w)n(e)g (pro)n(v)n(e)f(Theorem)h(2.13.)45 b(T)-7 b(o)30 b(this)h(purp)r(ose)f (w)n(e)g(will)h(de\014ne)g(suitable)456 1363 y(functions)22 b Fr(Q)874 1332 y Fs(+)874 1383 y Fo(a)951 1363 y Fq(\024)h Fr(q)j Fq(\024)d Fr(Q)1256 1332 y Fp(\000)1256 1383 y Fo(a)1311 1363 y Fw(,)g Fr(a)f Fw(a)f(small)h(parameter,)f(whic)n(h)h (are)e(close)h(to)h Fr(q)s Fw(,)h(see)e(\(7.10\))g(b)r(elo)n(w.)456 1462 y(W)-7 b(e)26 b(shall)f(pro)n(v)n(e)f(that)j(the)f(functions)g Fr(m)1760 1432 y Fs(+)1760 1483 y Fo(s)1841 1462 y Fw(\(resp.)g Fr(m)2144 1432 y Fp(\000)2144 1483 y Fo(s)2200 1462 y Fw(\))g(at)f(a)h(certain)f(time)i Fr(s)e Fw(are)g(ab)r(o)n(v)n(e)g Fr(Q)3389 1432 y Fp(\000)3389 1483 y Fo(a)456 1562 y Fw(\(resp.)33 b(b)r(elo)n(w)h Fr(Q)1001 1532 y Fs(+)1001 1582 y Fo(a)1056 1562 y Fw(\).)57 b(Then,)36 b(b)n(y)e(the)h (Comparison)d(Theorem)h(it)i(is)f(enough)g(to)g(study)g(the)456 1661 y(ev)n(olution)d(of)g Fr(Q)984 1631 y Fp(\006)984 1682 y Fo(a)1040 1661 y Fw(.)49 b(Using)32 b(the)g(sp)r(ectral)f(prop)r (erties)g(of)h(the)g(linear)f(op)r(erator)f Fr(L)p Fw(,)i(w)n(e)f(sho)n (w)456 1761 y(that,)g(for)g(a)f(time)h(in)n(terv)-5 b(al)30 b Fr(T)1409 1773 y Fo(a)1477 1761 y Fq(\030)e(j)14 b Fw(log)g Fr(a)p Fq(j)p Fw(,)32 b(the)f(ev)n(olution)f Fr(S)2410 1773 y Fo(T)2449 1781 y Fi(a)2489 1761 y Fw(\()p Fr(Q)2587 1731 y Fs(+)2587 1782 y Fo(a)2642 1761 y Fw(\))i(\(resp.)e Fr(S)2991 1773 y Fo(T)3030 1781 y Fi(a)3071 1761 y Fw(\()p Fr(Q)3169 1731 y Fp(\000)3169 1782 y Fo(a)3224 1761 y Fw(\)\))i(can)456 1861 y(b)r(e)22 b(b)r(ounded)h(from)f(ab)r(o)n(v)n(e) f(\(resp.)h(b)r(elo)n(w\))h(b)n(y)f(the)g(same)g(functions)h Fr(Q)2676 1831 y Fs(+)2676 1881 y Fo(a)2753 1861 y Fw(\(resp.)f Fr(Q)3045 1831 y Fp(\000)3045 1881 y Fo(a)3101 1861 y Fw(\))h(suitably)456 1960 y(translated)29 b(in)h(space,)g(see)g (Theorem)f(7.2)g(b)r(elo)n(w.)44 b(By)30 b(the)h(Comparison)d(Theorem)h (w)n(e)h(can)456 2060 y(iterate)j(the)h(argumen)n(t,)g(th)n(us)g (getting)f(b)r(ounds)h(at)f(longer)g(times)g(whic)n(h,)j(com)n(bined)d (with)456 2160 y(general)f(prop)r(erties)h(of)g(the)i(\015o)n(w)e Fr(S)1625 2172 y Fo(t)1654 2160 y Fw(,)i(lead)f(to)f(the)h(desired)g (result,)h(Corollary)c(7.3)i(b)r(elo)n(w,)456 2259 y(from)27 b(whic)n(h)g(Theorem)g(2.13)f(will)i(follo)n(w.)555 2383 y(In)c(the)f(sequel)g(w)n(e)g(shall)g(need)h(a)e(more)h(re\014ned)g Fu(apriori)f Fw(b)r(ound)h(on)g(the)h(ev)n(olution)e(around)456 2482 y(the)28 b(critical)f(droplet,)g(whic)n(h)g(is)h(the)g(con)n(ten)n (t)f(of)h(the)g(follo)n(wing)e(lemma.)456 2639 y Fz(Lemma)j(7.1.)41 b Fk(Ther)l(e)31 b(is)f Fr(K)f(>)23 b Fw(0)30 b Fk(such)g(that)g(if)h Fr(u)2061 2651 y Fo(t)2134 2592 y Fr(:)2113 2639 y Fw(=)23 b Fr(S)2252 2651 y Fo(t)2281 2639 y Fw(\()p Fr(q)f Fw(+)c Fr(u)2503 2651 y Fs(0)2540 2639 y Fw(\))h Fq(\000)f Fr(q)s Fk(,)31 b Fr(u)2818 2651 y Fs(0)2878 2639 y Fq(2)24 b Fr(L)3014 2651 y Fp(1)3084 2639 y Fw(\()p Fn(R)p Fw(\))q Fk(,)36 b(then,)456 2738 y(for)30 b(al)t(l)h Fw(\()p Fr(t;)14 b(x)p Fw(\))24 b Fq(2)f Fn(R)1041 2750 y Fs(+)1121 2738 y Fq(\002)18 b Fn(R)p Fk(,)704 2873 y Fm(\014)704 2922 y(\014)732 2943 y Fr(u)780 2955 y Fo(t)808 2943 y Fw(\()p Fr(x)p Fw(\))i Fq(\000)e Fr(e)1061 2909 y Fo(Lt)1135 2943 y Fr(u)1183 2955 y Fs(0)1220 2943 y Fw(\()p Fr(x)p Fw(\))1331 2873 y Fm(\014)1331 2922 y(\014)1383 2943 y Fq(\024)k Fr(K)1561 2830 y Fm(Z)1644 2851 y Fo(t)1607 3019 y Fs(0)1673 2943 y Fr(ds)14 b(e)1808 2909 y Fo(L)p Fs(\()p Fo(t)p Fp(\000)p Fo(s)p Fs(\))2031 2876 y Fm(\000)2069 2943 y Fr(J)27 b Fq(\003)18 b Fr(e)2241 2909 y Fo(Ls)2321 2943 y Fr(u)2369 2955 y Fs(0)2406 2876 y Fm(\001)2444 2890 y Fs(2)2495 2943 y Fw(\()p Fr(x)p Fw(\))i(+)e Fr(K)6 b Fq(R)2856 2955 y Fo(t)2885 2943 y Fw([)p Fr(u)2956 2955 y Fp(\001)2979 2943 y Fw(])p Fr(;)249 b Fw(\(7.1\))456 3132 y Fk(wher)l(e)567 3283 y Fq(R)637 3295 y Fo(t)667 3283 y Fw([)p Fr(u)738 3295 y Fp(\001)761 3283 y Fw(])24 b(=)e Fr(e)934 3249 y Fo(\025t)1050 3283 y Fw(sup)1016 3357 y Fo(s)p Fp(2)p Fs([0)p Fo(;t)p Fs(])1222 3191 y Fm(n)1277 3283 y Fq(k)p Fr(u)1367 3295 y Fo(s)1402 3283 y Fq(k)1443 3242 y Fs(3)1443 3308 y Fp(1)1532 3283 y Fw(+)c Fq(k)o Fr(u)1704 3295 y Fo(s)1739 3283 y Fq(k)1781 3308 y Fp(1)1865 3213 y Fm(\015)1865 3263 y(\015)1911 3283 y Fr(u)1959 3295 y Fo(s)2013 3283 y Fq(\000)g Fr(e)2135 3249 y Fo(Ls)2215 3283 y Fr(u)2263 3295 y Fs(0)2300 3213 y Fm(\015)2300 3263 y(\015)2346 3317 y Fp(1)2435 3283 y Fw(+)2518 3213 y Fm(\015)2518 3263 y(\015)2564 3283 y Fr(u)2612 3295 y Fo(s)2666 3283 y Fq(\000)g Fr(e)2788 3249 y Fo(Ls)2868 3283 y Fr(u)2916 3295 y Fs(0)2953 3213 y Fm(\015)2953 3263 y(\015)2999 3230 y Fs(2)2999 3317 y Fp(1)3070 3191 y Fm(o)3139 3283 y Fr(:)112 b Fw(\(7.2\))456 3531 y Fz(Pro)s(of.)55 b Fw(W)-7 b(e)31 b(recall)g Fr(u)1190 3543 y Fo(t)1250 3531 y Fw(solv)n(es)f(\(6.2\).)48 b(Expanding)30 b(the)i(tanh)g(app)r(earing)e(in)h(\(2.2\))g(around)456 3631 y Fr(\014)t Fw(\()p Fr(J)c Fq(\003)18 b Fr(q)j Fw(+)d Fr(h)p Fw(\))28 b(and)f(using)h(that)g Fr(q)i Fw(solv)n(es)c(\(1.7\))i (w)n(e)f(ha)n(v)n(e)710 3831 y Fr(f)22 b Fw(\()q Fr(q)f Fw(+)d Fr(u)995 3843 y Fo(s)1030 3831 y Fw(\))h Fq(\000)f Fr(f)9 b Fw(\()p Fr(q)s Fw(\))19 b Fq(\000)f Fr(Lu)1525 3843 y Fo(s)1582 3831 y Fw(=)23 b(\010)14 b(\()p Fr(J)26 b Fq(\003)18 b Fr(u)1956 3843 y Fo(s)1991 3831 y Fw(\))2024 3789 y Fs(2)2079 3831 y Fw(+)2172 3775 y Fr(\014)2223 3744 y Fs(3)p 2172 3812 89 4 v 2184 3888 a Fw(3!)2285 3831 y(tanh)2451 3794 y Fp(000)2512 3831 y Fw(\()p Fr(\022)2583 3843 y Fo(s)2619 3831 y Fw(\))c(\()p Fr(J)27 b Fq(\003)18 b Fr(u)2878 3843 y Fo(s)2913 3831 y Fw(\))2945 3789 y Fs(3)2996 3831 y Fr(;)255 b Fw(\(7.3\))456 4001 y(where)1426 4113 y(\010\()p Fr(x)p Fw(\))1642 4066 y Fr(:)1622 4113 y Fw(=)22 b Fq(\000)p Fr(\014)1825 4079 y Fs(2)1862 4113 y Fr(q)s Fw(\()p Fr(x)p Fw(\))2027 4046 y Fm(\000)2066 4113 y Fw(1)c Fq(\000)g Fr(q)s Fw(\()p Fr(x)p Fw(\))2360 4079 y Fs(2)2399 4046 y Fm(\001)2450 4113 y Fr(;)801 b Fw(\(7.4\))456 4243 y(while)21 b Fr(\022)705 4255 y Fo(s)761 4243 y Fw(is)f(a)h(n)n(um)n(b)r(er)f(in)h(the)g(in)n(terv)-5 b(al)20 b(with)i(end-p)r(oin)n(ts)e Fr(\014)t Fw([)p Fr(J)13 b Fq(\003)5 b Fr(q)j Fw(+)d Fr(h)p Fw(])19 b(and)i Fr(\014)t Fw([)p Fr(J)13 b Fq(\003)5 b Fw(\()p Fr(q)j Fw(+)d Fr(u)3211 4255 y Fo(s)3244 4243 y Fw(\))g(+)g Fr(h)p Fw(].)456 4342 y(Then)27 b(w)n(e)h(rewrite)e(\(6.2\))i(as)1004 4547 y Fr(u)1052 4559 y Fo(t)1104 4547 y Fw(=)23 b Fr(e)1231 4513 y Fo(Lt)1305 4547 y Fr(u)1353 4559 y Fs(0)1408 4547 y Fw(+)1491 4434 y Fm(Z)1574 4455 y Fo(t)1538 4623 y Fs(0)1604 4547 y Fr(ds)14 b(e)1739 4513 y Fo(L)p Fs(\()p Fo(t)p Fp(\000)p Fo(s)p Fs(\))1962 4455 y Fm(h)2001 4547 y Fw(\010)2075 4480 y Fm(\000)2113 4547 y Fr(J)27 b Fq(\003)18 b Fr(e)2285 4513 y Fo(Ls)2365 4547 y Fr(u)2413 4559 y Fs(0)2450 4480 y Fm(\001)2488 4494 y Fs(2)2544 4547 y Fw(+)g Fr(R)q Fw([)p Fr(u)2762 4559 y Fo(s)2797 4547 y Fw(])2820 4455 y Fm(i)2873 4547 y Fr(;)378 b Fw(\(7.5\))456 4741 y(where,)27 b(using)h(\(7.3\),)808 4941 y Fr(R)q Fw([)p Fr(u)943 4953 y Fo(s)978 4941 y Fw(])23 b(=)g(\010)1186 4849 y Fm(h)1225 4941 y Fw(\()p Fr(J)k Fq(\003)18 b Fr(u)1438 4953 y Fo(s)1473 4941 y Fw(\))1505 4899 y Fs(2)1561 4941 y Fq(\000)1644 4874 y Fm(\000)1682 4941 y Fr(J)26 b Fq(\003)18 b Fr(e)1853 4907 y Fo(Ls)1934 4941 y Fr(u)1982 4953 y Fs(0)2019 4874 y Fm(\001)2057 4888 y Fs(2)2094 4849 y Fm(i)2152 4941 y Fw(+)2245 4885 y Fr(\014)2296 4855 y Fs(3)p 2245 4922 V 2257 4998 a Fw(3!)2357 4941 y(tanh)2523 4904 y Fp(000)2585 4941 y Fw(\()p Fr(\022)2656 4953 y Fo(s)2691 4941 y Fw(\))c(\()q Fr(J)26 b Fq(\003)18 b Fr(u)2950 4953 y Fo(s)2985 4941 y Fw(\))3017 4899 y Fs(3)3069 4941 y Fr(:)182 b Fw(\(7.6\))555 5116 y(Since)34 b(\010)g(is)g(a)g(b)r (ounded)g(function)h(on)e Fn(R)p Fw(,)42 b(the)34 b(\014rst)g(in)n (tegral)f(on)h(the)g(r.h.s.)g(of)40 b(\(7.5\))34 b(is)456 5216 y(b)r(ounded)28 b(b)n(y)f(the)h(\014rst)f(term)h(on)f(the)h (r.h.s.)f(of)34 b(\(7.1\))28 b(with)g Fr(K)g Fw(=)23 b Fq(k)p Fw(\010)p Fq(k)2709 5228 y Fp(1)2778 5216 y Fw(.)p eop %%Page: 30 30 30 29 bop 456 251 a Fs(30)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)555 450 y Fw(W)e(e)28 b(next)g(rewrite)f(the)h(square)e(brac)n (k)n(et)g(on)i(the)g(r.h.s.)f(of)34 b(\(7.6\))27 b(as)476 612 y(\()p Fr(J)g Fq(\003)18 b Fr(u)689 624 y Fo(s)724 612 y Fw(\))756 570 y Fs(2)812 612 y Fq(\000)895 544 y Fm(\000)933 612 y Fr(J)26 b Fq(\003)18 b Fr(e)1104 577 y Fo(Ls)1185 612 y Fr(u)1233 624 y Fs(0)1270 544 y Fm(\001)1308 558 y Fs(2)1368 612 y Fw(=)1456 544 y Fm(\002)1490 612 y Fr(J)27 b Fq(\003)18 b Fw(\()p Fr(u)1703 624 y Fo(s)1757 612 y Fq(\000)g Fr(e)1879 577 y Fo(Ls)1959 612 y Fr(u)2007 624 y Fs(0)2044 612 y Fw(\))2076 544 y Fm(\003)2111 558 y Fs(2)2167 612 y Fw(+)g(2)2306 544 y Fm(\000)2343 612 y Fr(J)27 b Fq(\003)17 b Fr(e)2514 577 y Fo(Ls)2595 612 y Fr(u)2643 624 y Fs(0)2680 544 y Fm(\001)d(\002)2766 612 y Fr(J)27 b Fq(\003)18 b Fw(\()p Fr(u)2979 624 y Fo(s)3033 612 y Fq(\000)g Fr(e)3155 577 y Fo(Ls)3235 612 y Fr(u)3283 624 y Fs(0)3320 612 y Fw(\))3352 544 y Fm(\003)3401 612 y Fr(:)456 760 y Fw(Then)24 b(using)f(tanh)1048 723 y Fp(000)1134 760 y Fw(is)h(b)r(ounded)g(and)g Fr(J)32 b Fw(has)24 b(compact)f(supp)r(ort,)i(from)f(\(2.64\))f(w)n(e)g(ha)n(v) n(e,)h(for)456 859 y(an)n(y)i Fr(K)34 b Fw(large)26 b(enough,)1330 933 y Fm(\015)1330 983 y(\015)1330 1033 y(\015)1330 1083 y(\015)1376 941 y(Z)1459 961 y Fo(t)1423 1129 y Fs(0)1489 1054 y Fr(ds)14 b(e)1624 1019 y Fo(L)p Fs(\()p Fo(t)p Fp(\000)p Fo(s)p Fs(\))1833 1054 y Fr(R)q Fw([)p Fr(u)1968 1066 y Fo(s)2003 1054 y Fw(])2026 933 y Fm(\015)2026 983 y(\015)2026 1033 y(\015)2026 1083 y(\015)2072 1137 y Fp(1)2165 1054 y Fq(\024)23 b Fr(K)6 b Fq(R)2400 1066 y Fo(t)2429 1054 y Fw([)p Fr(u)2500 1066 y Fp(\001)2524 1054 y Fw(])p Fr(:)456 1244 y Fw(The)27 b(lemma)h(is)f(pro)n(v)n(ed.) 2136 b Fg(\003)456 1364 y Fk(Warning:)35 b Fw(F)-7 b(or)24 b(the)g(rest)f(of)h(the)g(section)g(w)n(e)f(shall)g(denote)h(b)n(y)g Fr(C)30 b Fw(a)23 b(generic)g(constan)n(t)g(whose)456 1464 y(n)n(umerical)28 b(v)-5 b(alue)30 b(ma)n(y)f(c)n(hange)f(from)h (line)h(to)f(line.)43 b(F)-7 b(rom)29 b(the)h(statemen)n(ts)g(it)g (will)f(app)r(ear)456 1564 y(clear)d(on)h(whic)n(h)h(parameters)e(it)i (dep)r(ends)g(on.)555 1684 y(Let)g Fr(\015)k Fw(and)c Fr(\025)g Fw(b)r(e)g(as)f(in)h(\(2.54\))e(and)i(\(2.61\))f(resp)r (ectiv)n(ely)-7 b(.)36 b(W)-7 b(e)28 b(\014x)f Fr(\016)k Fw(and)d Fr(R)3001 1696 y Fs(0)3065 1684 y Fw(suc)n(h)g(that)1329 1864 y(0)23 b Fr(<)g(\016)j(<)1642 1808 y Fw(1)p 1642 1845 42 4 v 1642 1921 a(8)1694 1864 y Fr(;)1907 1808 y Fw(3)p 1907 1845 V 1907 1921 a(2)1981 1864 y Fr(<)d(\015)5 b(R)2180 1876 y Fs(0)2240 1864 y Fr(<)23 b Fw(2)17 b Fq(\000)i Fw(4)p Fr(\016)o(;)702 b Fw(\(7.7\))456 2033 y(and)27 b(w)n(e)g(set,)h(for)f(an)n(y)g Fr(a)c Fq(2)g Fw(\(0)p Fr(;)14 b Fw(1],)993 2218 y Fr(T)1042 2230 y Fo(a)1126 2171 y Fr(:)1105 2218 y Fw(=)1207 2161 y Fr(\016)p 1203 2199 49 4 v 1203 2275 a(\025)1261 2218 y Fq(j)g Fw(log)g Fr(a)p Fq(j)p Fr(;)180 b(R)1752 2230 y Fo(a)1836 2171 y Fr(:)1815 2218 y Fw(=)23 b Fr(R)1966 2230 y Fs(0)2003 2218 y Fq(j)14 b Fw(log)g Fr(a)p Fq(j)p Fr(;)180 b Fw(\001)2500 2230 y Fo(a)2584 2171 y Fr(:)2563 2218 y Fw(=)23 b Fr(a)2695 2183 y Fs(1)p Fp(\000)p Fo(\016)r(=)p Fs(2)2884 2218 y Fr(:)367 b Fw(\(7.8\))456 2393 y(Recalling)25 b(\(2.61\))o(,)g (\(2.62\),)g(and)g(\(2.68\))o(,)g(there)g(exists)f(an)2305 2371 y(\026)2304 2393 y Fr(h)f Fq(2)g Fw(\(0)p Fr(;)14 b(h)2612 2363 y Fp(\003)2650 2393 y Fw(],)25 b Fr(h)2769 2363 y Fp(\003)2832 2393 y Fw(as)f(in)h(Prop)r(osition)456 2493 y(\(2.10\))o(,)j(suc)n(h)f(that)969 2677 y(\()p Fr(\015)1044 2689 y Fo(v)1102 2677 y Fq(\000)18 b Fr(\015)5 b Fw(\))p Fr(R)1328 2689 y Fs(0)1388 2677 y Fq(\024)1487 2621 y Fr(\016)p 1486 2658 42 4 v 1486 2734 a Fw(4)1648 2677 y(and)111 b Fr(\016)s(\025)1981 2643 y Fp(\000)p Fs(1)2070 2677 y Fr(!)26 b(>)d Fw(3)248 b Fq(8)14 b Fr(h)22 b Fq(2)h Fw([0)p Fr(;)2838 2655 y Fw(\026)2837 2677 y Fr(h)p Fw(])p Fr(:)343 b Fw(\(7.9\))555 2842 y(W)-7 b(e)28 b(de\014ne)g(the)g(symmetric)f(functions)1085 3003 y Fr(Q)1151 2969 y Fp(\006)1151 3024 y Fo(a)1207 3003 y Fw(\()p Fr(x)p Fw(\))1363 2956 y Fr(:)1342 3003 y Fw(=)c Fr(q)1470 2969 y Fp(\006)1467 3024 y Fo(a)1526 3003 y Fw(\()p Fr(x)p Fw(\))p Fz(1)1685 3018 y Fp(j)p Fo(x)p Fp(j\024)p Fo(R)1865 3026 y Fi(a)1923 3003 y Fw(+)2006 2911 y Fm(h)2046 3003 y Fr(m)2119 2968 y Fp(\000)2119 3028 y Fo(\014)s(;h)2240 3003 y Fq(\006)18 b Fr(a)2367 2969 y Fs(3)p Fo(=)p Fs(2)2472 2911 y Fm(i)2525 3003 y Fz(1)2572 3018 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)2752 3026 y Fi(a)2792 3003 y Fr(;)417 b Fw(\(7.10\))456 3165 y(where)728 3306 y Fr(q)768 3271 y Fs(+)765 3326 y Fo(a)824 3306 y Fw(\()p Fr(x)p Fw(\))979 3259 y Fr(:)959 3306 y Fw(=)22 b Fr(q)s Fw(\()p Fq(j)p Fr(x)p Fq(j)e Fw(+)e Fr(a)p Fw(\))p Fr(;)180 b(q)1633 3271 y Fp(\000)1630 3326 y Fo(a)1689 3306 y Fw(\()p Fr(x)p Fw(\))1845 3259 y Fr(:)1824 3306 y Fw(=)23 b Fr(q)s Fw(\(0\))p Fz(1)2105 3321 y Fp(j)p Fo(x)p Fp(j\024)p Fo(a)2293 3306 y Fw(+)18 b Fr(q)s Fw(\()p Fq(j)p Fr(x)p Fq(j)h(\000)f Fr(a)p Fw(\))p Fz(1)2767 3321 y Fp(j)p Fo(x)p Fp(j)p Fo(>a)2936 3306 y Fr(:)273 b Fw(\(7.11\))456 3447 y(The)27 b(main)h(result)f(of)h(this)g(section)f (is)g(the)h(con)n(ten)n(t)f(of)h(the)g(follo)n(wing)f(theorem.)456 3600 y Fz(Theorem)38 b(7.2.)43 b Fk(L)l(et)36 b Fr(h)e Fq(2)g Fw([0)p Fr(;)1485 3578 y Fw(\026)1485 3600 y Fr(h)o Fw(])i Fk(with)1778 3578 y Fw(\026)1777 3600 y Fr(h)g Fk(as)g(in)42 b Fw(\(7.9\))p Fk(.)57 b(Then)36 b(ther)l(e)g(is)g Fr(a)2914 3612 y Fs(0)2985 3600 y Fq(2)e Fw(\(0)p Fr(;)14 b Fw(1])35 b Fk(such)456 3700 y(that,)30 b(for)g(any)h Fr(a)22 b Fq(2)i Fw(\(0)p Fr(;)14 b(a)1243 3712 y Fs(0)1280 3700 y Fw(])p Fk(,)1439 3841 y Fr(S)1490 3853 y Fo(T)1529 3861 y Fi(a)1584 3774 y Fm(\000)1622 3841 y Fr(Q)1688 3806 y Fs(+)1688 3861 y Fo(a)1743 3774 y Fm(\001)1794 3841 y Fw(\()p Fr(x)p Fw(\))24 b Fq(\024)f Fr(Q)2083 3806 y Fs(+)2083 3861 y Fo(a)2138 3841 y Fw(\()p Fr(x)c Fw(+)f(\001)2388 3853 y Fo(a)2428 3841 y Fw(\))772 b(\(7.12\))456 3982 y Fk(and)1427 4088 y Fr(S)1478 4100 y Fo(T)1517 4108 y Fi(a)1571 4021 y Fm(\000)1609 4088 y Fr(Q)1675 4054 y Fp(\000)1675 4109 y Fo(a)1731 4021 y Fm(\001)1783 4088 y Fw(\()p Fr(x)p Fw(\))24 b Fq(\025)f Fr(Q)2072 4054 y Fp(\000)2072 4109 y Fo(a)2127 4088 y Fw(\()p Fr(x)c Fq(\000)f Fw(\001)2377 4100 y Fo(a)2418 4088 y Fw(\))p Fr(;)759 b Fw(\(7.13\))456 4212 y Fk(with)30 b Fr(T)685 4224 y Fo(a)754 4212 y Fk(and)h Fw(\001)985 4224 y Fo(a)1055 4212 y Fk(as)f(in)36 b Fw(\(7.8\))p Fk(.)456 4386 y Fz(Pro)s(of.)41 b Fw(F)-7 b(rom)25 b(Prop)r(osition)g(2.10)f(there)i(is)g(a)f(constan)n (t)g Fr(c)h Fw(suc)n(h)g(that)g Fq(j)p Fr(q)2773 4356 y Fp(00)2816 4386 y Fw(\()p Fr(x)p Fw(\))p Fq(j)e(\024)f Fr(c)p Fq(j)p Fr(q)3161 4356 y Fp(0)3184 4386 y Fw(\()p Fr(x)p Fw(\))p Fq(j)k Fw(for)456 4485 y(an)n(y)k Fq(j)p Fr(x)p Fq(j)g(\025)f Fw(1.)50 b(Then,)33 b(b)n(y)f(expanding)f(to)h (second)f(order)g Fr(q)2359 4455 y Fp(\006)2356 4506 y Fo(a)2415 4485 y Fw(\()p Fr(x)p Fw(\))i(around)e Fr(q)s Fw(\()p Fr(x)p Fw(\))i(for)f Fq(j)p Fr(x)p Fq(j)f(\025)f Fw(1,)456 4585 y(and)d(using)f Fr(q)873 4555 y Fs(+)870 4606 y Fo(a)929 4585 y Fw(\()p Fr(x)p Fw(\))e Fq(\024)e Fr(q)s Fw(\()p Fr(x)p Fw(\))i Fq(\024)f Fr(q)1454 4555 y Fp(\000)1451 4606 y Fo(a)1510 4585 y Fw(\()p Fr(x)p Fw(\))28 b(for)f(all)g Fr(x)c Fq(2)h Fn(R)33 b Fw(\(see)27 b(\(7.11\))o(\))h(w)n(e)f(ha)n(v)n(e,)f(for)g(an)n(y)h Fr(a)g Fw(small)456 4685 y(enough,)660 4842 y Fr(q)700 4808 y Fs(+)697 4863 y Fo(a)755 4842 y Fw(\()p Fr(x)p Fw(\))d Fq(\024)f Fr(q)s Fw(\()p Fr(x)p Fw(\))c(+)1241 4786 y Fr(a)p 1241 4823 44 4 v 1242 4899 a Fw(2)1295 4842 y Fr(q)1335 4808 y Fp(0)1358 4842 y Fw(\()p Fq(j)p Fr(x)p Fq(j)p Fw(\))p Fz(1)1564 4857 y Fp(j)p Fo(x)p Fp(j\025)p Fs(1)1730 4842 y Fr(;)180 b(q)1973 4808 y Fp(\000)1970 4863 y Fo(a)2029 4842 y Fw(\()p Fr(x)p Fw(\))25 b Fq(\025)d Fr(q)s Fw(\()p Fr(x)p Fw(\))e Fq(\000)2516 4786 y Fr(a)p 2516 4823 V 2517 4899 a Fw(2)2570 4842 y Fr(q)2610 4808 y Fp(0)2633 4842 y Fw(\()p Fq(j)p Fr(x)p Fq(j)p Fw(\))p Fz(1)2839 4857 y Fp(j)p Fo(x)p Fp(j\025)p Fs(1)3005 4842 y Fr(:)204 b Fw(\(7.14\))456 5012 y(Observing)26 b Fr(q)891 4981 y Fp(0)914 5012 y Fw(\()p Fq(j)p Fr(x)p Fq(j)p Fw(\))e(=)f Fq(\000j)p Fr(q)1311 4981 y Fp(0)1334 5012 y Fw(\()p Fr(x)p Fw(\))p Fq(j)29 b Fw(for)e(an)n(y)f Fr(x)e Fq(2)f Fn(R)p Fw(,)34 b(if)28 b(w)n(e)f(de\014ne)1552 5192 y Fr(')p Fw(\()p Fr(x)p Fw(\))1762 5145 y Fr(:)1741 5192 y Fw(=)1839 5136 y(1)p 1839 5173 42 4 v 1839 5249 a(2)1890 5192 y Fq(j)p Fr(q)1953 5158 y Fp(0)1976 5192 y Fw(\()p Fr(x)p Fw(\))p Fq(j)p Fz(1)2159 5207 y Fp(j)p Fo(x)p Fp(j\025)p Fs(1)2325 5192 y Fr(;)884 b Fw(\(7.15\))p eop %%Page: 31 31 31 30 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(31)456 450 y Fw(from)27 b(\(7.10\))g(and)g(\(7.14\))g(w)n (e)g(obtain:)719 622 y Fr(Q)785 588 y Fs(+)785 643 y Fo(a)840 622 y Fw(\()p Fr(x)p Fw(\))d Fq(\024)e Fr(q)s Fw(\()p Fr(x)p Fw(\))e Fq(\000)e Fr(a')p Fw(\()p Fr(x)p Fw(\))i(+)1628 530 y Fm(h)1667 622 y Fr(m)1740 587 y Fp(\000)1740 647 y Fo(\014)s(;h)1861 622 y Fq(\000)f Fr(q)s Fw(\()p Fr(x)p Fw(\))g(+)f Fr(a)2242 588 y Fs(3)p Fo(=)p Fs(2)2365 622 y Fw(+)g Fr(a')p Fw(\()p Fr(x)p Fw(\))2657 530 y Fm(i)2711 622 y Fz(1)2759 637 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)2939 645 y Fi(a)2978 622 y Fr(;)231 b Fw(\(7.16\))732 805 y Fr(Q)798 771 y Fp(\000)798 825 y Fo(a)853 805 y Fw(\()p Fr(x)p Fw(\))25 b Fq(\025)d Fr(q)s Fw(\()p Fr(x)p Fw(\))e(+)e Fr(a')p Fw(\()p Fr(x)p Fw(\))h(+)1641 713 y Fm(h)1681 805 y Fr(m)1754 769 y Fp(\000)1754 830 y Fo(\014)s(;h)1875 805 y Fq(\000)f Fr(q)s Fw(\()p Fr(x)p Fw(\))i Fq(\000)e Fr(a)2256 771 y Fs(3)p Fo(=)p Fs(2)2379 805 y Fq(\000)g Fr(a')p Fw(\()p Fr(x)p Fw(\))2671 713 y Fm(i)2725 805 y Fz(1)2773 820 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)2953 828 y Fi(a)2992 805 y Fr(:)217 b Fw(\(7.17\))456 977 y(Moreo)n(v)n(er,)25 b(from)i(\(2.55\))g(and)g(\(7.7\),)g(for)g(an)n(y)g Fr(a)h Fw(small)f(enough,)1095 1163 y Fq(j)p Fr(m)1191 1128 y Fp(\000)1191 1188 y Fo(\014)s(;h)1312 1163 y Fq(\000)18 b Fr(q)s Fw(\()p Fr(x)p Fw(\))p Fq(j)i Fw(+)e Fr(a)p Fq(j)p Fr(')p Fw(\()p Fr(x)p Fw(\))p Fq(j)24 b(\024)2049 1107 y Fw(1)p 2049 1144 42 4 v 2049 1220 a(2)2101 1163 y Fr(a)2145 1129 y Fs(3)p Fo(=)p Fs(2)2415 1163 y Fq(8)14 b(j)p Fr(x)p Fq(j)22 b Fr(>)h(R)2742 1175 y Fo(a)2782 1163 y Fr(;)456 1334 y Fw(so)k(that,)g(if)i(w)n(e)e(de\014ne)1340 1480 y Fr(U)1406 1445 y Fp(\006)1397 1502 y Fs(0)1461 1480 y Fw(\()p Fr(x)p Fw(\))1617 1433 y Fr(:)1597 1480 y Fw(=)22 b Fq(\007)p Fr(a')p Fw(\()p Fr(x)p Fw(\))d Fq(\006)2070 1424 y Fw(3)p 2070 1461 V 2070 1537 a(2)2122 1480 y Fr(a)2166 1446 y Fs(3)p Fo(=)p Fs(2)2270 1480 y Fz(1)2318 1495 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)2498 1503 y Fi(a)2537 1480 y Fr(;)672 b Fw(\(7.18\))456 1638 y(from)27 b(\(7.16\))g(and)g(\(7.17\))g(w)n(e)g(get,)h(for)f(an)n(y)g Fr(a)g Fw(small)g(enough,)1006 1786 y Fr(Q)1072 1752 y Fs(+)1072 1806 y Fo(a)1127 1786 y Fw(\()p Fr(x)p Fw(\))d Fq(\024)f Fr(q)s Fw(\()p Fr(x)p Fw(\))c(+)f Fr(U)1669 1750 y Fs(+)1660 1808 y(0)1724 1786 y Fw(\()p Fr(x)p Fw(\))p Fr(;)181 b(Q)2105 1752 y Fp(\000)2105 1806 y Fo(a)2161 1786 y Fw(\()p Fr(x)p Fw(\))24 b Fq(\025)e Fr(q)s Fw(\()p Fr(x)p Fw(\))e(+)e Fr(U)2703 1750 y Fp(\000)2694 1808 y Fs(0)2759 1786 y Fw(\()p Fr(x)p Fw(\))p Fr(:)339 b Fw(\(7.19\))555 1935 y(W)-7 b(e)31 b(shall)g(no)n(w)f(obtain)g(go)r (o)r(d)g(b)r(ounds)h(on)g Fr(S)2004 1947 y Fo(T)2043 1955 y Fi(a)2097 1868 y Fm(\000)2135 1935 y Fr(q)22 b Fw(+)c Fr(U)2343 1900 y Fp(\006)2334 1958 y Fs(0)2398 1868 y Fm(\001)2436 1935 y Fw(.)47 b(W)-7 b(e)31 b(can)f(apply)h(Lemma) f(7.1)456 2044 y(to)d Fr(U)623 2009 y Fp(\006)614 2065 y Fo(t)723 1997 y Fr(:)702 2044 y Fw(=)c Fr(S)841 2056 y Fo(t)884 1977 y Fm(\000)922 2044 y Fr(q)e Fw(+)d Fr(U)1129 2009 y Fp(\006)1120 2066 y Fs(0)1185 1977 y Fm(\001)1241 2044 y Fq(\000)g Fr(q)s Fw(,)28 b(getting:)748 2187 y Fm(\014)748 2237 y(\014)776 2257 y Fr(U)842 2222 y Fp(\006)833 2278 y Fo(t)916 2257 y Fq(\000)18 b Fr(e)1038 2223 y Fo(Lt)1112 2257 y Fr(U)1178 2222 y Fp(\006)1169 2280 y Fs(0)1234 2187 y Fm(\014)1234 2237 y(\014)1285 2257 y Fq(\024)k Fr(K)1463 2144 y Fm(Z)1546 2165 y Fo(t)1509 2333 y Fs(0)1575 2257 y Fr(ds)14 b(e)1710 2223 y Fo(L)p Fs(\()p Fo(t)p Fp(\000)p Fo(s)p Fs(\))1933 2190 y Fm(\000)1971 2257 y Fr(J)27 b Fq(\003)18 b Fr(e)2143 2223 y Fo(Ls)2224 2257 y Fr(U)2290 2222 y Fp(\006)2281 2280 y Fs(0)2345 2190 y Fm(\001)2383 2204 y Fs(2)2439 2257 y Fw(+)g Fr(K)6 b Fq(R)2669 2269 y Fo(t)2712 2190 y Fm(\002)2747 2257 y Fr(U)2813 2223 y Fp(\006)2804 2278 y(\001)2868 2190 y Fm(\003)2917 2257 y Fr(:)292 b Fw(\(7.20\))456 2460 y(W)-7 b(e)28 b(will)f(use)h(\(7.20\))f(to)g(estimate)h Fr(U)1639 2425 y Fp(\006)1630 2485 y Fo(T)1669 2493 y Fi(a)1709 2460 y Fw(,)g(b)n(y)f(analyzing)f(separately)g(the)i(v)-5 b(arious)27 b(terms.)456 2597 y Fk(Estimate)j(on)g Fr(e)963 2566 y Fo(LT)1048 2574 y Fi(a)1087 2597 y Fr(U)1153 2561 y Fp(\006)1144 2619 y Fs(0)1209 2597 y Fw(.)37 b(Since)28 b Fr(e)1525 2566 y Fo(\025T)1603 2574 y Fi(a)1667 2597 y Fw(=)22 b Fr(a)1798 2566 y Fp(\000)p Fo(\016)1887 2597 y Fw(,)27 b(see)g(\(7.8\),)h(w)n(e)f(ha)n(v)n(e)673 2784 y Fr(e)712 2750 y Fo(LT)797 2758 y Fi(a)837 2784 y Fr(U)903 2749 y Fp(\006)894 2806 y Fs(0)982 2784 y Fw(=)c Fq(\007)p Fr(a)1179 2750 y Fs(1)p Fp(\000)p Fo(\016)1300 2784 y Fr(\031)s Fw(\()p Fr(')p Fw(\))p Fr(v)f Fq(\007)c Fr(ae)1696 2750 y Fo(LT)1781 2758 y Fi(a)1835 2784 y Fw([)p Fr(')h Fq(\000)f Fr(\031)s Fw(\()p Fr(')p Fw(\))p Fr(v)s Fw(])i Fq(\006)2361 2728 y Fw(1)p 2361 2765 V 2361 2841 a(2)2412 2784 y Fr(a)2456 2750 y Fs(3)p Fo(=)p Fs(2)2560 2784 y Fr(e)2599 2750 y Fo(LT)2684 2758 y Fi(a)2724 2784 y Fz(1)2772 2799 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)2952 2807 y Fi(a)2991 2784 y Fr(;)218 b Fw(\(7.21\))456 2959 y(where,)27 b(recalling)g(\(2.66\))g(and)g(\(7.15\),)1388 3154 y Fr(\031)s Fw(\()p Fr(')p Fw(\))e(=)1668 3041 y Fm(Z)1751 3062 y Fp(1)1714 3230 y Fs(1)1821 3154 y Fr(dx)1936 3098 y(v)s Fw(\()p Fr(x)p Fw(\))p 1936 3135 156 4 v 1937 3211 a Fr(p)p Fw(\()p Fr(x)p Fw(\))2115 3154 y Fq(j)p Fr(q)2178 3120 y Fp(0)2201 3154 y Fw(\()p Fr(x)p Fw(\))p Fq(j)g Fr(>)d Fw(0)p Fr(:)720 b Fw(\(7.22\))456 3350 y(F)-7 b(rom)27 b(the)h(sp)r(ectral)f(gap)g(prop)r(ert)n(y)g(\(2.67\))g(and)g (\(7.9\),)1009 3430 y Fm(\015)1009 3480 y(\015)1056 3501 y Fr(e)1095 3466 y Fo(LT)1180 3474 y Fi(a)1233 3501 y Fw([)p Fr(')19 b Fq(\000)f Fr(\031)s Fw(\()p Fr(')p Fw(\))p Fr(v)s Fw(])1648 3430 y Fm(\015)1648 3480 y(\015)1694 3534 y Fp(1)1787 3501 y Fq(\024)23 b Fr(e)1914 3466 y Fp(\000)p Fo(!)r(T)2049 3474 y Fi(a)2089 3501 y Fq(k)p Fr(')18 b Fq(\000)g Fr(\031)s Fw(\()p Fr(')p Fw(\))p Fr(v)s Fq(k)2539 3513 y Fp(1)2633 3501 y Fq(\024)23 b Fr(C)6 b(a)2830 3466 y Fs(3)2868 3501 y Fr(:)341 b Fw(\(7.23\))456 3648 y(Analogously)26 b(w)n(e)h(estimate:)574 3798 y Fr(e)613 3763 y Fo(LT)698 3771 y Fi(a)738 3798 y Fz(1)786 3813 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)966 3821 y Fi(a)1028 3798 y Fw(=)c Fr(a)1160 3763 y Fp(\000)p Fo(\016)1248 3798 y Fr(\031)1312 3730 y Fm(\000)1350 3798 y Fz(1)1398 3813 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)1578 3821 y Fi(a)1618 3730 y Fm(\001)1670 3798 y Fr(v)e Fw(+)d Fr(e)1853 3763 y Fo(LT)1938 3771 y Fi(a)1992 3730 y Fm(\002)2027 3798 y Fz(1)2074 3813 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)2254 3821 y Fi(a)2312 3798 y Fq(\000)g Fr(\031)2459 3730 y Fm(\000)2497 3798 y Fz(1)2545 3813 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)2725 3821 y Fi(a)2765 3730 y Fm(\001)2816 3798 y Fr(v)2859 3730 y Fm(\003)2917 3798 y Fq(\024)23 b Fr(C)6 b(a)3114 3763 y Fs(3)p Fo(=)p Fs(2)p Fp(\000)p Fo(\016)3303 3798 y Fr(;)3232 3898 y Fw(\(7.24\))456 3997 y(where)27 b(w)n(e)h(used)f Fr(\031)1073 3930 y Fm(\000)1111 3997 y Fz(1)1158 4012 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)1338 4020 y Fi(a)1378 3930 y Fm(\001)1439 3997 y Fq(\024)c Fr(C)6 b(a)1636 3967 y Fo(\015)1671 3975 y Fi(v)1707 3967 y Fo(R)1757 3975 y Fl(0)1822 3997 y Fw(with)28 b Fr(\015)2054 4009 y Fo(v)2094 3997 y Fr(R)2157 4009 y Fs(0)2217 3997 y Fr(>)c(\015)5 b(R)2417 4009 y Fs(0)2477 3997 y Fr(>)23 b Fw(3)p Fr(=)p Fw(2.)36 b(F)-7 b(rom)28 b(\(7.21\))o(,)g(\(7.23\))456 4097 y(and)f(\(7.24\))g(w)n(e)g(obtain:)1358 4178 y Fm(\014)1358 4228 y(\014)1385 4249 y Fr(e)1424 4214 y Fo(LT)1509 4222 y Fi(a)1549 4249 y Fr(U)1615 4213 y Fp(\006)1606 4271 y Fs(0)1690 4249 y Fq(\006)18 b Fr(a)1817 4214 y Fs(1)p Fp(\000)p Fo(\016)1938 4249 y Fr(\031)s Fw(\()p Fr(')p Fw(\))p Fr(v)2149 4178 y Fm(\014)2149 4228 y(\014)2201 4249 y Fq(\024)23 b Fr(C)6 b(a)2398 4214 y Fs(3)p Fp(\000)p Fo(\016)2519 4249 y Fr(:)690 b Fw(\(7.25\))456 4446 y Fk(Estimate)31 b(on)926 4379 y Fm(R)982 4399 y Fo(T)1021 4407 y Fi(a)966 4475 y Fs(0)1061 4446 y Fr(ds)14 b(e)1196 4416 y Fo(L)p Fs(\()p Fo(T)1307 4424 y Fi(a)1343 4416 y Fp(\000)p Fo(s)p Fs(\))1470 4378 y Fm(\000)1508 4446 y Fr(J)27 b Fq(\003)18 b Fr(e)1680 4416 y Fo(Ls)1760 4446 y Fr(U)1826 4410 y Fp(\006)1817 4468 y Fs(0)1882 4378 y Fm(\001)1920 4395 y Fs(2)1957 4446 y Fk(.)43 b Fw(Using)30 b(\(2.64\))e(with)i Fr(\020)i Fw(=)25 b(0)k(and)g(\(7.24\)) o(,)h(w)n(e)456 4545 y(get,)d(for)g(an)n(y)g Fr(a)g Fw(small)h(enough,) 612 4636 y Fm(Z)695 4657 y Fo(T)734 4665 y Fi(a)658 4825 y Fs(0)774 4749 y Fr(ds)14 b(e)909 4715 y Fo(L)p Fs(\()p Fo(T)1020 4723 y Fi(a)1056 4715 y Fp(\000)p Fo(s)p Fs(\))1183 4682 y Fm(\000)1221 4749 y Fr(J)27 b Fq(\003)18 b Fr(e)1393 4715 y Fo(Ls)1473 4749 y Fr(U)1539 4714 y Fp(\006)1530 4772 y Fs(0)1595 4682 y Fm(\001)1633 4696 y Fs(2)1693 4749 y Fq(\024)23 b Fr(C)6 b(a)1890 4715 y Fs(2)1941 4636 y Fm(Z)2024 4657 y Fo(T)2063 4665 y Fi(a)1987 4825 y Fs(0)2104 4749 y Fr(ds)14 b(e)2239 4715 y Fo(L)p Fs(\()p Fo(T)2350 4723 y Fi(a)2386 4715 y Fp(\000)p Fo(s)p Fs(\))2499 4657 y Fm(h)2552 4682 y(\000)2590 4749 y Fr(J)26 b Fq(\003)18 b Fr(e)2761 4715 y Fo(Ls)2842 4749 y Fr(')2896 4682 y Fm(\001)2934 4696 y Fs(2)963 4954 y Fw(+)32 b Fr(a)1118 4887 y Fm(\000)1156 4954 y Fr(J)26 b Fq(\003)18 b Fr(e)1327 4919 y Fo(Ls)1408 4954 y Fz(1)1456 4969 y Fp(j)p Fo(x)p Fp(j\025)p Fo(R)1636 4977 y Fi(a)1675 4887 y Fm(\001)1713 4900 y Fs(2)1769 4954 y Fw(+)1852 4890 y Fq(p)p 1921 4890 44 4 v 64 x Fr(a)1979 4887 y Fm(\000)2017 4954 y Fr(J)26 b Fq(\003)18 b Fr(e)2188 4919 y Fo(Ls)2269 4954 y Fr(')2323 4887 y Fm(\001)c(\000)2413 4954 y Fr(J)27 b Fq(\003)18 b Fr(e)2585 4919 y Fo(Ls)2665 4954 y Fz(1)2713 4969 y Fp(j)p Fo(x)p Fp(j\025)p Fo(R)2893 4977 y Fi(a)2932 4887 y Fm(\001)2984 4862 y(i)895 5173 y Fq(\024)23 b Fr(C)6 b(a)1092 5139 y Fs(3)p Fp(\000)p Fs(2)p Fo(\016)1265 5173 y Fw(+)18 b Fr(C)6 b(a)1457 5139 y Fs(2)1508 5060 y Fm(Z)1591 5081 y Fo(T)1630 5089 y Fi(a)1554 5249 y Fs(0)1671 5173 y Fr(ds)14 b(e)1806 5139 y Fo(L)p Fs(\()p Fo(T)1917 5147 y Fi(a)1952 5139 y Fp(\000)p Fo(s)p Fs(\))2066 5173 y Fw(\()p Fr(J)26 b Fq(\003)18 b Fr(e)2269 5139 y Fo(Ls)2350 5173 y Fr(')p Fw(\))2450 5106 y Fm(\002)2485 5173 y Fw(\()p Fr(J)27 b Fq(\003)18 b Fr(e)2689 5139 y Fo(Ls)2769 5173 y Fr(')p Fw(\))h(+)f Fr(a)3001 5139 y Fs(2)p Fp(\000)p Fo(\016)3123 5106 y Fm(\003)3171 5173 y Fr(:)38 b Fw(\(7.26\))p eop %%Page: 32 32 32 31 bop 456 251 a Fs(32)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fw(No)n(w,)42 b(recalling)d(the)h(de\014nitions)g (\(7.15\))f(and)h(\(2.58\))o(,)j(from)d(the)g(asymptotics)f(\(2.55\))g (it)456 550 y(follo)n(ws)29 b(that)j Fr(')c Fq(2)h Fr(X)1150 562 y Fo(\015)1193 550 y Fw(.)46 b(Since)32 b Fr(\015)h(<)28 b(\015)1695 562 y Fo(v)1765 550 y Fw(w)n(e)j(can)f(use)h(\(2.64\))f (with)h Fr(\020)k Fw(=)28 b Fr(\015)5 b Fw(.)47 b(Hence,)32 b(since)e Fr(J)456 649 y Fw(has)d(compact)g(supp)r(ort,)1428 691 y Fm(\014)1428 741 y(\014)1456 694 y(\000)1494 761 y Fr(J)f Fq(\003)18 b Fr(e)1665 727 y Fo(Ls)1746 761 y Fr(')1800 694 y Fm(\001)1852 761 y Fw(\()p Fr(x)p Fw(\))1963 691 y Fm(\014)1963 741 y(\014)2015 761 y Fq(\024)23 b Fr(C)6 b(e)2207 727 y Fo(\025s)p Fp(\000)p Fo(\015)t Fp(j)p Fo(x)p Fp(j)2449 761 y Fr(:)456 888 y Fw(Therefore,)26 b(b)n(y)h(applying)g(again)g(\(2.64\))g(with)h Fr(\020)h Fw(=)23 b Fr(\015)5 b Fw(,)1167 981 y Fm(Z)1250 1002 y Fo(T)1289 1010 y Fi(a)1213 1170 y Fs(0)1330 1094 y Fr(ds)14 b(e)1465 1060 y Fo(L)p Fs(\()p Fo(T)1576 1068 y Fi(a)1611 1060 y Fp(\000)p Fo(s)p Fs(\))1739 1027 y Fm(\000)1777 1094 y Fr(J)26 b Fq(\003)18 b Fr(e)1948 1060 y Fo(Ls)2029 1094 y Fr(')2083 1027 y Fm(\001)2121 1041 y Fs(2)2181 1094 y Fq(\024)23 b Fr(C)6 b(a)2378 1060 y Fp(\000)p Fs(2)p Fo(\016)2500 1094 y Fr(e)2539 1060 y Fp(\000)p Fo(\015)t Fp(j)p Fo(x)p Fp(j)2710 1094 y Fr(;)456 1285 y Fw(so)27 b(that)g(from)h(\(7.26\))o(,)g(for)f(all)g Fr(a)h Fw(small)f(enough,)929 1378 y Fm(Z)1012 1399 y Fo(T)1051 1407 y Fi(a)975 1567 y Fs(0)1091 1491 y Fr(ds)14 b(e)1226 1457 y Fo(L)p Fs(\()p Fo(t)p Fp(\000)p Fo(s)p Fs(\))1450 1424 y Fm(\000)1488 1491 y Fr(J)26 b Fq(\003)18 b Fr(e)1659 1457 y Fo(Ls)1740 1491 y Fr(U)1806 1456 y Fp(\006)1797 1513 y Fs(0)1862 1424 y Fm(\001)1900 1438 y Fs(2)1960 1491 y Fq(\024)23 b Fr(C)2127 1399 y Fm(\020)2176 1491 y Fr(a)2220 1457 y Fs(2)p Fp(\000)p Fs(2)p Fo(\016)2375 1491 y Fr(e)2414 1457 y Fp(\000)p Fo(\015)t Fp(j)p Fo(x)p Fp(j)2603 1491 y Fw(+)18 b Fr(a)2730 1457 y Fs(3)p Fp(\000)p Fs(2)p Fo(\016)2885 1399 y Fm(\021)2948 1491 y Fr(:)261 b Fw(\(7.27\))456 1714 y Fk(Estimate)24 b(on)g Fq(R)982 1726 y Fo(t)1025 1714 y Fw([)p Fr(U)1114 1684 y Fp(\006)1105 1734 y(\001)1170 1714 y Fw(])p Fk(.)35 b Fw(W)-7 b(e)22 b(use)f(Lemma)g(6.1)f(to)h(obtain)g Fu(apriori)e Fw(b)r(ounds.)35 b(Since)22 b Fq(k)p Fr(U)3190 1678 y Fp(\006)3181 1736 y Fs(0)3245 1714 y Fq(k)3287 1726 y Fp(1)3380 1714 y Fq(\024)456 1813 y Fr(C)6 b(a)p Fw(,)26 b(comparing)f(the)h (de\014nitions)g(\(6.5\))g(and)f(\(7.8\))h(and)g(using)f Fr(\016)h(<)d Fw(1)j(w)n(e)f(conclude)h(that)g(for)456 1914 y(all)h Fr(a)g Fw(small)h(enough)f Fr(\033)s Fw(\()p Fr(U)1294 1878 y Fp(\006)1285 1936 y Fs(0)1350 1914 y Fw(\))d Fr(>)e(T)1542 1926 y Fo(a)1582 1914 y Fw(.)37 b(Therefore)26 b(from)h(\(6.6\))h(and)f(\(6.7\))1295 2064 y Fq(k)p Fr(U)1403 2028 y Fp(\006)1394 2084 y Fo(t)1458 2064 y Fq(k)1500 2076 y Fp(1)1593 2064 y Fq(\024)c Fw(\(1)18 b(+)g Fr(C)1915 2076 y Fs(1)1952 2064 y Fw(\))p Fr(a)2028 2029 y Fs(1)p Fp(\000)p Fo(\016)2316 2064 y Fq(8)c Fr(t)22 b Fq(\024)h Fr(T)2566 2076 y Fo(a)3232 2064 y Fw(\(7.28\))456 2208 y(and)1208 2247 y Fm(\015)1208 2297 y(\015)1254 2317 y Fr(U)1320 2282 y Fp(\006)1311 2338 y Fo(t)1394 2317 y Fq(\000)18 b Fr(e)1516 2283 y Fo(Lt)1591 2317 y Fr(U)1657 2282 y Fp(\006)1648 2340 y Fs(0)1712 2247 y Fm(\015)1712 2297 y(\015)1759 2351 y Fp(1)1852 2317 y Fq(\024)23 b Fr(N)9 b(a)2060 2283 y Fs(2)p Fp(\000)p Fs(2)p Fo(\016)2380 2317 y Fq(8)14 b Fr(t)22 b Fq(\024)g Fr(T)2629 2329 y Fo(a)2669 2317 y Fr(:)540 b Fw(\(7.29\))456 2448 y(Recalling)28 b(\(7.2\))o(,)g(from)f(\(7.28\))g(and)h(\(7.29\))e (w)n(e)i(get)1574 2597 y Fq(R)1644 2609 y Fo(T)1683 2617 y Fi(a)1738 2530 y Fm(\002)1772 2597 y Fr(U)1838 2563 y Fp(\006)1829 2617 y(\001)1894 2530 y Fm(\003)1952 2597 y Fq(\024)22 b Fr(C)6 b(a)2148 2563 y Fs(3)p Fp(\000)p Fs(4)p Fo(\016)2303 2597 y Fr(:)906 b Fw(\(7.30\))555 2763 y(Collecting)34 b(\(7.20\))o(,)g(\(7.25\),)g(\(7.27\))o(,)h(and)e (\(7.30\))o(,)h(w)n(e)f(conclude)g(that,)h(for)f(an)n(y)f Fr(a)h Fw(small)456 2863 y(enough,)985 2922 y Fm(\014)985 2972 y(\014)1013 2993 y Fr(U)1079 2957 y Fp(\006)1070 3017 y Fo(T)1109 3025 y Fi(a)1149 2993 y Fw(\()p Fr(x)p Fw(\))20 b Fq(\006)e Fr(a)1407 2959 y Fs(1)p Fp(\000)p Fo(\016)1528 2993 y Fr(\031)s Fw(\()p Fr(')p Fw(\))p Fr(v)s Fw(\()p Fr(x)p Fw(\))1850 2922 y Fm(\014)1850 2972 y(\014)1903 2993 y Fq(\024)23 b Fr(C)2070 2901 y Fm(\020)2120 2993 y Fr(a)2164 2959 y Fs(2)p Fp(\000)p Fs(2)p Fo(\016)2318 2993 y Fr(e)2357 2959 y Fp(\000)p Fo(\015)t Fp(j)p Fo(x)p Fp(j)2547 2993 y Fw(+)18 b Fr(a)2674 2959 y Fs(3)p Fp(\000)p Fs(4)p Fo(\016)2828 2901 y Fm(\021)2892 2993 y Fr(:)317 b Fw(\(7.31\))456 3154 y(Therefore,)20 b(from)h(the)f(Comparison)f(Theorem)h(and)g(\(7.19\))o(,)i(recalling)e Fr(U)2776 3118 y Fp(\006)2767 3174 y Fo(t)2855 3154 y Fw(=)i Fr(S)2993 3166 y Fo(t)3036 3087 y Fm(\000)3074 3154 y Fr(q)g Fw(+)c Fr(U)3282 3118 y Fp(\006)3273 3176 y Fs(0)3338 3087 y Fm(\001)3380 3154 y Fq(\000)456 3254 y Fr(q)s Fw(,)27 b(w)n(e)h(\014nally)f(get)782 3418 y Fr(S)833 3430 y Fo(T)872 3438 y Fi(a)926 3351 y Fm(\000)964 3418 y Fr(Q)1030 3384 y Fs(+)1030 3439 y Fo(a)1085 3351 y Fm(\001)1137 3418 y Fw(\()p Fr(x)p Fw(\))d Fq(\024)f Fr(q)s Fw(\()p Fr(x)p Fw(\))c Fq(\000)f Fr(C)1692 3326 y Fm(h)1732 3418 y Fr(a)1776 3384 y Fs(1)p Fp(\000)p Fo(\016)1897 3418 y Fr(v)s Fw(\()p Fr(x)p Fw(\))i Fq(\000)e Fr(a)2198 3384 y Fs(2)p Fp(\000)p Fs(2)p Fo(\016)2352 3418 y Fr(e)2391 3384 y Fp(\000)p Fo(\015)t Fp(j)p Fo(x)p Fp(j)2581 3418 y Fq(\000)g Fr(a)2708 3384 y Fs(3)p Fp(\000)p Fs(4)p Fo(\016)2862 3326 y Fm(i)2915 3418 y Fr(;)294 b Fw(\(7.32\))795 3601 y Fr(S)846 3613 y Fo(T)885 3621 y Fi(a)939 3534 y Fm(\000)977 3601 y Fr(Q)1043 3567 y Fp(\000)1043 3621 y Fo(a)1099 3534 y Fm(\001)1151 3601 y Fw(\()p Fr(x)p Fw(\))24 b Fq(\025)f Fr(q)s Fw(\()p Fr(x)p Fw(\))c(+)f Fr(C)1706 3509 y Fm(h)1745 3601 y Fr(a)1789 3567 y Fs(1)p Fp(\000)p Fo(\016)1911 3601 y Fr(v)s Fw(\()p Fr(x)p Fw(\))h Fq(\000)f Fr(a)2211 3567 y Fs(2)p Fp(\000)p Fs(2)p Fo(\016)2366 3601 y Fr(e)2405 3567 y Fp(\000)p Fo(\015)t Fp(j)p Fo(x)p Fp(j)2595 3601 y Fq(\000)g Fr(a)2722 3567 y Fs(3)p Fp(\000)p Fs(4)p Fo(\016)2876 3509 y Fm(i)2929 3601 y Fr(:)280 b Fw(\(7.33\))555 3774 y(Next)35 b(w)n(e)f(shall)f(\014nd)i(go)r(o)r(d)f(b)r(ounds)g(on)g Fr(Q)1960 3744 y Fp(\006)1960 3794 y Fo(a)2016 3774 y Fw(\()p Fr(x)23 b Fq(\006)g Fw(\001)2275 3786 y Fo(a)2315 3774 y Fw(\).)57 b(W)-7 b(e)35 b(\014rst)f(observ)n(e)e(that,)37 b(since)456 3873 y(\001)525 3885 y Fo(a)588 3873 y Fq(\024)23 b Fw(1,)f Fq(j)p Fr(x)p Fq(j)h(\024)g Fr(R)1030 3885 y Fo(a)1076 3873 y Fq(\000)6 b Fw(1)20 b(implies)h Fq(j)p Fr(x)6 b Fw(+)g(\001)1700 3885 y Fo(a)1739 3873 y Fq(j)23 b(\024)g Fr(R)1936 3885 y Fo(a)1997 3873 y Fw(while)f Fq(j)p Fr(x)p Fq(j)h Fr(>)g(R)2475 3885 y Fo(a)2521 3873 y Fw(+)6 b(1)20 b(implies)h Fq(j)p Fr(x)6 b Fw(+)g(\001)3145 3885 y Fo(a)3184 3873 y Fq(j)23 b Fr(>)g(R)3381 3885 y Fo(a)3421 3873 y Fw(.)456 3976 y(Moreo)n(v)n(er,)28 b(from)j(\(2.55\))o(,)h(\(7.7\))o(,)g(and)f(\(7.11\))o(,)g Fq(j)p Fr(q)2033 3946 y Fp(\006)2030 3997 y Fo(a)2090 3976 y Fw(\()p Fr(x)21 b Fq(\006)f Fw(\001)2344 3988 y Fo(a)2384 3976 y Fw(\))h Fq(\000)f Fr(m)2595 3941 y Fp(\000)2595 4001 y Fo(\014)s(;h)2699 3976 y Fq(j)28 b(\024)g Fr(a)2887 3946 y Fs(3)p Fo(=)p Fs(2)3022 3976 y Fw(if)j Fr(R)3164 3988 y Fo(a)3225 3976 y Fq(\000)20 b Fw(1)28 b Fr(<)456 4083 y Fq(j)p Fr(x)p Fq(j)23 b(\024)g Fr(R)723 4095 y Fo(a)782 4083 y Fw(+)18 b(1)27 b(and)g Fr(a)h Fw(is)f(small)g(enough.)37 b(Hence:)701 4252 y Fr(Q)767 4218 y Fs(+)767 4273 y Fo(a)822 4252 y Fw(\()p Fr(x)19 b Fw(+)f(\001)1072 4264 y Fo(a)1113 4252 y Fw(\))23 b Fq(\025)g Fr(q)1296 4218 y Fs(+)1293 4273 y Fo(a)1351 4252 y Fw(\()p Fr(x)c Fw(+)f(\001)1601 4264 y Fo(a)1641 4252 y Fw(\))p Fz(1)1721 4267 y Fp(j)p Fo(x)p Fp(j\024)p Fo(R)1901 4275 y Fi(a)1937 4267 y Fs(+1)2043 4252 y Fw(+)2126 4160 y Fm(h)2166 4252 y Fr(m)2239 4217 y Fp(\000)2239 4277 y Fo(\014)s(;h)2360 4252 y Fw(+)g Fr(a)2487 4218 y Fs(3)p Fo(=)p Fs(2)2591 4160 y Fm(i)2645 4252 y Fz(1)2692 4267 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)2872 4275 y Fi(a)2908 4267 y Fs(+1)2996 4252 y Fr(;)713 4435 y(Q)779 4401 y Fp(\000)779 4456 y Fo(a)835 4435 y Fw(\()p Fr(x)h Fq(\000)f Fw(\001)1085 4447 y Fo(a)1126 4435 y Fw(\))23 b Fq(\024)g Fr(q)1309 4401 y Fp(\000)1306 4456 y Fo(a)1365 4435 y Fw(\()p Fr(x)c Fq(\000)f Fw(\001)1615 4447 y Fo(a)1655 4435 y Fw(\))p Fz(1)1735 4450 y Fp(j)p Fo(x)p Fp(j\024)p Fo(R)1915 4458 y Fi(a)1950 4450 y Fs(+1)2057 4435 y Fw(+)2140 4343 y Fm(h)2179 4435 y Fr(m)2252 4400 y Fp(\000)2252 4460 y Fo(\014)s(;h)2374 4435 y Fq(\000)g Fr(a)2501 4401 y Fs(3)p Fo(=)p Fs(2)2605 4343 y Fm(i)2658 4435 y Fz(1)2706 4450 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)2886 4458 y Fi(a)2921 4450 y Fs(+1)3010 4435 y Fr(:)456 4613 y Fw(No)n(w)28 b(w)n(e)g(notice)h Fr(q)1055 4583 y Fs(+)1052 4634 y Fo(a)1110 4613 y Fw(\()p Fr(x)20 b Fw(+)e(\001)1361 4625 y Fo(a)1402 4613 y Fw(\))25 b Fq(\025)f Fr(q)1588 4578 y Fs(+)1585 4638 y Fo(a)p Fs(+\001)1727 4646 y Fi(a)1767 4613 y Fw(\()p Fr(x)p Fw(\))30 b(and)e Fr(q)2110 4583 y Fp(\000)2107 4634 y Fo(a)2166 4613 y Fw(\()p Fr(x)20 b Fq(\000)f Fw(\001)2418 4625 y Fo(a)2458 4613 y Fw(\))25 b Fq(\024)g Fr(q)2645 4578 y Fp(\000)2642 4638 y Fo(a)p Fs(+\001)2784 4646 y Fi(a)2824 4613 y Fw(\()p Fr(x)p Fw(\))30 b(for)e(all)g Fr(x)d Fq(2)g Fn(R)p Fw(.)456 4717 y(Moreo)n(v)n(er,)f(since)j Fq(j)p Fr(q)1109 4687 y Fp(00)1151 4717 y Fw(\()p Fr(x)p Fw(\))p Fq(j)e(\024)d Fr(c)p Fq(j)p Fr(q)1496 4687 y Fp(0)1520 4717 y Fw(\()p Fr(x)p Fw(\))p Fq(j)28 b Fw(for)e Fq(j)p Fr(x)p Fq(j)e(\025)f Fw(1,)j(b)n(y)h(expanding)g(to)f(the)i(second)e(order)g(for)456 4817 y Fq(j)p Fr(x)p Fq(j)d Fr(>)g Fw(1)k(and)g(to)h(the)g(\014rst)f (one)g(for)h Fq(j)p Fr(x)p Fq(j)23 b(\024)g Fw(1,)k(w)n(e)g(get,)h(if)g Fr(a)f Fw(is)h(small)f(enough,)547 4962 y Fr(q)587 4926 y Fs(+)584 4986 y Fo(a)p Fs(+\001)726 4994 y Fi(a)766 4962 y Fw(\()p Fr(x)p Fw(\))d Fq(\025)f Fr(q)s Fw(\()p Fr(x)p Fw(\))d Fq(\000)e Fw(\()p Fr(a)g Fw(+)g(\001)1489 4974 y Fo(a)1530 4962 y Fw(\))p Fr( )s Fw(\()p Fr(x)p Fw(\))p Fr(;)181 b(q)1974 4926 y Fp(\000)1971 4986 y Fo(a)p Fs(+\001)2113 4994 y Fi(a)2153 4962 y Fw(\()p Fr(x)p Fw(\))24 b Fq(\024)f Fr(q)s Fw(\()p Fr(x)p Fw(\))c(+)f(\()p Fr(a)h Fw(+)f(\001)2876 4974 y Fo(a)2916 4962 y Fw(\))p Fr( )s Fw(\()p Fr(x)p Fw(\))p Fr(;)93 b Fw(\(7.34\))456 5106 y(where)1353 5216 y Fr( )s Fw(\()p Fr(x)p Fw(\))1565 5169 y Fr(:)1545 5216 y Fw(=)22 b(2)1688 5148 y Fm(\002)1722 5216 y Fq(j)p Fr(q)1785 5181 y Fp(0)1809 5216 y Fw(\()p Fr(x)p Fw(\))p Fq(j)d Fw(+)f Fq(k)p Fr(q)2127 5181 y Fp(0)2150 5216 y Fq(k)2192 5228 y Fp(1)2262 5216 y Fz(1)2310 5231 y Fp(j)p Fo(x)p Fp(j\024)p Fs(1)2476 5148 y Fm(\003)2524 5216 y Fr(;)685 b Fw(\(7.35\))p eop %%Page: 33 33 33 32 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(33)456 450 y Fw(hence)509 604 y Fr(Q)575 570 y Fs(+)575 624 y Fo(a)630 604 y Fw(\()p Fr(x)19 b Fw(+)f(\001)880 616 y Fo(a)921 604 y Fw(\))23 b Fq(\025)g Fw([)p Fr(q)s Fw(\()p Fr(x)p Fw(\))c Fq(\000)f Fw(\()p Fr(a)h Fw(+)f(\001)1587 616 y Fo(a)1627 604 y Fw(\))p Fr( )s Fw(\()p Fr(x)p Fw(\)])e Fz(1)1913 619 y Fp(j)p Fo(x)p Fp(j\024)p Fo(R)2093 627 y Fi(a)2129 619 y Fs(+1)2235 604 y Fw(+)2318 512 y Fm(h)2357 604 y Fr(m)2430 568 y Fp(\000)2430 629 y Fo(\014)s(;h)2552 604 y Fw(+)i Fr(a)2679 570 y Fs(3)p Fo(=)p Fs(2)2783 512 y Fm(i)2836 604 y Fz(1)2884 619 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)3064 627 y Fi(a)3099 619 y Fs(+1)3188 604 y Fr(;)j Fw(\(7.36\))522 786 y Fr(Q)588 752 y Fp(\000)588 807 y Fo(a)644 786 y Fw(\()p Fr(x)e Fq(\000)f Fw(\001)894 798 y Fo(a)935 786 y Fw(\))23 b Fq(\024)g Fw([)p Fr(q)s Fw(\()p Fr(x)p Fw(\))c(+)f(\()p Fr(a)h Fw(+)f(\001)1601 798 y Fo(a)1641 786 y Fw(\))p Fr( )s Fw(\()p Fr(x)p Fw(\)])d Fz(1)1927 801 y Fp(j)p Fo(x)p Fp(j\024)p Fo(R)2107 809 y Fi(a)2142 801 y Fs(+1)2249 786 y Fw(+)2332 694 y Fm(h)2371 786 y Fr(m)2444 751 y Fp(\000)2444 811 y Fo(\014)s(;h)2566 786 y Fq(\000)j Fr(a)2693 752 y Fs(3)p Fo(=)p Fs(2)2797 694 y Fm(i)2850 786 y Fz(1)2898 801 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)3078 809 y Fi(a)3113 801 y Fs(+1)3202 786 y Fr(:)7 b Fw(\(7.37\))555 956 y(W)-7 b(e)41 b(can)g(no)n(w)f(conclude)g(the)i(pro)r(of)e(of)h (the)g(theorem.)76 b(W)-7 b(e)41 b(consider)f(\014rst)g(the)h(case)456 1056 y Fq(j)p Fr(x)p Fq(j)28 b(\024)g Fr(R)733 1068 y Fo(a)793 1056 y Fw(+)20 b(1.)46 b(Since)30 b Fr(v)k Fw(is)d(strictly)f (p)r(ositiv)n(e)g(and)h(ob)r(eys)f(the)h(asymptotics)e(\(2.63\),)i(and) f Fr(q)456 1156 y Fw(satis\014es)c(\(2.55\),)h(from)h(\(7.9\))f(and)g (\(7.35\))g(w)n(e)g(ha)n(v)n(e)1276 1312 y Fr(v)s Fw(\()p Fr(x)p Fw(\))e Fq(\025)d Fr(C)6 b(a)1651 1278 y Fo(\016)r(=)p Fs(4)1755 1312 y Fr( )s Fw(\()p Fr(x)p Fw(\))167 b Fq(8)14 b(j)p Fr(x)p Fq(j)23 b(\024)f Fr(R)2417 1324 y Fo(a)2476 1312 y Fw(+)c(1)p Fr(:)608 b Fw(\(7.38\))456 1462 y(On)27 b(the)h(other)f(hand,)h(using)g(\(7.7\),)1052 1619 y Fr(a)1096 1584 y Fs(2)p Fp(\000)p Fs(2)p Fo(\016)1250 1619 y Fr(e)1289 1584 y Fp(\000)p Fo(\015)t Fp(j)p Fo(x)p Fp(j)1479 1619 y Fw(+)18 b Fr(a)1606 1584 y Fs(3)p Fp(\000)p Fs(4)p Fo(\016)1783 1619 y Fq(\024)23 b Fr(C)6 b(a )s Fw(\()p Fr(x)p Fw(\))167 b Fq(8)14 b(j)p Fr(x)p Fq(j)23 b(\024)f Fr(R)2642 1631 y Fo(a)2701 1619 y Fw(+)c(1)p Fr(:)456 1768 y Fw(Therefore,)26 b(for)h(an)n(y)g Fr(a)g Fw(small)h(enough,)624 1920 y Fr(a)668 1886 y Fs(1)p Fp(\000)p Fo(\016)789 1920 y Fr(v)s Fw(\()p Fr(x)p Fw(\))20 b Fq(\000)e Fr(a)1090 1886 y Fs(2)p Fp(\000)p Fs(2)p Fo(\016)1244 1920 y Fr(e)1283 1886 y Fp(\000)p Fo(\015)t Fp(j)p Fo(x)p Fp(j)1473 1920 y Fq(\000)g Fr(a)1600 1886 y Fs(3)p Fp(\000)p Fs(4)p Fo(\016)1777 1920 y Fq(\025)23 b Fr(C)6 b(a)1974 1886 y Fs(1)p Fp(\000)p Fs(3)p Fo(\016)r(=)p Fs(4)2196 1920 y Fr( )s Fw(\()p Fr(x)p Fw(\))167 b Fq(8)14 b(j)p Fr(x)p Fq(j)22 b(\024)h Fr(R)2858 1932 y Fo(a)2917 1920 y Fw(+)18 b(1)p Fr(:)167 b Fw(\(7.39\))456 2073 y(Since)27 b(\()p Fr(a)19 b Fw(+)f(\001)919 2085 y Fo(a)959 2073 y Fw(\))p Fr(a)1035 2043 y Fp(\000)p Fs(1+3)p Fo(\016)r(=)p Fs(4)1335 2073 y Fw(v)-5 b(anishes)27 b(as)g Fr(a)c Fq(#)g Fw(0,)k(\(7.12\))g(and)g(\(7.13\))g(for)g Fq(j)p Fr(x)p Fq(j)c(\024)g Fr(R)3022 2085 y Fo(a)3080 2073 y Fw(+)18 b(1)27 b(follo)n(w)456 2173 y(from)g(\(7.32\))o(,)h(\(7.33\))o(,)g (\(7.36\))o(,)g(\(7.37\))o(,)f(and)h(\(7.39\))o(.)555 2272 y(Finally)k(w)n(e)f(consider)f(the)i(case)e Fq(j)p Fr(x)p Fq(j)h Fr(>)e(R)1902 2284 y Fo(a)1963 2272 y Fw(+)20 b(1.)48 b(Using)32 b Fr(\015)2442 2284 y Fo(v)2481 2272 y Fr(R)2544 2284 y Fs(0)2611 2272 y Fr(>)d(\015)5 b(R)2816 2284 y Fs(0)2884 2272 y Fw(and)32 b(\(7.7\))o(,)h(from)456 2372 y(\(7.32\))26 b(and)i(\(7.33\))f(w)n(e)g(get)1134 2546 y(lim)1140 2601 y Fo(a)p Fp(#)p Fs(0)1263 2546 y Fr(a)1307 2512 y Fp(\000)p Fs(3)p Fo(=)p Fs(2)1564 2546 y Fw(sup)1477 2620 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)1657 2628 y Fi(a)1692 2620 y Fs(+1)1790 2451 y Fm(\014)1790 2501 y(\014)1790 2551 y(\014)1818 2546 y Fr(S)1869 2558 y Fo(T)1908 2566 y Fi(a)1962 2479 y Fm(\000)2000 2546 y Fr(Q)2066 2512 y Fp(\006)2066 2567 y Fo(a)2122 2479 y Fm(\001)2174 2546 y Fw(\()p Fr(x)p Fw(\))19 b Fq(\000)f Fr(m)2460 2511 y Fp(\000)2460 2571 y Fo(\014)s(;h)2563 2451 y Fm(\014)2563 2501 y(\014)2563 2551 y(\014)2614 2546 y Fw(=)23 b(0)p Fr(:)465 b Fw(\(7.40\))456 2763 y(Then)34 b(\(7.12\))g(and)h(\(7.13\))f(for)g Fq(j)p Fr(x)p Fq(j)h Fr(>)g(R)1767 2775 y Fo(a)1830 2763 y Fw(+)23 b(1)34 b(and)h Fr(a)g Fw(small)f(enough)g(follo)n(w)g(from)g(\(7.36\))o (,)456 2863 y(\(7.37\))o(,)28 b(and)f(\(7.40\))o(.)2265 b Fg(\003)456 3046 y Fz(Corollary)35 b(7.3.)41 b Fk(In)32 b(the)f(same)h(hyp)l(othesis)i(of)e(The)l(or)l(em)h(7.2,)h(ther)l(e)d (is)h Fr(a)2893 3058 y Fs(1)2957 3046 y Fq(2)27 b Fw(\(0)p Fr(;)14 b(a)3194 3058 y Fs(0)3231 3046 y Fw(])31 b Fk(such)456 3146 y(that,)f(for)g(any)h Fr(a)22 b Fq(2)i Fw(\(0)p Fr(;)14 b(a)1243 3158 y Fs(1)1280 3146 y Fw(])p Fk(,)1432 3296 y Fw(lim)1386 3346 y Fo(t)p Fp(!)p Fs(+)p Fp(1)1608 3225 y Fm(\015)1608 3275 y(\015)1654 3296 y Fr(S)1705 3308 y Fo(t)1748 3229 y Fm(\000)1786 3296 y Fr(Q)1852 3262 y Fs(+)1852 3317 y Fo(a)1907 3229 y Fm(\001)1963 3296 y Fq(\000)k Fr(m)2119 3260 y Fp(\000)2119 3321 y Fo(\014)s(;h)2223 3225 y Fm(\015)2223 3275 y(\015)2269 3329 y Fp(1)2362 3296 y Fw(=)23 b(0)p Fr(;)717 b Fw(\(7.41\))1302 3506 y(lim)1256 3556 y Fo(t)p Fp(!)p Fs(+)p Fp(1)1478 3506 y Fr(S)1529 3518 y Fo(t)1572 3439 y Fm(\000)1610 3506 y Fr(Q)1676 3471 y Fp(\000)1676 3526 y Fo(a)1732 3439 y Fm(\001)1784 3506 y Fw(\()p Fr(x)p Fw(\))24 b(=)e Fr(m)2079 3470 y Fs(+)2079 3531 y Fo(\014)s(;h)2352 3506 y Fq(8)14 b Fr(x)23 b Fq(2)g Fn(R)p Fr(:)594 b Fw(\(7.42\))555 3695 y(T)-7 b(o)27 b(pro)n(v)n(e)f(the)i(ab)r(o)n(v)n(e)f(Corollary)e (w)n(e)i(need)h(the)g(follo)n(wing)e(Barrier)g(Lemma.)456 3853 y Fz(Lemma)34 b(7.4.)43 b(\(The)37 b(Barrier)h(Lemma,)e Fw([8)o(])p Fz(\))e Fk(Ther)l(e)h(ar)l(e)f Fr(V)53 b Fk(and)35 b Fr(C)2819 3823 y Fp(\003)2891 3853 y Fk(p)l(ositive)g(so)f (that)456 3953 y(if)h Fr(m;)29 b Fw(~)-57 b Fr(m)30 b Fq(2)h Fr(L)897 3965 y Fp(1)967 3953 y Fw(\()p Fn(R)p Fw(;)14 b([)p Fq(\000)p Fw(1)p Fr(;)g Fw(1]\))39 b Fk(and,)d(for)f (some)g Fr(x)1986 3965 y Fs(0)2054 3953 y Fq(2)c Fn(R)40 b Fk(and)35 b Fr(T)42 b(>)30 b Fw(0)p Fk(,)35 b Fr(m)p Fw(\()p Fr(x)p Fw(\))d(=)46 b(~)-58 b Fr(m)p Fw(\()p Fr(x)p Fw(\))35 b Fk(for)g(al)t(l)456 4052 y Fq(j)p Fr(x)19 b Fq(\000)f Fr(x)675 4064 y Fs(0)712 4052 y Fq(j)23 b(\024)g Fr(V)c(T)12 b Fk(,)29 b(then:)1319 4205 y Fq(j)p Fr(S)1393 4217 y Fo(t)1423 4205 y Fw(\()p Fr(m)p Fw(\)\()p Fr(x)1639 4217 y Fs(0)1677 4205 y Fw(\))19 b Fq(\000)f Fr(S)1862 4217 y Fo(t)1891 4205 y Fw(\()e(~)-58 b Fr(m)p Fw(\)\()p Fr(x)2107 4217 y Fs(0)2145 4205 y Fw(\))p Fq(j)24 b(\024)e Fr(C)2376 4171 y Fp(\003)2415 4205 y Fr(e)2454 4171 y Fp(\000)p Fo(T)2558 4205 y Fr(:)456 4388 y Fz(Pro)s(of)37 b(of)h(Corollary)g(7.3.)51 b Fw(W)-7 b(e)33 b(\014rst)g(pro)n(v)n(e)e (\(7.41\))o(.)53 b(By)33 b(\(7.12\))f(and)h(the)g(Comparison)456 4488 y(Theorem,)26 b(for)i(an)n(y)e(in)n(teger)h Fr(n)p Fw(,)1165 4638 y Fr(S)1216 4650 y Fo(nT)1296 4658 y Fi(a)1350 4570 y Fm(\000)1388 4638 y Fr(Q)1454 4603 y Fs(+)1454 4658 y Fo(a)1509 4570 y Fm(\001)1561 4638 y Fw(\()p Fr(x)p Fw(\))d Fq(\024)f Fr(Q)1850 4603 y Fs(+)1850 4658 y Fo(a)1904 4638 y Fw(\()p Fr(x)d Fw(+)e Fr(n)p Fw(\001)2205 4650 y Fo(a)2245 4638 y Fw(\))166 b Fq(8)14 b Fr(x)23 b Fq(2)g Fn(R)p Fr(:)456 4787 y Fw(F)-7 b(rom)32 b(\(7.10\))f(the)i(function)g (on)f(the)h(r.h.s.)f(of)g(the)h(ab)r(o)n(v)n(e)e(inequalit)n(y)h(is)g (iden)n(tically)g(equal)456 4890 y(to)26 b Fr(m)629 4854 y Fp(\000)629 4915 y Fo(\014)s(;h)748 4890 y Fw(+)16 b Fr(a)873 4860 y Fs(3)p Fo(=)p Fs(2)1003 4890 y Fw(for)26 b(all)g Fr(x)e(>)e(R)1464 4902 y Fo(a)1520 4890 y Fq(\000)16 b Fr(n)p Fw(\001)1720 4902 y Fo(a)1760 4890 y Fw(.)37 b(On)26 b(the)g(other)g(hand)g Fr(S)2571 4902 y Fo(nT)2651 4910 y Fi(a)2706 4890 y Fw(\()p Fr(Q)2804 4860 y Fs(+)2804 4910 y Fo(a)2859 4890 y Fw(\))h(is)f(a)g(symmetric)456 4992 y(function)i(for)f(all)g(in)n(teger)g Fr(n)p Fw(,)g(then)982 5179 y Fr(S)1033 5191 y Fo(nT)1113 5199 y Fi(a)1168 5112 y Fm(\000)1206 5179 y Fr(Q)1272 5145 y Fs(+)1272 5200 y Fo(a)1326 5112 y Fm(\001)1378 5179 y Fw(\()p Fr(x)p Fw(\))d Fq(\024)f Fr(m)1674 5144 y Fp(\000)1674 5204 y Fo(\014)s(;h)1796 5179 y Fw(+)18 b Fr(a)1923 5145 y Fs(3)p Fo(=)p Fs(2)2193 5179 y Fq(8)c Fr(x)23 b Fq(2)g Fn(R)89 b Fq(8)14 b Fr(n)22 b(>)2778 5123 y(R)2841 5135 y Fo(a)p 2775 5160 110 4 v 2775 5236 a Fw(\001)2844 5248 y Fo(a)2895 5179 y Fr(:)p eop %%Page: 34 34 34 33 bop 456 251 a Fs(34)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fw(Using)26 b(again)f(the)h(Comparison)f(Theorem)h (and)g(recalling)f Fr(m)2424 415 y Fp(\000)2424 475 y Fo(\014)s(;h)2550 450 y Fq(\024)e Fr(Q)2704 420 y Fs(+)2704 471 y Fo(a)2758 450 y Fw(,)k(w)n(e)f(conclude)g(that:)765 658 y Fr(m)838 623 y Fp(\000)838 683 y Fo(\014)s(;h)965 658 y Fq(\024)c Fr(S)1103 670 y Fo(t)1146 591 y Fm(\000)1184 658 y Fr(Q)1250 624 y Fs(+)1250 679 y Fo(a)1305 591 y Fm(\001)1366 658 y Fq(\024)h Fr(S)1505 670 y Fo(t)1534 591 y Fm(\000)1572 658 y Fr(m)1645 623 y Fp(\000)1645 683 y Fo(\014)s(;h)1767 658 y Fw(+)18 b Fr(a)1894 624 y Fs(3)p Fo(=)p Fs(2)1998 591 y Fm(\001)2202 658 y Fq(8)c Fr(t)22 b(>)2403 541 y Fm(\022)2464 658 y Fw(1)c(+)2620 602 y Fr(R)2683 614 y Fo(a)p 2617 639 110 4 v 2617 715 a Fw(\001)2686 727 y Fo(a)2736 541 y Fm(\023)2811 658 y Fr(T)2860 670 y Fo(a)2900 658 y Fr(:)309 b Fw(\(7.43\))555 863 y(W)-7 b(e)28 b(no)n(w)f(observ)n(e)f(that)i Fr(S)1398 875 y Fo(t)1427 795 y Fm(\000)1465 863 y Fr(m)1538 827 y Fp(\000)1538 888 y Fo(\014)s(;h)1660 863 y Fw(+)18 b Fr(a)1787 833 y Fs(3)p Fo(=)p Fs(2)1891 795 y Fm(\001)1957 863 y Fw(solv)n(es)26 b(the)i(homogeneous)e(equation:)1324 1013 y Fr(d\032)p Fw(\()p Fr(t)p Fw(\))p 1324 1050 181 4 v 1378 1126 a Fr(dt)1538 1069 y Fw(=)d Fq(\000)p Fr(\032)p Fw(\()p Fr(t)p Fw(\))18 b(+)g(tanh)p Fq(f)p Fr(\014)t Fw([)p Fr(\032)p Fw(\()p Fr(t)p Fw(\))h(+)f Fr(h)p Fw(])p Fq(g)p Fr(;)646 b Fw(\(7.44\))456 1251 y(with)42 b(initial)h(datum)g Fr(m)1265 1216 y Fp(\000)1265 1276 y Fo(\014)s(;h)1396 1251 y Fw(+)28 b Fr(a)1533 1221 y Fs(3)p Fo(=)p Fs(2)1637 1251 y Fw(.)81 b(Since)43 b(the)f(free)g(energy)f(densit)n(y)j(\(1.1\)) e(is)g(a)g(Ly)n(a-)456 1358 y(puno)n(v)25 b(functional)h(for)f(the)h (\015o)n(w)f(ev)n(olution)g(\(7.44\))o(,)i(it)f(is)g(easy)e(to)i(v)n (erify)f(that)h(the)g(in)n(terv)-5 b(als)456 1459 y(\()p Fq(\000)p Fw(1)p Fr(;)14 b(m)705 1429 y Fs(0)705 1482 y Fo(\014)s(;h)807 1459 y Fw(\))32 b(and)f(\()p Fr(m)1141 1429 y Fs(0)1141 1482 y Fo(\014)s(;h)1245 1459 y Fr(;)14 b Fw(1\))31 b(are)f(basins)h(of)h(attraction)e(of)i(the)g(stationary)d (solutions)i Fr(m)3341 1423 y Fp(\000)3341 1484 y Fo(\014)s(;h)456 1575 y Fw(and)c Fr(m)690 1539 y Fs(+)690 1600 y Fo(\014)s(;h)793 1575 y Fw(.)37 b(Then)28 b(for)f(an)n(y)g Fr(a)g Fw(small)h(enough,) 1439 1741 y(lim)1393 1791 y Fo(t)p Fp(!)p Fs(+)p Fp(1)1615 1741 y Fr(S)1666 1753 y Fo(t)1695 1673 y Fm(\000)1733 1741 y Fr(m)1806 1705 y Fp(\000)1806 1766 y Fo(\014)s(;h)1928 1741 y Fw(+)18 b Fr(a)2055 1706 y Fs(3)p Fo(=)p Fs(2)2159 1673 y Fm(\001)2220 1741 y Fw(=)23 b Fr(m)2381 1705 y Fp(\000)2381 1766 y Fo(\014)s(;h)2484 1741 y Fr(:)725 b Fw(\(7.45\))456 1920 y(F)-7 b(rom)27 b(\(7.43\))g(and)g(\(7.45\))g(w) n(e)g(get)g(\(7.41\).)555 2019 y(W)-7 b(e)34 b(shall)f(next)h(pro)n(v)n (e)e(\(7.42\))o(.)55 b(W)-7 b(e)34 b(need)g(to)f(sho)n(w)g(that)h(for)f (an)n(y)f Fr(x)2813 2031 y Fs(0)2884 2019 y Fq(2)i Fn(R)39 b Fw(and)34 b Fr(")e(>)h Fw(0)456 2119 y(there)27 b(is)g Fr(T)800 2131 y Fo(";x)889 2139 y Fl(0)953 2119 y Fw(so)g(that)1200 2194 y Fm(\014)1200 2244 y(\014)1200 2294 y(\014)1227 2290 y Fr(S)1278 2302 y Fo(t)1321 2222 y Fm(\000)1359 2290 y Fr(Q)1425 2255 y Fp(\000)1425 2310 y Fo(a)1481 2222 y Fm(\001)1533 2290 y Fw(\()p Fr(x)1612 2302 y Fs(0)1650 2290 y Fw(\))19 b Fq(\000)f Fr(m)1857 2254 y Fs(+)1857 2315 y Fo(\014)s(;h)1960 2194 y Fm(\014)1960 2244 y(\014)1960 2294 y(\014)2011 2290 y Fr(<)k(")166 b Fq(8)14 b Fr(t)22 b(>)h(T)2553 2302 y Fo(";x)2642 2310 y Fl(0)2677 2290 y Fr(:)532 b Fw(\(7.46\))456 2458 y(By)28 b(\(7.13\))f(and)g(the)h (Comparison)e(Theorem,)h(for)g(an)n(y)g(in)n(teger)g Fr(n)p Fw(,)1164 2601 y Fr(S)1215 2613 y Fo(nT)1295 2621 y Fi(a)1349 2533 y Fm(\000)1387 2601 y Fr(Q)1453 2566 y Fp(\000)1453 2621 y Fo(a)1509 2533 y Fm(\001)1561 2601 y Fw(\()p Fr(x)p Fw(\))d Fq(\025)f Fr(Q)1850 2566 y Fp(\000)1850 2621 y Fo(a)1905 2601 y Fw(\()p Fr(x)d Fq(\000)e Fr(n)p Fw(\001)2206 2613 y Fo(a)2246 2601 y Fw(\))166 b Fq(8)14 b Fr(x)23 b Fq(2)g Fn(R)p Fr(:)456 2743 y Fw(Recalling)k(the)h (de\014nition)f(\(7.10\))g(w)n(e)h(ha)n(v)n(e)1078 2886 y Fr(S)1129 2898 y Fo(nT)1209 2906 y Fi(a)1263 2819 y Fm(\000)1301 2886 y Fr(Q)1367 2852 y Fp(\000)1367 2907 y Fo(a)1423 2819 y Fm(\001)1475 2886 y Fw(\()p Fr(x)p Fw(\))c Fq(\025)f Fr(q)s Fw(\()p Fr(x)c Fq(\000)f Fr(n)p Fw(\001)2038 2898 y Fo(a)2097 2886 y Fq(\000)g Fr(a)p Fw(\))166 b Fq(8)14 b Fr(x)22 b(>)h(n)p Fw(\001)2759 2898 y Fo(a)2799 2886 y Fr(:)456 3033 y Fw(Since)k Fr(Q)738 3003 y Fp(\000)738 3054 y Fo(a)822 3033 y Fw(is)g(symmetric)g(and)h (non)f(increasing)f(for)i Fr(x)23 b(>)g Fw(0,)k(from)g(Lemma)h(6.5,) 1226 3176 y Fr(S)1277 3188 y Fo(nT)1357 3196 y Fi(a)1412 3109 y Fm(\000)1450 3176 y Fr(Q)1516 3142 y Fp(\000)1516 3197 y Fo(a)1571 3109 y Fm(\001)1623 3176 y Fw(\()p Fr(x)p Fw(\))c Fq(\025)f Fr(q)s Fw(\(0\))166 b Fq(8)14 b Fr(x)23 b Fq(2)g Fw([0)p Fr(;)14 b(n)p Fw(\001)2588 3188 y Fo(a)2628 3176 y Fw(])p Fr(:)456 3319 y Fw(Hence:)963 3427 y Fr(S)1014 3439 y Fo(nT)1094 3447 y Fi(a)1148 3360 y Fm(\000)1186 3427 y Fr(Q)1252 3393 y Fp(\000)1252 3448 y Fo(a)1308 3360 y Fm(\001)1360 3427 y Fw(\()p Fr(x)p Fw(\))24 b Fq(\025)e Fr(q)s Fw(\(0\))p Fz(1)1776 3442 y Fp(j)p Fo(x)p Fp(j\024)p Fo(n)p Fs(\001)2002 3450 y Fi(a)2060 3427 y Fw(+)c Fr(q)s Fw(\()p Fq(j)p Fr(x)p Fq(j)h(\000)f Fr(n)p Fw(\001)2529 3439 y Fo(a)2569 3427 y Fw(\))p Fz(1)2649 3442 y Fp(j)p Fo(x)p Fp(j\025)p Fo(n)p Fs(\001)2875 3450 y Fi(a)2914 3427 y Fr(:)456 3553 y Fw(W)-7 b(e)28 b(conclude)f(that)h (for)f(an)n(y)g Fr(R)c(>)g Fw(0)k(w)n(e)g(can)h(\014nd)g(a)f(time)h Fr(T)2395 3565 y Fo(R)2477 3553 y Fw(so)f(that)1161 3696 y Fr(S)1212 3708 y Fo(t)1255 3628 y Fm(\000)1293 3696 y Fr(Q)1359 3661 y Fp(\000)1359 3716 y Fo(a)1415 3628 y Fm(\001)1467 3696 y Fw(\()p Fr(x)p Fw(\))d Fq(\025)e Fr(q)s Fw(\(0\))166 b Fq(8)14 b(j)p Fr(x)p Fq(j)23 b(\024)g Fr(R)83 b Fq(8)14 b Fr(t)22 b(>)h(T)2662 3708 y Fo(R)2716 3696 y Fr(:)493 b Fw(\(7.47\))555 3846 y(W)-7 b(e)26 b(can)f(no)n(w)g(pro)n(v)n(e)e(\(7.46\))o(.)37 b(Giv)n(en)25 b(an)n(y)f Fr(")f(>)g Fw(0)i(w)n(e)g(c)n(ho)r(ose)f Fr(T)2549 3858 y Fo(")2609 3846 y Fw(so)h(large)f(that)h Fr(C)3152 3816 y Fp(\003)3191 3846 y Fr(e)3230 3816 y Fp(\000)p Fo(T)3321 3824 y Fi(")3380 3846 y Fr(<)456 3946 y("=)p Fw(2)35 b(and)h Fr(R)i Fq(\025)f(j)p Fr(x)1057 3958 y Fs(0)1094 3946 y Fq(j)25 b Fw(+)e Fr(V)c(T)1346 3958 y Fo(")1381 3946 y Fw(,)39 b Fr(C)1508 3916 y Fp(\003)1546 3946 y Fw(,)g Fr(V)55 b Fw(as)35 b(in)i(the)f(Barrier)e(Lemma)i(7.4.)62 b(Hence)36 b(from)g(the)456 4045 y(Comparison)26 b(Theorem,)g (\(7.47\),)h(and)h(the)g(Barrier)d(Lemma)j(it)g(follo)n(ws)e(that)1094 4210 y Fr(S)1145 4222 y Fo(t)1188 4142 y Fm(\000)1226 4210 y Fr(Q)1292 4175 y Fp(\000)1292 4230 y Fo(a)1348 4142 y Fm(\001)1400 4210 y Fw(\()p Fr(x)1479 4222 y Fs(0)1517 4210 y Fw(\))d Fr(>)g(S)1711 4222 y Fo(t)1754 4210 y Fw(\()p Fr(q)s Fw(\(0\)\))c Fq(\000)2077 4153 y Fr(")p 2076 4190 42 4 v 2076 4267 a Fw(2)2293 4210 y Fq(8)14 b Fr(t)22 b(>)h(T)2543 4222 y Fo(R)2616 4210 y Fw(+)18 b Fr(T)2748 4222 y Fo(")2783 4210 y Fr(:)426 b Fw(\(7.48\))456 4390 y(On)28 b(the)h(other)f(hand)h(since)g Fr(q)s Fw(\(0\))f(b)r (elongs)h(to)f(the)h(basin)g(of)f(attraction)g(of)h Fr(m)2963 4354 y Fs(+)2963 4415 y Fo(\014)s(;h)3095 4390 y Fw(w.r.t.)40 b(the)456 4506 y(dynamics)27 b(\(7.44\))g(\(in)h(fact)f Fr(m)1428 4476 y Fs(0)1428 4529 y Fo(\014)s(;h)1555 4506 y Fr(<)22 b Fw(0)h Fr(<)g(q)s Fw(\(0\))g Fr(<)f(m)2124 4470 y Fs(+)2124 4531 y Fo(\014)s(;h)2228 4506 y Fw(\),)28 b(there)f(is)2623 4485 y(\026)2607 4506 y Fr(T)38 b Fw(suc)n(h)28 b(that)1320 4594 y Fm(\014)1320 4644 y(\014)1320 4693 y(\014)1348 4689 y Fr(S)1399 4701 y Fo(t)1442 4689 y Fw(\()p Fr(q)s Fw(\(0\)\))19 b Fq(\000)f Fr(m)1827 4654 y Fs(+)1827 4714 y Fo(\014)s(;h)1930 4594 y Fm(\014)1930 4644 y(\014)1930 4693 y(\014)1981 4689 y Fr(<)2080 4633 y(")p 2079 4670 V 2079 4746 a Fw(2)2296 4689 y Fq(8)c Fr(t)22 b(>)2513 4668 y Fw(\026)2497 4689 y Fr(T)11 b(:)652 b Fw(\(7.49\))456 4869 y(Recalling)25 b Fr(Q)881 4839 y Fp(\000)881 4890 y Fo(a)959 4869 y Fq(\024)e Fr(m)1120 4834 y Fs(+)1120 4894 y Fo(\014)s(;h)1223 4869 y Fw(,)k(from)e(the)h (Comparison)f(Theorem,)g(\(7.48\))o(,)h(and)g(\(7.49\))f(w)n(e)h (\014nally)456 4969 y(get)1000 5077 y Fr(m)1073 5041 y Fs(+)1073 5102 y Fo(\014)s(;h)1194 5077 y Fq(\000)18 b Fr(")23 b(<)g(S)1478 5089 y Fo(t)1521 5010 y Fm(\000)1559 5077 y Fr(Q)1625 5043 y Fp(\000)1625 5097 y Fo(a)1680 5010 y Fm(\001)1741 5077 y Fq(\024)g Fr(m)1902 5041 y Fs(+)1902 5102 y Fo(\014)s(;h)2171 5077 y Fq(8)14 b Fr(t)22 b(>)2388 5056 y Fw(\026)2372 5077 y Fr(T)30 b Fq(_)18 b Fw(\()p Fr(T)2605 5089 y Fo(")2659 5077 y Fw(+)g Fr(T)2791 5089 y Fo(R)2845 5077 y Fw(\))p Fr(;)456 5216 y Fw(whic)n(h)27 b(implies)h(\(7.46\))f(with)h Fr(T)1453 5228 y Fo(";x)1542 5236 y Fl(0)1601 5216 y Fw(=)1705 5195 y(\026)1688 5216 y Fr(T)i Fq(_)19 b Fw(\()p Fr(T)1922 5228 y Fo(")1975 5216 y Fw(+)f Fr(T)2107 5228 y Fo(R)2161 5216 y Fw(\).)1164 b Fg(\003)p eop %%Page: 35 35 35 34 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(35)456 450 y Fz(Pro)s(of)38 b(of)h(Theorem)e(2.13.)53 b Fw(Let)33 b Fr(w)1720 420 y Fp(\006)1718 471 y Fo(s)1777 450 y Fw(,)i Fr(s)e Fq(\024)f Fw(0,)j(b)r(e)f(as)f(in)g(Theorem)g(6.2.) 54 b(W)-7 b(e)34 b(will)f(next)456 550 y(pro)n(v)n(e)26 b(that)h(for)g(an)n(y)g Fr(a)c(>)g Fw(0)k(small)g(enough)g(there)h(is)f Fr(s)2205 562 y Fo(a)2268 550 y Fr(<)c Fw(0)k(suc)n(h)g(that)984 706 y Fr(w)1045 672 y Fp(\000)1043 727 y Fo(s)1074 735 y Fi(a)1115 706 y Fw(\()p Fr(x)p Fw(\))d Fq(\024)f Fr(Q)1404 672 y Fs(+)1404 727 y Fo(a)1459 706 y Fw(\()p Fr(x)p Fw(\))84 b(and)e Fr(Q)1936 672 y Fp(\000)1936 727 y Fo(a)1992 706 y Fw(\()p Fr(x)p Fw(\))24 b Fq(\024)f Fr(w)2276 672 y Fs(+)2274 727 y Fo(s)2305 735 y Fi(a)2346 706 y Fw(\()p Fr(x)p Fw(\))167 b Fq(8)14 b Fr(x)23 b Fq(2)g Fn(R)p Fr(:)322 b Fw(\(7.50\))456 863 y(Theorem)18 b(2.13)g(will)h(then)h (follo)n(ws)e(from)g(the)i(Comparison)d(Theorem,)j(Corollary)d(7.3,)j (\(2.74\))o(,)456 963 y(and)32 b(\(7.50\))h(\(recall)f(that)h(the)g (relation)f(b)r(et)n(w)n(een)h Fr(w)2157 933 y Fp(\006)2155 983 y Fo(s)2247 963 y Fw(and)g Fr(m)2487 933 y Fp(\006)2487 983 y Fo(s)2576 963 y Fw(is)f(only)h(a)f(time)i(shift,)h(see)456 1062 y(\(6.39\))o(\).)555 1162 y(T)-7 b(o)44 b(pro)n(v)n(e)f(\(7.50\))h (w)n(e)g(need)h(a)f(more)g(accurate)f(estimate)i(on)f(the)h (di\013erence)f Fr(w)3293 1132 y Fp(\006)3291 1183 y Fo(s)3380 1162 y Fq(\000)456 1194 y Fm(\000)494 1262 y Fr(q)21 b Fq(\006)d Fr(e)674 1231 y Fo(\025s)749 1262 y Fr(\032v)835 1194 y Fm(\001)873 1262 y Fw(.)37 b(This)27 b(is)h(the)g(con)n(ten)n(t)f(of)h(Prop)r(osition)e(7.5)g(b)r(elo)n(w.) 456 1428 y Fz(Prop)s(osition)j(7.5.)39 b Fk(L)l(et)28 b Fr(w)1363 1397 y Fp(\006)1361 1448 y Fo(s)1449 1428 y Fk(b)l(e)h(as)g(in)g(The)l(or)l(em)h(6.2)g(and)f Fr(\015)34 b Fk(as)29 b(in)35 b Fw(\(2.54\))o Fk(.)k(Then)29 b(ther)l(e)g(is)456 1527 y(a)h(c)l(onstant)878 1506 y Fw(\026)859 1527 y Fr(C)36 b Fk(so)30 b(that,)g(for)h(al)t(l)f Fr(x)24 b Fq(2)f Fn(R)36 b Fk(and)30 b Fr(s)23 b Fq(\024)g Fw(0)p Fk(,)1086 1634 y Fm(\014)1086 1683 y(\014)1114 1704 y Fr(w)1175 1670 y Fp(\006)1173 1725 y Fo(s)1232 1704 y Fw(\()p Fr(x)p Fw(\))c Fq(\000)f Fr(q)j Fq(\007)e Fr(e)1626 1670 y Fo(\025s)1700 1704 y Fr(\032v)s Fw(\()p Fr(x)p Fw(\))1897 1634 y Fm(\014)1897 1683 y(\014)1949 1704 y Fq(\024)2055 1683 y Fw(\026)2037 1704 y Fr(C)2116 1612 y Fm(\020)2165 1704 y Fr(e)2204 1670 y Fs(2)p Fo(\025s)p Fp(\000)p Fo(\015)t Fp(j)p Fo(x)p Fp(j)2498 1704 y Fw(+)f Fr(e)2620 1670 y Fs(3)p Fo(\025s)2727 1612 y Fm(\021)2791 1704 y Fr(:)418 b Fw(\(7.51\))456 1915 y Fz(Pro)s(of.)41 b Fw(W)-7 b(e)28 b(apply)g(Lemma)f(7.1)g(with)h Fr(u)1795 1927 y Fs(0)1855 1915 y Fw(=)22 b Fq(\006)p Fr("v)s Fw(,)28 b(getting)490 2054 y Fm(\014)490 2104 y(\014)517 2125 y Fr(S)568 2137 y Fo(t)611 2125 y Fw(\()p Fr( )697 2137 y Fp(\006)p Fo(")785 2125 y Fw(\))19 b Fq(\000)f Fr(q)j Fq(\007)d Fr(e)1099 2091 y Fo(\025s)1174 2125 y Fr("v)s Fw(\()p Fr(x)p Fw(\))1367 2054 y Fm(\014)1367 2104 y(\014)1418 2125 y Fq(\024)23 b Fr(K)6 b(")1622 2091 y Fs(2)1672 2012 y Fm(Z)1755 2032 y Fo(t)1718 2201 y Fs(0)1785 2125 y Fr(dt)1858 2091 y Fp(0)1895 2125 y Fr(e)1934 2091 y Fs(2)p Fo(\025t)2031 2066 y Fj(0)2058 2125 y Fr(e)2097 2091 y Fo(L)p Fs(\()p Fo(t)p Fp(\000)p Fo(t)2271 2066 y Fj(0)2292 2091 y Fs(\))2322 2125 y Fw(\()p Fr(J)27 b Fq(\003)18 b Fr(v)s Fw(\))2562 2091 y Fs(2)2618 2125 y Fw(+)g Fr(K)6 b Fq(R)2848 2137 y Fo(t)2891 2125 y Fw([)p Fr(S)2965 2137 y Fp(\001)3003 2125 y Fw(\()p Fr( )3089 2137 y Fp(\006)p Fo(")3177 2125 y Fw(\))18 b Fq(\000)g Fr(q)s Fw(])c Fr(:)3232 2271 y Fw(\(7.52\))456 2371 y(W)-7 b(e)24 b(b)r(ound)g(\()p Fr(J)18 b Fq(\003)10 b Fr(v)s Fw(\))1070 2341 y Fs(2)1131 2371 y Fw(b)n(y)24 b Fq(k)p Fr(v)s Fq(k)1370 2383 y Fp(1)1439 2371 y Fr(J)19 b Fq(\003)10 b Fr(v)s Fw(;)25 b(w)n(e)e(next)h(observ)n(e)e(that)i(since)f Fr(J)32 b Fw(has)23 b(compact)g(supp)r(ort)456 2470 y(and)j Fr(v)k Fw(satis\014es)c(\(2.63\))g(with)h Fr(\015)1461 2482 y Fo(v)1524 2470 y Fr(>)c(\015)5 b Fw(,)26 b(hence)h Fr(J)e Fq(\003)16 b Fr(v)26 b Fq(2)e Fr(X)2282 2482 y Fo(\015)2324 2470 y Fw(,)j(see)g(\(2.58\))o(.)36 b(Then,)28 b(b)n(y)e(applying)456 2570 y(\(2.64\))g(with)j Fr(\020)g Fw(=)23 b Fr(\015)5 b Fw(,)634 2784 y Fr(K)h(")750 2750 y Fs(2)800 2671 y Fm(Z)883 2692 y Fo(t)846 2860 y Fs(0)913 2784 y Fr(dt)986 2750 y Fp(0)1023 2784 y Fr(e)1062 2750 y Fs(2)p Fo(\025t)1159 2725 y Fj(0)1186 2784 y Fr(e)1225 2750 y Fo(L)p Fs(\()p Fo(t)p Fp(\000)p Fo(t)1399 2725 y Fj(0)1420 2750 y Fs(\))1450 2784 y Fw(\()p Fr(J)27 b Fq(\003)18 b Fr(v)s Fw(\))1690 2750 y Fs(2)1751 2784 y Fq(\024)23 b Fr(K)6 b(C)1975 2796 y Fs(1)2012 2784 y Fq(k)p Fr(J)26 b Fq(\003)18 b Fr(v)s Fq(k)2271 2796 y Fo(\015)2313 2784 y Fq(k)p Fr(v)s Fq(k)2440 2796 y Fp(1)2509 2784 y Fr(\025)2557 2750 y Fp(\000)p Fs(1)2647 2784 y Fr(")2686 2750 y Fs(2)2723 2784 y Fr(e)2762 2750 y Fs(2)p Fo(\025t)p Fp(\000)p Fo(\015)t Fp(j)p Fo(x)p Fp(j)3031 2784 y Fr(:)178 b Fw(\(7.53\))456 2988 y(W)-7 b(e)35 b(no)n(w)g(observ)n(e)e(that)j(for)f Fr(t)h Fw(=)f Fr(\034)9 b Fw(\()p Fr(\032;)14 b(")p Fw(\))24 b(+)g Fr(s)p Fw(,)37 b Fr(s)f Fq(\024)f Fw(0,)i(the)f(r.h.s.)f(of)41 b(\(7.53\))35 b(is)g(b)r(ounded,)456 3090 y(uniformly)29 b(as)g Fr(")d Fq(#)g Fw(0,)k(b)n(y)f(const)14 b Fr(e)1528 3060 y Fs(2)p Fo(\025s)p Fp(\000)p Fo(\015)t Fp(j)p Fo(x)p Fp(j)1803 3090 y Fw(.)43 b(Analogously)-7 b(,)28 b(from)h(\(6.32\),)h (\(6.35\))o(,)g(and)g(\(7.2\))o(,)456 3190 y(w)n(e)c(get)h(that)h Fr(K)6 b Fq(R)1042 3205 y Fo(\034)h Fs(\()p Fo(\032;")p Fs(\)+)p Fo(s)1316 3190 y Fw([)p Fr(S)1390 3202 y Fp(\001)1428 3190 y Fw(\()p Fr( )1514 3202 y Fp(\006)p Fo(")1602 3190 y Fw(\))19 b Fq(\000)f Fr(q)s Fw(])27 b(is)g(b)r(ounded)g(b)n(y)g (const)14 b Fr(e)2605 3160 y Fs(3)p Fo(\025s)2712 3190 y Fw(.)37 b(The)27 b(Prop)r(osition)f(is)456 3290 y(pro)n(v)n(ed.)2657 b Fg(\003)456 3418 y Fz(Pro)s(of)37 b(of)h(\(7.50\).)50 b Fw(W)-7 b(e)33 b(\014rst)f(observ)n(e)f(that,)j(arguing)e(as)f(in)i (getting)h(\(7.36\))e(and)g(\(7.37\))o(,)456 3517 y(for)27 b(all)g Fr(a)h Fw(small)f(enough)g(w)n(e)g(ha)n(v)n(e:)837 3694 y Fr(Q)903 3660 y Fs(+)903 3715 y Fo(a)957 3694 y Fw(\()p Fr(x)p Fw(\))d Fq(\025)f Fw([)p Fr(q)s Fw(\()p Fr(x)p Fw(\))d Fq(\000)e Fr(a )s Fw(\()p Fr(x)p Fw(\)])c Fz(1)1754 3709 y Fp(j)p Fo(x)p Fp(j\024)p Fo(R)1934 3717 y Fi(a)1992 3694 y Fw(+)2075 3602 y Fm(h)2114 3694 y Fr(m)2187 3659 y Fp(\000)2187 3719 y Fo(\014)s(;h)2309 3694 y Fw(+)k Fr(a)2436 3660 y Fs(3)p Fo(=)p Fs(2)2540 3602 y Fm(i)2593 3694 y Fz(1)2641 3709 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)2821 3717 y Fi(a)2861 3694 y Fr(;)348 b Fw(\(7.54\))849 3877 y Fr(Q)915 3843 y Fp(\000)915 3898 y Fo(a)971 3877 y Fw(\()p Fr(x)p Fw(\))24 b Fq(\024)f Fw([)p Fr(q)s Fw(\()p Fr(x)p Fw(\))c(+)f Fr(a )s Fw(\()p Fr(x)p Fw(\)])d Fz(1)1768 3892 y Fp(j)p Fo(x)p Fp(j\024)p Fo(R)1948 3900 y Fi(a)2006 3877 y Fw(+)2089 3785 y Fm(h)2128 3877 y Fr(m)2201 3842 y Fp(\000)2201 3902 y Fo(\014)s(;h)2323 3877 y Fq(\000)j Fr(a)2450 3843 y Fs(3)p Fo(=)p Fs(2)2554 3785 y Fm(i)2607 3877 y Fz(1)2655 3892 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)2835 3900 y Fi(a)2874 3877 y Fr(;)335 b Fw(\(7.55\))456 4059 y(where)27 b Fr( )j Fw(is)e(de\014ned)g(in)g (\(7.35\))o(.)37 b(W)-7 b(e)28 b(then)g(set)966 4259 y Fr(s)1005 4271 y Fo(a)1089 4212 y Fr(:)1068 4259 y Fw(=)1170 4202 y Fr(r)p 1166 4240 49 4 v 1166 4316 a(\025)1238 4259 y Fw(log)14 b Fr(a)166 b Fw(with)1768 4202 y(1)p 1768 4240 42 4 v 1768 4316 a(2)1842 4259 y Fr(<)23 b(r)j(<)c Fw(1)c Fq(\000)2234 4202 y Fr(\016)p 2233 4240 V 2233 4316 a Fw(4)2313 4259 y(and)27 b Fr(\016)k Fw(as)c(in)g(\(7.7\))p Fr(:)298 b Fw(\(7.56\))456 4443 y(Recalling)28 b(\(2.63\))f(and)g(that) h Fr(\015)1442 4455 y Fo(v)1481 4443 y Fr(R)1544 4455 y Fs(0)1605 4443 y Fr(>)22 b(\015)5 b(R)24 b(>)e Fw(3)p Fr(=)p Fw(2,)k(from)i(\(7.51\))f(it)h(follo)n(ws)1288 4625 y(lim)1294 4679 y Fo(a)p Fp(#)p Fs(0)1417 4625 y Fr(a)1461 4591 y Fp(\000)p Fs(3)p Fo(=)p Fs(2)1676 4625 y Fw(sup)1631 4698 y Fp(j)p Fo(x)p Fp(j)p Fo(>R)1811 4706 y Fi(a)1860 4529 y Fm(\014)1860 4579 y(\014)1860 4629 y(\014)1888 4625 y Fr(w)1949 4591 y Fp(\006)1947 4645 y Fo(s)1978 4653 y Fi(a)2019 4625 y Fw(\()p Fr(x)p Fw(\))20 b Fq(\000)e Fr(m)2306 4589 y Fp(\000)2306 4650 y Fo(\014)s(;h)2409 4529 y Fm(\014)2409 4579 y(\014)2409 4629 y(\014)2460 4625 y Fw(=)k(0)p Fr(:)620 b Fw(\(7.57\))456 4849 y(On)27 b(the)h(other)f(hand,)h(b)n(y)f(using)h(\(7.38\))o(,)g(w)n (e)f(also)g(ha)n(v)n(e,)f(if)j Fr(a)e Fw(is)h(small)f(enough,)1385 5010 y Fr(e)1424 4975 y Fo(\025s)1494 4983 y Fi(a)1535 5010 y Fr(\032v)s Fw(\()p Fr(x)p Fw(\))d Fr(>)f(a )s Fw(\()p Fr(x)p Fw(\))167 b Fq(8)14 b Fr(x)22 b Fq(2)i Fn(R)p Fr(:)723 b Fw(\(7.58\))456 5166 y(Then)27 b(\(7.50\))g(follo)n (ws)g(from)g(\(7.51\))o(,)h(\(7.54\))o({\(7.58\))o(.)37 b(Theorem)27 b(2.13)f(is)h(pro)n(v)n(ed.)334 b Fg(\003)p eop %%Page: 36 36 36 35 bop 456 251 a Fs(36)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)674 450 y Fw(8.)41 b Fv(The)32 b(quasi-inv)-7 b(ariant)31 b(manif)n(old)g(and)g(the)g(spectral)h(estima)-6 b(tes)555 600 y Fw(In)24 b(this)f(section)g(w)n(e)g(pro)n(v)n(e)f (Theorem)g(2.2,)i(Prop)r(osition)d(2.3,)j(Lemma)f(2.4,)g(and)g(Theorem) 456 699 y(2.5.)39 b(Most)28 b(of)h(the)g(statemen)n(ts)f(follo)n(w)g (from)h(the)g(results)f(pro)n(v)n(ed)f(in)i([6)o(])g(and)g([7)o(])g(as) f(w)n(e)g(are)456 799 y(going)e(to)h(explain.)555 898 y(The)i(\014rst)f(step)h(is)f(the)h(de\014nition)g(of)f(the)h(function) g Fr(n)2289 910 y Fo(\030)2354 898 y Fw(whic)n(h)g(c)n(haracterizes)d (the)i(quasi-)456 998 y(in)n(v)-5 b(arian)n(t)31 b(manifold,)j(see)e (\(2.22\))o(.)51 b(As)33 b(w)n(e)f(already)f(men)n(tioned)h(in)h(the)g (In)n(tro)r(duction,)g(the)456 1098 y(existence)22 b(of)g(the)h (critical)f(droplet)g(has)g(b)r(een)h(pro)n(v)n(ed)e(in)i([7)o(])g(b)n (y)f(using)g(the)h(Newton)g(metho)r(d:)456 1197 y Fr(n)506 1209 y Fo(\030)569 1197 y Fw(will)28 b(b)r(e)g(a)f(sligh)n(t)h(mo)r (di\014cation)f(of)h(the)g(starting)e(p)r(oin)n(t)i(of)g(the)g(Newton)f (map.)555 1297 y(Recalling)22 b(the)h(de\014nition)f(\(2.11\))g(and)g (the)h(prop)r(erties)e(of)h(the)h(instan)n(ton)f(stated)g(in)h(\(2.9\)) o(,)456 1397 y(w)n(e)k(consider)f(the)i(follo)n(wing)f(symmetric)g (function:)1414 1561 y Fr(m)1487 1527 y Fs(0)1487 1582 y Fo(\030)1524 1561 y Fw(\()p Fr(x)p Fw(\))1680 1514 y Fr(:)1659 1561 y Fw(=)38 b(\026)-57 b Fr(m)1820 1573 y Fo(\030)1856 1561 y Fw(\()p Fr(x)p Fw(\))19 b Fq(\000)f Fr(ae)2152 1527 y Fp(\000)p Fo(\013)p Fs(\()p Fo(\030)r Fs(+)p Fp(j)p Fo(x)p Fp(j)p Fs(\))2463 1561 y Fr(:)788 b Fw(\(8.1\))456 1731 y(The)27 b(op)r(erator)g Fr(L)1019 1743 y Fo(m)1109 1731 y Fw(\(see)h(\(2.5\))o(\))h(for)e Fr(m)h Fw(in)g(a)f(neigh)n(b)r(or)g(of)h Fr(m)2411 1701 y Fs(0)2411 1755 y Fo(\030)2476 1731 y Fw(has)f(b)r(een)h(studied)h(in) f(details)456 1837 y(in)d([6,)h(Thms.)f(2.1,)h(2.3,)f(2.4],)g(see)g (also)f([7,)i(Thm.)g(3.2],)f(and)g(for)g(the)h(reader)e(con)n(v)n (enience)g(w)n(e)456 1936 y(no)n(w)j(recall)f(these)i(results.)555 2036 y(W)-7 b(e)28 b(\014rst)g(de\014ne)g(the)g(neigh)n(b)r(or)e(of)i Fr(m)1762 2006 y Fs(0)1762 2059 y Fo(\030)1827 2036 y Fw(as)f(the)h(set)g Fr(G)p Fw(\()p Fr(c)2335 2006 y Fp(\003)2374 2036 y Fr(;)14 b(\030)t(;)g(\016)2528 2006 y Fp(\003)2566 2036 y Fw(\),)28 b Fr(\030)f(>)c Fw(1,)28 b Fr(c)2929 2006 y Fp(\003)2967 2036 y Fr(;)14 b(\016)3044 2006 y Fp(\003)3105 2036 y Fr(>)23 b Fw(0)k(of)h(all)456 2141 y(functions)g Fr(m)22 b Fq(2)i Fr(C)1053 2111 y Fs(sym)1173 2141 y Fw(\()p Fn(R)p Fr(;)14 b Fw([)p Fq(\000)p Fw(1)p Fr(;)g Fw(1]\))33 b(suc)n(h)27 b(that)1019 2307 y Fm(\014)1019 2357 y(\014)1046 2378 y Fr(m)p Fw(\()p Fr(x)p Fw(\))20 b Fq(\000)e Fr(m)1406 2344 y Fs(0)1406 2398 y Fo(\030)1443 2378 y Fw(\()p Fr(x)p Fw(\))1554 2307 y Fm(\014)1554 2357 y(\014)1606 2378 y Fq(\024)k Fr(c)1729 2344 y Fp(\003)1781 2236 y Fm(\()1848 2321 y Fr(e)1887 2291 y Fp(\000)p Fs(2)p Fo(\013\030)2051 2321 y Fr(e)2090 2291 y Fo(\013)p Fs(\()p Fo(\030)r Fp(\000)p Fo(x)p Fs(\))2394 2321 y Fw(for)27 b(0)22 b Fq(\024)h Fr(x)h Fq(\024)e Fr(\030)1848 2441 y(e)1887 2411 y Fp(\000)p Fs(2)p Fo(\013\030)2394 2441 y Fw(for)27 b Fr(\030)g(<)c(x)3274 2378 y Fw(\(8.2\))456 2610 y(and)641 2793 y Fq(\000)720 2680 y Fm(Z)766 2869 y Fp(j)p Fo(x)p Fp(\000)p Fo(\030)r Fp(j\024)980 2825 y(p)p 1033 2825 33 3 v 1033 2869 a Fo(\030)1070 2793 y Fr(dx)14 b Fw([)p Fr(m)p Fw(\()p Fr(x)p Fw(\))20 b Fq(\000)e Fr(m)1557 2759 y Fs(0)1557 2814 y Fo(\030)1594 2793 y Fw(\()p Fr(x)p Fw(\)])f(\026)-59 b Fr(m)1801 2759 y Fp(0)1825 2793 y Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))2078 2759 y Fs(2)2132 2793 y Fw(\026)-58 b Fr(m)p Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))24 b Fq(\025)e(\000)p Fr(c)2654 2759 y Fp(\003)2692 2793 y Fr(e)2731 2759 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)2968 2734 y Fj(\003)3003 2759 y Fs(\))p Fo(\030)3065 2793 y Fr(:)186 b Fw(\(8.3\))456 3048 y Fz(Sp)s(ectral)31 b(analysis)f(in)h Fr(G)p Fw(\()p Fr(c)1430 3017 y Fp(\003)1468 3048 y Fr(;)14 b(\030)t(;)g(\016)1622 3017 y Fp(\003)1661 3048 y Fw(\).)37 b(Some)26 b(of)h(the)g(results)f(that)h(w)n(e)g(need)g (are)e(true)i(for)f Fr(\030)456 3147 y Fw(su\016cien)n(tly)h(large,)f (so)g(w)n(e)h(assume)g(this)g(condition)g(ev)n(en)g(if)h(it)g(is)f(not) g(alw)n(a)n(ys)e(necessary)-7 b(.)35 b(In)456 3247 y([6)o(])30 b(the)f(results)g(are)f(stated)i(in)f(the)h(in)n(terv)-5 b(al)28 b([0)p Fr(;)14 b(`)p Fw(])29 b(with)h(Neumann)f(b)r(oundary)g (conditions,)456 3346 y(but)f(since)f(the)h(estimates)f(are)g(uniform)h (in)g Fr(`)f Fw(they)h(include)g(the)g(case)e Fr(`)d Fw(=)g Fq(1)k Fw(treated)g(here.)555 3446 y(F)-7 b(or)33 b(all)g Fr(m)g Fq(2)g Fr(G)p Fw(\()p Fr(c)1158 3416 y Fp(\003)1197 3446 y Fr(;)14 b(\030)t(;)g(\016)1351 3416 y Fp(\003)1389 3446 y Fw(\))34 b(the)g(op)r(erator)e Fr(L)2002 3458 y Fo(m)2098 3446 y Fw(has)h(a)g(strictly)g(p)r(ositiv)n (e)g(eigen)n(v)-5 b(alue)33 b Fr(\025)3381 3458 y Fo(m)456 3546 y Fw(with)25 b(strictly)g(p)r(ositiv)n(e)f(righ)n(t)h(and)g(left)g (eigenfunctions)g(denoted,)h(resp)r(ectiv)n(ely)-7 b(,)25 b(b)n(y)f Fr(v)3225 3516 y Fp(\003)3222 3566 y Fo(m)3311 3546 y Fw(and)456 3645 y Fr(v)496 3657 y Fo(m)588 3645 y Fw(=)k Fr(p)723 3657 y Fo(m)786 3645 y Fr(v)829 3615 y Fp(\003)826 3666 y Fo(m)921 3645 y Fw(\(recall)i(the)i(de\014nition)f (of)g Fr(p)1837 3657 y Fo(m)1931 3645 y Fw(in)h(\(2.5\))o(\),)h(whic)n (h)e(w)n(e)g(assume)f(normalized)g(in)456 3745 y(suc)n(h)d(a)g(w)n(a)n (y)f(that:)1198 3842 y Fm(Z)1281 3863 y Fp(1)1244 4031 y Fs(0)1351 3955 y Fr(dx)1465 3899 y(v)1505 3911 y Fo(m)1569 3899 y Fw(\()p Fr(x)p Fw(\))1680 3869 y Fs(2)p 1465 3936 253 4 v 1483 4012 a Fr(p)1525 4024 y Fo(m)1588 4012 y Fw(\()p Fr(x)p Fw(\))1751 3955 y Fq(\021)1839 3842 y Fm(Z)1922 3863 y Fp(1)1885 4031 y Fs(0)1992 3955 y Fr(dx)14 b(v)2139 3921 y Fp(\003)2136 3976 y Fo(m)2200 3955 y Fw(\()p Fr(x)p Fw(\))p Fr(v)2351 3967 y Fo(m)2415 3955 y Fw(\()p Fr(x)p Fw(\))24 b(=)f(1)p Fr(:)571 b Fw(\(8.4\))456 4166 y(In)27 b([6,)h(Thm.)g(2.3])f(it)h(has)f(b)r(een)h(pro)n(v)n(ed)e (that)i(there)f(are)g Fr(c)2322 4178 y Fp(\006)2405 4166 y Fw(and)h Fr(c)2603 4136 y Fp(0)2654 4166 y Fw(all)f(p)r(ositiv)n(e)g (so)g(that:)970 4332 y Fr(c)1006 4344 y Fp(\000)1062 4332 y Fr(e)1101 4298 y Fp(\000)p Fs(2)p Fo(\013\030)1288 4332 y Fq(\024)c Fr(\025)1424 4344 y Fo(m)1510 4332 y Fq(\024)g Fr(c)1634 4344 y Fs(+)1689 4332 y Fr(e)1728 4298 y Fp(\000)p Fs(2)p Fo(\013\030)1892 4332 y Fr(;)1359 b Fw(\(8.5\))970 4469 y Fr(v)1010 4481 y Fo(m)1073 4469 y Fw(\()p Fr(x)p Fw(\))24 b Fq(\024)f Fr(c)1332 4481 y Fs(+)1387 4469 y Fr(e)1426 4434 y Fp(\000)p Fo(\013)1521 4409 y Fj(0)1543 4434 y Fp(j)p Fo(\030)r Fp(\000)p Fo(x)p Fp(j)1708 4469 y Fr(;)346 b(\013)2130 4434 y Fp(0)2177 4469 y Fw(=)23 b Fr(\013)c Fq(\000)f Fr(c)2456 4434 y Fp(0)2479 4469 y Fr(e)2518 4434 y Fp(\000)p Fs(2)p Fo(\013\030)2682 4469 y Fr(;)569 b Fw(\(8.6\))970 4600 y Fq(j)p Fr(v)1033 4612 y Fo(m)1096 4600 y Fw(\()p Fr(x)p Fw(\))20 b Fq(\000)33 b Fw(~)-57 b Fr(m)1383 4566 y Fp(0)1383 4621 y Fo(\030)1419 4600 y Fw(\()p Fr(x)p Fw(\))p Fq(j)24 b(\024)f Fr(c)1701 4612 y Fs(+)1756 4600 y Fr(e)1795 4566 y Fp(\000)p Fs(2)p Fo(\013\030)1959 4600 y Fr(e)1998 4566 y Fo(\013)p Fp(j)p Fo(\030)r Fp(\000)p Fo(x)p Fp(j)2206 4600 y Fr(\030)2246 4566 y Fs(4)2477 4600 y Fw(for)k Fq(j)p Fr(\030)c Fq(\000)18 b Fr(x)p Fq(j)23 b(\024)g Fr(\030)t(=)p Fw(2)p Fr(:)177 b Fw(\(8.7\))456 4765 y(where)25 b Fr(\026)h Fw(is)g(de\014ned)g(in)g (\(2.10\))f(and)42 b(~)-58 b Fr(m)1702 4777 y Fo(\030)1738 4765 y Fw(\()p Fr(x)p Fw(\))27 b(in)f(\(2.27\))o(.)37 b(Let)26 b(us)g(remark)e(that)i(b)n(y)g(c)n(ho)r(osing)e Fr(\030)456 4865 y Fw(large)i(enough)h(the)h(parameter)e Fr(\013)1538 4835 y Fp(0)1589 4865 y Fw(can)h(b)r(e)h(done)g(as)f (close)f(to)i Fr(\013)g Fw(as)f(needed.)555 4964 y(F)-7 b(or)27 b(an)n(y)g Fr(m)c Fq(2)g Fr(G)p Fw(\()p Fr(c)1168 4934 y Fp(\003)1207 4964 y Fr(;)14 b(\030)t(;)g(\016)1361 4934 y Fp(\003)1399 4964 y Fw(\))28 b(w)n(e)f(de\014ne)h(the)g(linear)f (functional)h Fr(\031)2632 4976 y Fo(m)2723 4964 y Fw(on)f Fr(L)2895 4934 y Fs(sym)2895 4985 y Fp(1)3015 4964 y Fw(\()p Fn(R)p Fw(\))34 b(as:)1458 5173 y Fr(\031)1505 5185 y Fo(m)1568 5173 y Fw(\()p Fr( )s Fw(\))1733 5126 y Fr(:)1713 5173 y Fw(=)1800 5060 y Fm(Z)1883 5081 y Fp(1)1846 5249 y Fs(0)1954 5173 y Fr(dx)14 b(v)2101 5139 y Fp(\003)2098 5194 y Fo(m)2161 5173 y Fw(\()p Fr(x)p Fw(\))p Fr( )s Fw(\()p Fr(x)p Fw(\))834 b(\(8.8\))p eop %%Page: 37 37 37 36 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(37)456 450 y Fw(\(observ)n(e)26 b(that)h Fr(\031)1010 462 y Fo(m)1074 450 y Fw(\()p Fr(v)1146 462 y Fo(m)1209 450 y Fw(\))d(=)e(1)27 b(b)n(y)i(\(8.4\))o(\).)37 b(An)28 b(easy)f(consequence)f(of)34 b(\(8.6\))27 b(and)h(\(8.7\))f(is) g(the)456 550 y(existence)g(of)g(a)h(constan)n(t)f Fr(c)1344 562 y Fs(3)1404 550 y Fr(>)22 b Fw(0)28 b(suc)n(h)f(that:)1260 686 y Fq(j)p Fr(\031)1330 698 y Fo(m)1393 686 y Fw(\()p Fr( )s Fw(\))p Fq(j)d(\024)e Fr(c)1684 698 y Fs(3)1722 686 y Fq(k)p Fr( )s Fq(k)1863 698 y Fp(1)2098 686 y Fq(8)14 b Fr( )25 b Fq(2)e Fr(L)2373 651 y Fs(sym)2373 706 y Fp(1)2493 686 y Fw(\()p Fn(R)p Fw(\))p Fr(:)640 b Fw(\(8.9\))456 822 y(In)29 b([6,)g(Thm.)h(2.4])f(it)g(is)h(pro)n(v)n(ed)d(that)j (there)f(exists)g Fr(\030)2169 834 y Fs(0)2232 822 y Fr(>)d Fw(1)j(suc)n(h)f(that)i(for)f(an)n(y)f Fr(\030)i(>)c(\030)3244 834 y Fs(0)3311 822 y Fw(and)456 921 y Fr(m)d Fq(2)g Fr(G)p Fw(\()p Fr(c)763 891 y Fp(\003)802 921 y Fr(;)14 b(\030)t(;)g(\016)956 891 y Fp(\003)994 921 y Fw(\))28 b(there)f(exist)h(constan)n(ts)e Fr(d)1873 933 y Fp(\006)1953 921 y Fr(>)c Fw(0)27 b(so)g(that:)1115 1061 y Fq(k)p Fr(e)1196 1027 y Fo(L)1242 1035 y Fi(m)1296 1027 y Fo(t)1325 1061 y Fr( )s Fq(k)1424 1073 y Fp(1)1517 1061 y Fq(\024)c Fr(d)1648 1073 y Fs(+)1703 1061 y Fr(e)1742 1027 y Fo(\025)1781 1035 y Fi(m)1837 1027 y Fo(t)1866 1061 y Fq(k)p Fr( )s Fq(k)2007 1073 y Fp(1)2242 1061 y Fq(8)14 b Fr( )25 b Fq(2)f Fr(L)2518 1027 y Fs(sym)2518 1082 y Fp(1)2637 1061 y Fw(\()p Fn(R)p Fw(\))q Fr(;)453 b Fw(\(8.10\))456 1197 y(and,)27 b(if)h Fr( )j Fw(is)c(suc)n(h)h(that)g Fr(\031)1299 1209 y Fo(m)1362 1197 y Fw(\()p Fr( )s Fw(\))c(=)e(0,)1438 1337 y Fq(k)p Fr(e)1519 1303 y Fo(L)1565 1311 y Fi(m)1618 1303 y Fo(t)1648 1337 y Fr( )s Fq(k)1747 1349 y Fp(1)1840 1337 y Fq(\024)g Fr(d)1970 1349 y Fs(+)2026 1337 y Fr(e)2065 1303 y Fp(\000)p Fo(d)2152 1311 y Fj(\000)2200 1303 y Fo(t)2229 1337 y Fq(k)p Fr( )s Fq(k)2370 1349 y Fp(1)2439 1337 y Fr(:)770 b Fw(\(8.11\))456 1473 y(Moreo)n(v)n(er)24 b(there)k(is)f Fr(c)1152 1485 y Fs(4)1213 1473 y Fr(>)22 b Fw(0)27 b(suc)n(h)h(that)g(the)f(in)n(v)n(erse)g Fr(L)2212 1443 y Fp(\000)p Fs(1)2212 1494 y Fo(m)2328 1473 y Fw(exists)g(and)1152 1610 y Fq(k)p Fr(L)1251 1576 y Fp(\000)p Fs(1)1251 1631 y Fo(m)1339 1610 y Fr( )s Fq(k)1438 1622 y Fp(1)1531 1610 y Fq(\024)c Fr(c)1655 1622 y Fs(4)1692 1610 y Fr(\025)1740 1576 y Fp(\000)p Fs(1)1740 1631 y Fo(m)1829 1610 y Fq(k)p Fr( )s Fq(k)1970 1622 y Fp(1)2206 1610 y Fq(8)14 b Fr( )25 b Fq(2)e Fr(L)2481 1576 y Fs(sym)2481 1631 y Fp(1)2601 1610 y Fw(\()p Fn(R)p Fw(\))p Fr(:)490 b Fw(\(8.12\))456 1746 y(F)-7 b(or)27 b Fr( )j Fw(suc)n(h)e(that)f Fr(\031)1103 1758 y Fo(m)1167 1746 y Fw(\()p Fr( )s Fw(\))d(=)e(0)27 b(w)n(e)h(also)e(ha)n(v)n(e:)1007 1940 y Fr(L)1064 1906 y Fp(\000)p Fs(1)1064 1961 y Fo(m)1153 1940 y Fr( )g Fw(=)d Fq(\000)1400 1827 y Fm(Z)1482 1848 y Fs(+)p Fp(1)1445 2016 y Fs(0)1604 1940 y Fr(dt)14 b(e)1730 1906 y Fo(L)1776 1914 y Fi(m)1830 1906 y Fo(t)1859 1940 y Fr( )s(;)180 b Fq(k)p Fr(L)2218 1906 y Fp(\000)p Fs(1)2218 1961 y Fo(m)2306 1940 y Fr( )s Fq(k)2405 1952 y Fp(1)2498 1940 y Fq(\024)23 b Fr(c)2622 1952 y Fs(4)2659 1940 y Fq(k)p Fr( )s Fq(k)2800 1952 y Fp(1)2869 1940 y Fr(:)340 b Fw(\(8.13\))555 2128 y(In)30 b(the)g(case)e Fr(m)f Fw(=)e Fr(m)1246 2098 y Fs(0)1246 2151 y Fo(\030)1313 2128 y Fw(stronger)j(results)h(hold.)42 b(In)30 b(order)e(to)h(a)n(v)n(oid)f(hea)n(vy)h(notation)g(w)n(e)456 2229 y(abbreviate:)644 2344 y(\026)634 2365 y Fr(L)691 2377 y Fo(\030)750 2365 y Fw(=)22 b Fr(L)894 2383 y Fo(m)953 2363 y Fl(0)953 2403 y Fi(\030)989 2365 y Fr(;)104 b Fw(\026)-49 b Fr(p)1151 2377 y Fo(\030)1210 2365 y Fw(=)23 b Fr(p)1340 2383 y Fo(m)1399 2363 y Fl(0)1399 2403 y Fi(\030)1436 2365 y Fr(;)1559 2343 y Fw(\026)1555 2365 y Fr(\025)1603 2377 y Fo(\030)1663 2365 y Fw(=)g Fr(\025)1799 2383 y Fo(m)1858 2363 y Fl(0)1858 2403 y Fi(\030)1895 2365 y Fr(;)100 b Fw(\026)-45 b Fr(v)2058 2330 y Fp(\003)2055 2385 y Fo(\030)2119 2365 y Fw(=)23 b Fr(v)2250 2330 y Fp(\003)2247 2390 y Fo(m)2306 2370 y Fl(0)2306 2411 y Fi(\030)2343 2365 y Fr(;)100 b Fw(\026)-45 b Fr(v)2503 2377 y Fo(\030)2562 2365 y Fw(=)23 b Fr(v)2690 2383 y Fo(m)2749 2363 y Fl(0)2749 2403 y Fi(\030)2786 2365 y Fr(;)101 b Fw(\026)-46 b Fr(\031)2953 2377 y Fo(\030)3013 2365 y Fw(=)22 b Fr(\031)3147 2383 y Fo(m)3206 2363 y Fl(0)3206 2403 y Fi(\030)3243 2365 y Fr(:)456 2526 y Fw(In)j([6)o(,)h(Thm.)g(11.1])d(it)j(is)f(pro)n(v)n(ed)e(that)j(there)f (are)f Fr(\021)i Fq(2)d Fw(\()p Fr(\013;)14 b Fw(3)p Fr(\013=)p Fw(2\),)26 b Fr(\030)2605 2538 y Fs(1)2665 2526 y Fr(>)d(\030)2789 2538 y Fs(0)2827 2526 y Fw(,)i(and)g Fr(c)3070 2538 y Fs(5)3131 2526 y Fr(>)d Fw(0)j(suc)n(h)456 2625 y(that,)j(for)f(all)g Fr(\030)g(>)c(\030)1088 2637 y Fs(1)1125 2625 y Fw(,)870 2691 y Fm(\014)870 2740 y(\014)897 2761 y Fr(@)941 2773 y Fo(\030)981 2761 y Fw(\026)-45 b Fr(v)1018 2773 y Fo(\030)1054 2761 y Fw(\()p Fr(x)p Fw(\))20 b Fq(\000)34 b Fw(~)-58 b Fr(m)1341 2727 y Fp(00)1341 2782 y Fo(\030)1383 2761 y Fw(\()p Fr(x)p Fw(\))1494 2691 y Fm(\014)1494 2740 y(\014)1606 2761 y Fq(\024)83 b Fr(c)1790 2773 y Fs(5)1827 2761 y Fr(e)1866 2727 y Fp(\000)p Fo(\013\030)1997 2761 y Fr(\030)2037 2727 y Fs(4)2323 2761 y Fw(for)g Fq(j)p Fr(\030)22 b Fq(\000)c Fr(x)p Fq(j)24 b(\024)f Fr(\030)t(=)p Fw(2)p Fr(;)233 b Fw(\(8.14\))1208 2898 y Fq(j)p Fr(@)1275 2910 y Fo(\030)1314 2898 y Fw(\026)-45 b Fr(v)1351 2910 y Fo(\030)1388 2898 y Fw(\()p Fr(x)p Fw(\))p Fq(j)84 b(\024)f Fr(c)1790 2910 y Fs(5)1827 2898 y Fr(\030)1867 2864 y Fs(2)1907 2898 y Fw(\026)-45 b Fr(v)1944 2910 y Fo(\030)1981 2898 y Fw(\()p Fr(x)p Fw(\))250 b(for)82 b Fq(j)p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fq(j)24 b(\025)e Fr(\030)t(=)p Fw(2)p Fr(;)215 b Fw(\(8.15\))1320 2960 y Fm(\014)1320 3010 y(\014)1320 3059 y(\014)1320 3109 y(\014)1357 3024 y Fr(d)1403 3002 y Fw(\026)1400 3024 y Fr(\025)1448 3036 y Fo(\030)p 1357 3061 128 4 v 1380 3137 a Fr(d\030)1495 2960 y Fm(\014)1495 3010 y(\014)1495 3059 y(\014)1495 3109 y(\014)1606 3080 y Fq(\024)83 b Fr(c)1790 3092 y Fs(5)1827 3080 y Fr(e)1866 3046 y Fp(\000)p Fo(\021)r(\030)1990 3080 y Fr(:)1219 b Fw(\(8.16\))456 3260 y(Setting)1059 3404 y Fr(@)1103 3416 y Fo(\030)1144 3404 y Fw(\026)-47 b Fr(\031)1186 3416 y Fo(\030)1223 3404 y Fw(\()p Fr( )s Fw(\))23 b(=)1455 3291 y Fm(Z)1538 3311 y Fp(1)1501 3479 y Fs(0)1608 3404 y Fr(dx)14 b(@)1756 3416 y Fo(\030)1797 3404 y Fw(\026)-46 b Fr(v)1836 3369 y Fp(\003)1833 3424 y Fo(\030)1875 3404 y Fw(\()p Fr(x)p Fw(\))p Fr( )s Fw(\()p Fr(x)p Fw(\))p Fr(;)182 b( )26 b Fq(2)d Fr(L)2574 3369 y Fs(sym)2574 3424 y Fp(1)2694 3404 y Fw(\()p Fn(R)p Fw(\))p Fr(;)397 b Fw(\(8.17\))456 3569 y(from)27 b(\(8.14\))g(and)g (\(8.15\))g(it)h(follo)n(ws)f(that,)h(for)f(some)g Fr(c)2223 3581 y Fs(6)2283 3569 y Fr(>)22 b Fw(0,)1575 3705 y Fq(j)p Fr(@)1642 3717 y Fo(\030)1683 3705 y Fw(\026)-46 b Fr(\031)1726 3717 y Fo(\030)1763 3705 y Fw(\()p Fr( )s Fw(\))p Fq(j)23 b(\024)g Fr(c)2054 3717 y Fs(6)2091 3705 y Fq(k)p Fr( )s Fq(k)2232 3717 y Fp(1)2302 3705 y Fr(:)907 b Fw(\(8.18\))456 3841 y(F)-7 b(urthermore)29 b(from)i([6,)h(Prop.)d(6.3])i(\(see)f(also) g([7,)i(Eq.)e(\(6.38\)]\),)i(for)e(an)n(y)g Fr(\020)35 b Fq(2)29 b Fw(\(0)p Fr(;)14 b(\013)p Fw(\))32 b(there)456 3940 y(exists)27 b Fr(c)721 3952 y Fs(7)781 3940 y Fr(>)c Fw(0)k(so)g(that,)h(for)f(an)n(y)g Fr(t;)14 b(x;)g(y)26 b(>)c Fw(0,)1396 4084 y Fr(e)1443 4035 y Fs(\026)1435 4050 y Fo(L)1481 4059 y Fi(\030)1513 4050 y Fo(t)1542 4084 y Fw(\()p Fr(x;)14 b(y)s Fw(\))24 b Fq(\024)f Fr(c)1882 4096 y Fs(7)1922 4062 y Fw(\026)1919 4084 y Fr(\025)1967 4096 y Fo(\030)2004 4084 y Fr(t)14 b(e)2090 4035 y Fs(\026)2087 4050 y Fo(\025)2126 4059 y Fi(\030)2158 4050 y Fo(t)2187 4084 y Fr(e)2226 4050 y Fp(\000)p Fo(\020)t Fp(j)p Fo(x)p Fp(\000)p Fo(y)r Fp(j)2481 4084 y Fr(:)728 b Fw(\(8.19\))555 4238 y(W)-7 b(e)30 b(no)n(w)e(come)h(bac)n(k)f(to)h(the)h(construction) e(of)h Fr(n)2154 4250 y Fo(\030)2190 4238 y Fw(.)42 b(The)29 b(de\014nition)g(\(8.1\))g(is)g(motiv)-5 b(ated)456 4338 y(b)n(y)33 b(the)g(fact)h(that)f Fr(f)9 b Fw(\()p Fr(m)1235 4308 y Fs(0)1235 4362 y Fo(\030)1272 4338 y Fw(\))34 b(\(see)f(\(2.2\))o(\))h(is)f(small)g(for)g Fr(h)g Fw(small)g(and)g Fr(\030)k Fw(large.)53 b(With)34 b(this)f(in)456 4443 y(mind)28 b(and)f(follo)n(wing)g([7)o(])h(w)n(e)f(de\014ne:)1223 4581 y Fr(b)1259 4593 y Fo(\030)1295 4581 y Fw(\()p Fr(x)p Fw(\))1451 4533 y Fr(:)1430 4581 y Fw(=)c Fq(\000)p Fr(m)1656 4546 y Fs(0)1656 4601 y Fo(\030)1693 4581 y Fw(\()p Fr(x)p Fw(\))c(+)f(tanh)2086 4513 y Fm(\010)2135 4581 y Fr(\014)t Fw(\()p Fr(J)27 b Fq(\003)18 b Fr(m)2424 4546 y Fs(0)2424 4601 y Fo(\030)2461 4581 y Fw(\)\()p Fr(x)p Fw(\))2604 4513 y Fm(\011)2654 4581 y Fr(;)555 b Fw(\(8.20\))456 4717 y(so)27 b(that)1463 4818 y Fr(f)9 b Fw(\()p Fr(m)1618 4783 y Fs(0)1618 4838 y Fo(\030)1655 4818 y Fw(\))23 b(=)g Fr(b)1834 4830 y Fo(\030)1888 4818 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)2061 4830 y Fo(\030)2116 4818 y Fw(+)18 b Fr(O)r Fw(\()p Fr(h)2344 4783 y Fs(2)2382 4818 y Fw(\))p Fr(:)456 4939 y Fw(In)27 b([7,)g(Lemma)g(5.1])g(it)h(is) f(sho)n(wn)g(that)g(giv)n(en)g Fr(\013)1994 4951 y Fs(0)2054 4939 y Fr(>)c(\013)28 b Fw(as)e(in)i(\(2.9\))f(there)g(is)h Fr(c)2951 4951 y Fs(8)3011 4939 y Fr(>)22 b Fw(0)27 b(so)g(that,)456 5039 y(for)g(all)g Fr(\030)g(>)c Fw(1)k(and)g Fr(x)d Fq(\025)f Fw(0,)960 5124 y Fm(\014)960 5174 y(\014)988 5195 y Fr(b)1024 5207 y Fo(\030)1060 5195 y Fw(\()p Fr(x)p Fw(\))c(+)f Fr(e)1312 5161 y Fp(\000)p Fs(2)p Fo(\013\030)1476 5195 y Fr(k)1522 5161 y Fs(0)1519 5215 y Fo(\030)1559 5195 y Fw(\()p Fr(x)p Fw(\))1670 5124 y Fm(\014)1670 5174 y(\014)1722 5195 y Fq(\024)23 b Fr(c)1846 5207 y Fs(8)1897 5103 y Fm(\020)1946 5195 y Fr(e)1985 5161 y Fp(\000)p Fo(\013)2080 5169 y Fl(0)2113 5161 y Fo(\030)2149 5195 y Fz(1)2197 5207 y Fs(0)p Fp(\024)p Fo(x)p Fp(\024)p Fs(1)2427 5195 y Fw(+)18 b Fr(e)2549 5161 y Fp(\000)p Fs(2)p Fo(\013)p Fs(\()p Fo(\030)r Fs(+)p Fo(x)p Fs(\))2853 5103 y Fm(\021)2917 5195 y Fr(;)292 b Fw(\(8.21\))p eop %%Page: 38 38 38 37 bop 456 251 a Fs(38)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fw(where)27 b Fr(k)742 420 y Fs(0)739 474 y Fo(\030)779 450 y Fw(\()p Fr(x)p Fw(\))d(=)e Fr(k)1047 420 y Fs(0)1085 450 y Fw(\()p Fr(\030)g Fq(\000)c Fr(x)p Fw(\))29 b(and)1380 646 y Fr(k)1426 612 y Fs(0)1463 646 y Fw(\()p Fr(y)s Fw(\))1615 599 y Fr(:)1594 646 y Fw(=)1733 590 y Fr(a)14 b(e)1830 560 y Fo(\013y)p 1692 627 261 4 v 1692 703 a Fw(1)k Fq(\000)g Fr(m)1908 674 y Fs(2)1908 728 y Fo(\014)1976 579 y Fm(\002)2011 646 y Fr(m)2084 612 y Fs(2)2084 666 y Fo(\014)2147 646 y Fq(\000)34 b Fw(\026)-58 b Fr(m)2303 612 y Fs(2)2340 646 y Fw(\()p Fr(y)s Fw(\))2448 579 y Fm(\003)2497 646 y Fr(:)712 b Fw(\(8.22\))456 851 y(Observ)n(e)26 b(that)h(from)h(\(2.9\))f(it)h (follo)n(ws)f(that)h(there)f(is)g Fr(c)2216 863 y Fs(9)2277 851 y Fr(>)22 b Fw(0)27 b(suc)n(h)h(that)1728 993 y Fq(k)p Fr(k)1816 959 y Fs(0)1853 993 y Fq(k)1895 1005 y Fp(1)1988 993 y Fq(\024)22 b Fr(c)2111 1005 y Fs(9)2149 993 y Fr(:)1060 b Fw(\(8.23\))456 1134 y(F)-7 b(rom)29 b(the)i(estimate)e(\(8.21\))h (the)g(leading)f(term)h(in)h(the)f(pro)5 b(jection)29 b(of)h Fr(f)9 b Fw(\()p Fr(m)2927 1104 y Fs(0)2927 1158 y Fo(\030)2964 1134 y Fw(\))30 b(is)g Fr(V)19 b Fw(\()p Fr(\030)t Fw(\))p Fr(=)3325 1083 y Fq(p)p 3394 1083 51 4 v 51 x Fr(\026)456 1240 y Fw(\(with)28 b Fr(V)19 b Fw(\()p Fr(\030)t Fw(\))28 b(as)f(in)h(\(2.21\))o(\):)37 b(this)28 b(is)g(the)g(con)n(ten)n(t)f(of)h(the)g(next)f(lemma.)456 1393 y Fz(Lemma)36 b(8.1.)44 b Fk(Ther)l(e)37 b(is)f Fr(`)1370 1405 y Fs(0)1440 1393 y Fr(>)d Fw(1)i Fk(such)h(that)f(for)i (any)f Fr(h)d(<)g(e)2507 1363 y Fp(\000)p Fs(2)p Fo(\013`)2663 1371 y Fl(0)2735 1393 y Fk(the)i(fol)t(lowing)j(holds.)456 1493 y(Ther)l(e)30 b(exists)g Fr(\016)959 1505 y Fs(3)1019 1493 y Fr(>)22 b Fw(0)30 b Fk(such)g(that,)g(for)g(any)g Fr(\024)23 b(>)g Fw(0)29 b Fk(and)h(any)h Fr(\030)c Fq(2)c Fw(\000)2597 1505 y Fo(`)2625 1513 y Fl(0)2657 1505 y Fo(;\024)2720 1493 y Fk(,)1212 1573 y Fm(\014)1212 1623 y(\014)1239 1588 y Fq(p)p 1309 1588 V 1309 1644 a Fr(\026)18 b Fw(\026)-47 b Fr(\031)1419 1656 y Fo(\030)1456 1644 y Fw(\()p Fr(b)1524 1656 y Fo(\030)1579 1644 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)1752 1656 y Fo(\030)1788 1644 y Fw(\))18 b Fq(\000)g Fr(V)h Fw(\()p Fr(\030)t Fw(\))2092 1573 y Fm(\014)2092 1623 y(\014)2144 1644 y Fq(\024)j Fr(C)6 b(e)2335 1610 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)2570 1618 y Fl(3)2603 1610 y Fs(\))p Fo(\030)2665 1644 y Fr(:)544 b Fw(\(8.24\))456 1818 y Fz(Pro)s(of.)42 b Fw(W)-7 b(e)28 b(c)n(ho)r(ose)f Fr(`)1199 1830 y Fs(0)1259 1818 y Fr(>)c(\030)1383 1830 y Fs(1)1448 1818 y Fw(so)k(large)g(that)h Fr(\013)1987 1830 y Fs(0)2043 1818 y Fw(+)18 b Fr(\013)2179 1788 y Fp(0)2226 1818 y Fr(>)23 b Fw(2)p Fr(\013)k Fw(\()p Fr(\013)2521 1788 y Fp(0)2573 1818 y Fw(as)g(in)h(\(8.6\)\).)38 b(Then,)28 b(from)456 1918 y(\(8.6\))f(and)g(\(8.21\),)g(it)h(is)g(easy)e(to)i(c)n (hec)n(k)f(w)n(e)g(can)g(\014nd)h Fr(\016)2230 1930 y Fs(4)2290 1918 y Fr(>)23 b Fw(0)k(suc)n(h)g(that:)1260 1995 y Fm(\014)1260 2045 y(\014)1293 2066 y Fw(\026)-47 b Fr(\031)1335 2078 y Fo(\030)1372 2066 y Fw(\()p Fr(b)1440 2078 y Fo(\030)1476 2066 y Fw(\))19 b(+)f Fr(e)1649 2032 y Fp(\000)p Fs(2)p Fo(\013\030)1817 2066 y Fw(\026)-46 b Fr(\031)1860 2078 y Fo(\030)1897 2066 y Fw(\()p Fr(k)1975 2032 y Fs(0)1972 2086 y Fo(\030)2012 2066 y Fw(\))2044 1995 y Fm(\014)2044 2045 y(\014)2095 2066 y Fq(\024)23 b Fr(C)6 b(e)2287 2032 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)2522 2040 y Fl(4)2554 2032 y Fs(\))p Fo(\030)2616 2066 y Fr(:)593 b Fw(\(8.25\))456 2210 y(F)-7 b(rom)27 b(\(8.6\))g(and)h(\(8.23\))e(w)n(e)i(ha)n(v)n(e:)1130 2416 y(\026)-46 b Fr(\031)1173 2428 y Fo(\030)1210 2416 y Fw(\()p Fr(k)1288 2382 y Fs(0)1285 2437 y Fo(\030)1325 2416 y Fw(\))23 b(=)1468 2303 y Fm(Z)1551 2324 y Fs(3)p Fo(\030)r(=)p Fs(2)1514 2492 y Fo(\030)r(=)p Fs(2)1688 2416 y Fr(dx)1806 2360 y Fw(\026)-45 b Fr(v)1843 2372 y Fo(\030)1879 2360 y Fw(\()p Fr(x)p Fw(\))p 1802 2397 191 4 v 1809 2473 a(\026)c Fr(p)1844 2485 y Fo(\030)1880 2473 y Fw(\()p Fr(x)p Fw(\))2002 2416 y Fr(k)2048 2382 y Fs(0)2045 2437 y Fo(\030)2085 2416 y Fw(\()p Fr(x)p Fw(\))20 b(+)e Fr(O)2378 2324 y Fm(\020)2428 2416 y Fr(e)2467 2382 y Fp(\000)p Fo(\013)2562 2357 y Fj(0)2584 2382 y Fo(\030)r(=)p Fs(2)2687 2324 y Fm(\021)2751 2416 y Fr(:)456 2624 y Fw(On)24 b(the)h(other)e(hand,)i(b)n(y)g(the)f(de\014nitions)h (\(2.14\))o(,)g(\(8.22\))o(,)h(recalling)d(that)i Fr(p)2905 2636 y Fs(\026)-46 b Fo(m)2977 2624 y Fw(=)23 b Fr(\014)t Fw(\(1)12 b Fq(\000)28 b Fw(\026)-58 b Fr(m)3352 2594 y Fs(2)3389 2624 y Fw(\),)456 2723 y(and)27 b(using)g(the)h(estimates)g (\(2.9\))f(and)g(\(8.23\))o(,)h(w)n(e)f(also)g(ha)n(v)n(e:)687 2874 y Fq(p)p 756 2874 51 4 v 56 x Fr(\026)14 b(K)29 b Fw(=)1007 2874 y Fq(p)p 1076 2874 V 56 x Fr(\026)1140 2817 y Fm(Z)1223 2930 y Fr(dx)1358 2874 y Fw(\026)-58 b Fr(m)1415 2844 y Fp(0)1438 2874 y Fw(\()p Fr(x)p Fw(\))p 1338 2911 217 4 v 1338 2987 a Fr(p)1393 2999 y Fs(\026)-46 b Fo(m)1443 2987 y Fw(\()p Fr(x)p Fw(\))1565 2930 y Fr(k)1611 2896 y Fs(0)1648 2930 y Fw(\()p Fr(x)p Fw(\))24 b(=)1870 2817 y Fm(Z)1953 2837 y Fs(3)p Fo(\030)r(=)p Fs(2)1916 3005 y Fo(\030)r(=)p Fs(2)2090 2930 y Fr(dx)2234 2863 y Fw(~)-58 b Fr(m)2291 2833 y Fp(0)2291 2886 y Fo(\030)2328 2863 y Fw(\()p Fr(x)p Fw(\))p 2204 2911 250 4 v 2204 2987 a Fr(p)2259 2999 y Fs(\026)-46 b Fo(m)2305 3008 y Fi(\030)2342 2987 y Fw(\()p Fr(x)p Fw(\))2464 2930 y Fr(k)2510 2896 y Fs(0)2507 2950 y Fo(\030)2547 2930 y Fw(\()p Fr(x)p Fw(\))19 b(+)f Fr(O)2840 2838 y Fm(\020)2889 2930 y Fr(e)2928 2896 y Fp(\000)p Fo(\013\030)r(=)p Fs(2)3127 2838 y Fm(\021)3190 2930 y Fr(:)456 3132 y Fw(F)-7 b(rom)27 b(\(8.1\))g(and)h(\(8.7\))f(w)n(e)g(conclude)g(that)1422 3214 y Fm(\014)1422 3264 y(\014)1454 3285 y Fw(\026)-46 b Fr(\031)1497 3297 y Fo(\030)1533 3285 y Fw(\()p Fr(k)1611 3251 y Fs(0)1608 3305 y Fo(\030)1649 3285 y Fw(\))18 b Fq(\000)1782 3229 y(p)p 1852 3229 51 4 v 1852 3285 a Fr(\026)13 b(K)1992 3214 y Fm(\014)1992 3264 y(\014)2043 3285 y Fq(\024)22 b Fr(C)6 b(e)2234 3251 y Fp(\000)p Fo(\013)2329 3226 y Fj(0)2352 3251 y Fo(\030)r(=)p Fs(2)2455 3285 y Fr(:)754 b Fw(\(8.26\))456 3426 y(W)-7 b(e)28 b(\014nally)f(observ)n(e)f(that)740 3625 y(\026)-47 b Fr(\031)782 3637 y Fo(\030)819 3625 y Fw(\()7 b(\026)-49 b Fr(p)893 3637 y Fo(\030)930 3625 y Fw(\))83 b(=)1192 3512 y Fm(Z)1275 3532 y Fp(1)1239 3700 y Fs(0)1346 3625 y Fr(dx)17 b Fw(\026)-45 b Fr(v)1490 3637 y Fo(\030)1527 3625 y Fw(\()p Fr(x)p Fw(\))24 b(=)1750 3512 y Fm(Z)1833 3532 y Fo(\030)1796 3700 y Fp(\0001)1918 3625 y Fr(dx)30 b Fw(~)-58 b Fr(m)2095 3590 y Fp(0)2119 3625 y Fw(\()p Fr(x)p Fw(\))19 b(+)2332 3512 y Fm(Z)2415 3532 y Fp(1)2378 3700 y Fs(0)2485 3625 y Fr(dx)14 b Fw(\()s(\026)-45 b Fr(v)2661 3637 y Fo(\030)2717 3625 y Fq(\000)34 b Fw(~)-58 b Fr(m)2873 3590 y Fp(0)2873 3645 y Fo(\030)2910 3625 y Fw(\)\()p Fr(x)p Fw(\))24 b(=)1045 3858 y(=)82 b(2)p Fr(m)1307 3870 y Fo(\014)1352 3802 y Fq(p)p 1421 3802 V 56 x Fr(\026)18 b Fq(\000)1572 3745 y Fm(Z)1655 3765 y Fp(1)1618 3933 y Fo(\030)1726 3858 y Fr(dx)30 b Fw(~)-58 b Fr(m)1903 3824 y Fp(0)1926 3858 y Fw(\()p Fr(x)p Fw(\))20 b(+)2140 3745 y Fm(Z)2223 3765 y Fp(1)2186 3933 y Fs(0)2293 3858 y Fr(dx)14 b Fw(\()s(\026)-45 b Fr(v)2469 3870 y Fo(\030)2525 3858 y Fq(\000)34 b Fw(~)-58 b Fr(m)2681 3824 y Fp(0)2681 3878 y Fo(\030)2717 3858 y Fw(\)\()p Fr(x)p Fw(\))1045 4073 y(=)82 b(2)p Fr(m)1307 4085 y Fo(\014)1352 4018 y Fq(p)p 1421 4018 V 55 x Fr(\026)18 b Fw(+)g Fr(O)1652 3981 y Fm(\020)1701 4073 y Fr(e)1740 4039 y Fp(\000)p Fo(\013)1835 4014 y Fj(0)1858 4039 y Fo(\030)r(=)p Fs(2)1961 3981 y Fm(\021)2024 4073 y Fr(;)1185 b Fw(\(8.27\))456 4239 y(where)23 b(in)h(the)g(last)g(equalit)n(y)f(w)n (e)g(used)h(again)e(\(2.9\),)j(\(8.6\))o(,)g(and)f(\(8.7\))o(.)36 b(F)-7 b(rom)23 b(\(8.25\))o(,)i(\(8.26\))o(,)456 4339 y(and)i(\(8.27\))g(w)n(e)g(get)g(\(8.24\))g(for)g(a)h(suitable)f Fr(\016)1900 4351 y Fs(3)1960 4339 y Fr(>)c Fw(0.)1267 b Fg(\003)555 4459 y Fw(W)-7 b(e)30 b(no)n(w)f(construct)g(the)h (function)g Fr(n)1764 4471 y Fo(\030)1830 4459 y Fw(in)g(suc)n(h)f(a)g (w)n(a)n(y)f(that)i(the)g(main)g(con)n(tribution)f(to)456 4559 y Fr(f)9 b Fw(\()p Fr(n)588 4571 y Fo(\030)624 4559 y Fw(\))28 b(is)f(giv)n(en)g(b)n(y)32 b(\026)-47 b Fr(\031)1146 4571 y Fo(\030)1183 4559 y Fw(\()p Fr(b)1251 4571 y Fo(\030)1306 4559 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)1479 4571 y Fo(\030)1515 4559 y Fw(\))s(\026)k Fr(v)1587 4571 y Fo(\030)1623 4559 y Fw(.)456 4712 y Fz(De\014nition)28 b(8.2.)38 b Fk(Given)29 b Fr(\030)e(>)22 b(`)1513 4724 y Fs(0)1578 4712 y Fk(we)28 b(de\014ne)g(the)g(symmetric)g(function)g Fr( )2856 4724 y Fo(\030)2920 4712 y Fk(as)g(the)g(solution)456 4812 y(of:)1267 4897 y Fw(\026)1257 4918 y Fr(L)1314 4930 y Fo(\030)1350 4918 y Fr( )1404 4930 y Fo(\030)1459 4918 y Fw(+)18 b Fr(b)1578 4930 y Fo(\030)1633 4918 y Fw(+)g Fr(h)7 b Fw(\026)-49 b Fr(p)1806 4930 y Fo(\030)1860 4918 y Fq(\000)22 b Fw(\026)-46 b Fr(\031)1990 4930 y Fo(\030)2027 4918 y Fw(\()p Fr(b)2095 4930 y Fo(\030)2149 4918 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)2322 4930 y Fo(\030)2358 4918 y Fw(\))s(\026)k Fr(v)2430 4930 y Fo(\030)2490 4918 y Fw(=)23 b(0)p Fr(:)589 b Fw(\(8.28\))456 5042 y Fk(The)30 b(quasi-invariant)h(manifold)41 b Fw(\(2.22\))28 b Fk(is)j(then)e(de\014ne)l(d)h(via)h(the)f(symmetric)g(functions:)1220 5184 y Fr(n)1270 5196 y Fo(\030)1306 5184 y Fw(\()p Fr(x)p Fw(\))1462 5137 y Fr(:)1442 5184 y Fw(=)22 b Fr(m)1602 5149 y Fs(0)1602 5204 y Fo(\030)1639 5184 y Fw(\()p Fr(x)p Fw(\))e(+)e Fr( )1907 5196 y Fo(\030)1943 5184 y Fw(\()p Fr(x)p Fw(\))p Fr(;)185 b(\030)27 b Fq(2)d Fw(\000)2456 5196 y Fo(`)2484 5204 y Fl(0)2516 5196 y Fo(;)p Fs(+)p Fp(1)2657 5184 y Fr(:)552 b Fw(\(8.29\))p eop %%Page: 39 39 39 38 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(39)555 450 y Fw(W)-7 b(e)28 b(note)g(that)g(the)g (function)g Fr( )1585 462 y Fo(\030)1649 450 y Fw(is)f(w)n(ell)h (de\014ned,)g(indeed)g(from)f(\(8.13\))o(,)1342 656 y Fr( )1396 668 y Fo(\030)1455 656 y Fw(=)c Fq(\000)1622 543 y Fm(Z)1704 564 y Fs(+)p Fp(1)1667 732 y Fs(0)1826 656 y Fr(dt)14 b(e)1960 607 y Fs(\026)1952 622 y Fo(L)1998 631 y Fi(\030)2029 622 y Fo(t)2075 635 y Fw(\026)2059 656 y Fr(T)2108 668 y Fo(\030)2157 656 y Fw(\()q Fr(b)2226 668 y Fo(\030)2280 656 y Fw(+)k Fr(h)7 b Fw(\026)-49 b Fr(p)2453 668 y Fo(\030)2489 656 y Fw(\))14 b Fr(;)456 854 y Fw(where)712 833 y(\026)696 854 y Fr(T)745 866 y Fo(\030)808 854 y Fw(is)28 b(the)g(pro)5 b(jection)26 b(acting)h(on)h Fr(L)1851 824 y Fs(sym)1851 875 y Fp(1)1970 854 y Fw(\()p Fn(R)p Fw(\))34 b(de\014ned)28 b(b)n(y:)1608 979 y(\026)1592 1000 y Fr(T)1641 1012 y Fo(\030)1677 1000 y Fr( )e Fw(=)d Fr( )e Fq(\000)h Fw(\026)-46 b Fr(\031)2050 1012 y Fo(\030)2087 1000 y Fw(\()p Fr( )s Fw(\))s(\026)h Fr(v)2248 1012 y Fo(\030)2285 1000 y Fr(:)456 1146 y Fw(Observ)n(e)26 b(that,)h(b)n(y)i(\(8.6\))e(and)h(\(8.9\))o(,)1598 1291 y Fq(k)1655 1270 y Fw(\026)1640 1291 y Fr(T)1689 1303 y Fo(\030)1724 1291 y Fr( )s Fq(k)1823 1303 y Fp(1)1916 1291 y Fq(\024)23 b Fr(C)6 b Fq(k)p Fr( )s Fq(k)2210 1303 y Fp(1)2279 1291 y Fr(:)930 b Fw(\(8.30\))555 1436 y(In)27 b(the)h(next)f(lemma)g(w)n(e)g(estimate)g(the)g(sup-norm)g(of)g (b)r(oth)g Fr( )2563 1448 y Fo(\030)2627 1436 y Fw(and)g(its)g(deriv)-5 b(ativ)n(e)26 b(with)456 1536 y(resp)r(ect)h(to)g Fr(\030)t Fw(.)456 1692 y Fz(Lemma)j(8.3.)40 b Fk(In)30 b(the)g(same)g(hyp)l (othesis)i(of)f(L)l(emma)f(8.1,)i(ther)l(e)f(exists)e Fr(\016)2869 1704 y Fs(5)2930 1692 y Fr(>)23 b Fw(0)30 b Fk(such)g(that,)456 1791 y(for)g(any)g Fr(\024)23 b(>)g Fw(0)29 b Fk(and)h(any)g Fr(\030)e Fq(2)23 b Fw(\000)1491 1803 y Fo(`)1519 1811 y Fl(0)1551 1803 y Fo(;\024)1614 1791 y Fk(,)1040 1947 y Fq(k)p Fr( )1136 1959 y Fo(\030)1172 1947 y Fq(k)1214 1959 y Fp(1)1307 1947 y Fq(\024)f Fr(C)6 b(e)1498 1913 y Fp(\000)p Fs(\()p Fo(\013)p Fs(+)p Fo(\016)1700 1921 y Fl(5)1733 1913 y Fs(\))p Fo(\030)1795 1947 y Fr(;)184 b Fq(k)p Fr(@)2088 1959 y Fo(\030)2124 1947 y Fr( )2178 1959 y Fo(\030)2214 1947 y Fq(k)2256 1959 y Fp(1)2349 1947 y Fq(\024)22 b Fr(C)6 b(e)2540 1913 y Fp(\000)p Fs(\()p Fo(\013)p Fs(+)p Fo(\016)2742 1921 y Fl(5)2775 1913 y Fs(\))p Fo(\030)2837 1947 y Fr(:)372 b Fw(\(8.31\))456 2126 y Fz(Pro)s(of.)41 b Fw(F)-7 b(rom)28 b(\(8.21\))e(and)i(\(8.23\))o (,)1432 2275 y Fq(k)p Fr(b)1510 2287 y Fo(\030)1546 2275 y Fq(k)1588 2287 y Fp(1)1681 2275 y Fq(\024)22 b Fr(C)1848 2208 y Fm(\000)1886 2275 y Fr(e)1925 2241 y Fp(\000)p Fs(2)p Fo(\013\030)2107 2275 y Fw(+)c Fr(e)2229 2241 y Fp(\000)p Fo(\013)2324 2249 y Fl(0)2356 2241 y Fo(\030)2393 2208 y Fm(\001)2445 2275 y Fr(:)456 2420 y Fw(By)28 b(\(2.16\))o(,)g (since)34 b(\026)-49 b Fr(p)1095 2432 y Fo(\030)1155 2420 y Fq(\024)22 b Fr(\014)t Fw(,)1319 2573 y Fq(k)p Fr(b)1397 2585 y Fo(\030)1451 2573 y Fw(+)c Fr(h)7 b Fw(\026)-49 b Fr(p)1624 2585 y Fo(\030)1660 2573 y Fq(k)1702 2585 y Fp(1)1794 2573 y Fq(\024)23 b Fr(C)1961 2506 y Fm(\000)1999 2573 y Fr(e)2038 2539 y Fp(\000)p Fs(2)p Fo(\013\030)2221 2573 y Fw(+)18 b Fr(e)2343 2539 y Fp(\000)p Fo(\013)2438 2547 y Fl(0)2470 2539 y Fo(\030)2506 2506 y Fm(\001)2558 2573 y Fr(;)651 b Fw(\(8.32\))456 2718 y(so)27 b(that,)g(b)n(y)i(\(8.13\))e(and)g(\(8.30\))o(,)1423 2868 y Fq(k)p Fr( )1519 2880 y Fo(\030)1555 2868 y Fq(k)1597 2880 y Fp(1)1690 2868 y Fq(\024)c Fr(C)1857 2800 y Fm(\000)1895 2868 y Fr(e)1934 2833 y Fp(\000)p Fs(2)p Fo(\013\030)2116 2868 y Fw(+)18 b Fr(e)2238 2833 y Fp(\000)p Fo(\013)2333 2841 y Fl(0)2366 2833 y Fo(\030)2402 2800 y Fm(\001)2454 2868 y Fr(:)755 b Fw(\(8.33\))555 3013 y(T)-7 b(o)27 b(compute)g(the)h(deriv)-5 b(ativ)n(e)26 b(of)h Fr( )1684 3025 y Fo(\030)1748 3013 y Fw(w)n(e)g(di\013eren)n(tiate)g(\(8.28\))f (and,)h(recalling)g(\(8.17\))o(,)h(w)n(e)456 3113 y(get:)765 3237 y(\026)755 3258 y Fr(L)812 3270 y Fo(\030)848 3258 y Fr(@)892 3270 y Fo(\030)929 3258 y Fr( )983 3270 y Fo(\030)1037 3258 y Fw(+)18 b Fr(@)1164 3270 y Fo(\030)1208 3258 y Fw(\026)-49 b Fr(p)1243 3270 y Fo(\030)1279 3258 y Fr(J)26 b Fq(\003)18 b Fr( )1465 3270 y Fo(\030)1585 3258 y Fw(=)82 b Fq(\000)14 b Fr(@)1855 3270 y Fo(\030)1891 3258 y Fw(\()p Fr(b)1959 3270 y Fo(\030)2014 3258 y Fw(+)k Fr(h)7 b Fw(\026)-49 b Fr(p)2187 3270 y Fo(\030)2223 3258 y Fw(\))19 b(+)j(\026)-46 b Fr(\031)2404 3270 y Fo(\030)2440 3191 y Fm(\000)2478 3258 y Fr(@)2522 3270 y Fo(\030)2559 3258 y Fw(\()p Fr(b)2627 3270 y Fo(\030)2682 3258 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)2855 3270 y Fo(\030)2891 3258 y Fw(\))2923 3191 y Fm(\001)2964 3258 y Fw(\026)k Fr(v)3001 3270 y Fo(\030)1732 3383 y Fw(+)14 b Fr(@)1855 3395 y Fo(\030)1896 3383 y Fw(\026)-47 b Fr(\031)1938 3395 y Fo(\030)1975 3383 y Fw(\()p Fr(b)2043 3395 y Fo(\030)2098 3383 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)2271 3395 y Fo(\030)2307 3383 y Fw(\))s(\026)k Fr(v)2379 3395 y Fo(\030)2434 3383 y Fw(+)22 b(\026)-46 b Fr(\031)2564 3395 y Fo(\030)2601 3383 y Fw(\()p Fr(b)2669 3395 y Fo(\030)2723 3383 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)2896 3395 y Fo(\030)2932 3383 y Fw(\))p Fr(@)3008 3395 y Fo(\030)3048 3383 y Fw(\026)k Fr(v)3085 3395 y Fo(\030)3122 3383 y Fr(:)87 b Fw(\(8.34\))456 3528 y(W)-7 b(e)29 b(no)n(w)f(pro)r(ceed)g(as)g(in)h(the)g(pro)r(of)f(of)g(Prop)r (osition)f(4.2.)39 b(Recalling)28 b(the)h(de\014nition)g(of)g(the)456 3628 y(op)r(erator)807 3607 y(\026)790 3628 y Fr(T)839 3640 y Fo(\030)875 3628 y Fw(,)f(w)n(e)f(rewrite)g(\(8.34\))g(as)g (follo)n(ws:)582 3755 y(\026)572 3776 y Fr(L)629 3788 y Fo(\030)665 3776 y Fr(@)709 3788 y Fo(\030)746 3776 y Fr( )800 3788 y Fo(\030)919 3776 y Fw(=)83 b Fq(\000)1161 3755 y Fw(\026)1146 3776 y Fr(T)1195 3788 y Fo(\030)1230 3776 y Fw(\()p Fr(@)1306 3788 y Fo(\030)1350 3776 y Fw(\026)-49 b Fr(p)1385 3788 y Fo(\030)1421 3776 y Fr(J)26 b Fq(\003)18 b Fr( )1607 3788 y Fo(\030)1644 3776 y Fw(\))g Fq(\000)1794 3755 y Fw(\026)1777 3776 y Fr(T)1826 3788 y Fo(\030)1862 3709 y Fm(\000)1900 3776 y Fr(@)1944 3788 y Fo(\030)1981 3776 y Fw(\()p Fr(b)2049 3788 y Fo(\030)2104 3776 y Fw(+)g Fr(h)7 b Fw(\026)-49 b Fr(p)2277 3788 y Fo(\030)2312 3776 y Fw(\))2344 3709 y Fm(\001)2401 3776 y Fw(+)23 b(\026)-47 b Fr(\031)2531 3788 y Fo(\030)2568 3776 y Fw(\()p Fr(b)2636 3788 y Fo(\030)2691 3776 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)2864 3788 y Fo(\030)2900 3776 y Fw(\))2948 3755 y(\026)2932 3776 y Fr(T)2981 3788 y Fo(\030)3017 3776 y Fr(@)3061 3788 y Fo(\030)3100 3776 y Fw(\026)k Fr(v)3137 3788 y Fo(\030)1067 3909 y Fq(\000)1146 3842 y Fm(\002)1184 3909 y Fw(\026)f Fr(\031)1227 3921 y Fo(\030)1263 3909 y Fw(\()p Fr(@)1339 3921 y Fo(\030)1383 3909 y Fw(\026)d Fr(p)1418 3921 y Fo(\030)1454 3909 y Fr(J)27 b Fq(\003)18 b Fr( )1641 3921 y Fo(\030)1677 3909 y Fw(\))h Fq(\000)f Fr(@)1855 3921 y Fo(\030)1896 3909 y Fw(\026)-47 b Fr(\031)1938 3921 y Fo(\030)1975 3909 y Fw(\()p Fr(b)2043 3921 y Fo(\030)2098 3909 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)2271 3921 y Fo(\030)2307 3909 y Fw(\))18 b Fq(\000)23 b Fw(\026)-47 b Fr(\031)2487 3921 y Fo(\030)2524 3909 y Fw(\()p Fr(b)2592 3921 y Fo(\030)2647 3909 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)2820 3921 y Fo(\030)2856 3909 y Fw(\))t(\026)j Fr(\031)2935 3921 y Fo(\030)2972 3909 y Fw(\()p Fr(@)3048 3921 y Fo(\030)3088 3909 y Fw(\026)g Fr(v)3124 3921 y Fo(\030)3161 3909 y Fw(\))3193 3842 y Fm(\003)3231 3909 y Fw(\026)h Fr(v)3268 3921 y Fo(\030)3304 3909 y Fr(:)456 4054 y Fw(F)-7 b(rom)27 b(the)h(de\014nition)g(of)f Fr( )1334 4066 y Fo(\030)1371 4054 y Fw(,)1297 4200 y Fr(b)1333 4212 y Fo(\030)1387 4200 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)1560 4212 y Fo(\030)1619 4200 y Fw(=)23 b Fq(\000)1782 4179 y Fw(\026)1772 4200 y Fr(L)1829 4212 y Fo(\030)1864 4200 y Fr( )1918 4212 y Fo(\030)1973 4200 y Fw(+)f(\026)-46 b Fr(\031)2103 4212 y Fo(\030)2140 4200 y Fw(\()p Fr(b)2208 4212 y Fo(\030)2262 4200 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)2435 4212 y Fo(\030)2471 4200 y Fw(\))s(\026)k Fr(v)2543 4212 y Fo(\030)2580 4200 y Fr(;)629 b Fw(\(8.35\))456 4346 y(then)954 4456 y Fr(@)998 4468 y Fo(\030)1038 4456 y Fw(\026)-46 b Fr(\031)1081 4468 y Fo(\030)1118 4456 y Fw(\()p Fr(b)1186 4468 y Fo(\030)1240 4456 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)1413 4468 y Fo(\030)1449 4456 y Fw(\))24 b(=)e Fq(\000)p Fr(@)1701 4468 y Fo(\030)1742 4456 y Fw(\026)-47 b Fr(\031)1784 4468 y Fo(\030)1821 4456 y Fw(\()1863 4435 y(\026)1853 4456 y Fr(L)1910 4468 y Fo(\030)1946 4456 y Fr( )2000 4468 y Fo(\030)2037 4456 y Fw(\))18 b(+)23 b(\026)-47 b Fr(\031)2217 4468 y Fo(\030)2254 4456 y Fw(\()p Fr(b)2322 4468 y Fo(\030)2377 4456 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)2550 4468 y Fo(\030)2586 4456 y Fw(\))p Fr(@)2662 4468 y Fo(\030)2703 4456 y Fw(\026)i Fr(\031)2745 4468 y Fo(\030)2782 4456 y Fw(\()s(\026)i Fr(v)2854 4468 y Fo(\030)2891 4456 y Fw(\))p Fr(:)286 b Fw(\(8.36\))456 4584 y(W)-7 b(e)31 b(next)h(observ)n(e)e(that,)j(b)n (y)f(\(8.4\),)37 b(\026)-47 b Fr(\031)1694 4596 y Fo(\030)1731 4584 y Fw(\()p Fr(@)1807 4596 y Fo(\030)1847 4584 y Fw(\026)i Fr(v)1884 4596 y Fo(\030)1920 4584 y Fw(\))22 b(+)e Fr(@)2103 4596 y Fo(\030)2144 4584 y Fw(\026)-46 b Fr(\031)2187 4596 y Fo(\030)2223 4584 y Fw(\()s(\026)h Fr(v)2295 4596 y Fo(\030)2332 4584 y Fw(\))30 b(=)f(0.)48 b(F)-7 b(rom)31 b(\(8.35\))g(and)g(\(8.36\))456 4684 y(w)n(e)c(then)h(get:)780 4808 y(\026)770 4829 y Fr(L)827 4841 y Fo(\030)863 4829 y Fr(@)907 4841 y Fo(\030)943 4829 y Fr( )997 4841 y Fo(\030)1117 4829 y Fw(=)82 b Fq(\000)1359 4808 y Fw(\026)1343 4829 y Fr(T)1392 4841 y Fo(\030)1428 4762 y Fm(\002)1462 4829 y Fr(@)1506 4841 y Fo(\030)1543 4829 y Fw(\()p Fr(b)1611 4841 y Fo(\030)1665 4829 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)1838 4841 y Fo(\030)1874 4829 y Fw(\))19 b(+)f Fr(@)2052 4841 y Fo(\030)2095 4829 y Fw(\026)-49 b Fr(p)2130 4841 y Fo(\030)2167 4829 y Fr(J)26 b Fq(\003)18 b Fr( )2353 4841 y Fo(\030)2408 4829 y Fq(\000)k Fw(\026)-46 b Fr(\031)2538 4841 y Fo(\030)2574 4829 y Fw(\()p Fr(b)2642 4841 y Fo(\030)2697 4829 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)2870 4841 y Fo(\030)2906 4829 y Fw(\))p Fr(@)2982 4841 y Fo(\030)3022 4829 y Fw(\026)k Fr(v)3059 4841 y Fo(\030)3095 4762 y Fm(\003)1264 4962 y Fq(\000)1343 4895 y Fm(\002)1382 4962 y Fw(\026)e Fr(\031)1424 4974 y Fo(\030)1461 4962 y Fw(\()p Fr(@)1537 4974 y Fo(\030)1581 4962 y Fw(\026)e Fr(p)1616 4974 y Fo(\030)1652 4962 y Fr(J)26 b Fq(\003)18 b Fr( )1838 4974 y Fo(\030)1875 4962 y Fw(\))g(+)g Fr(@)2052 4974 y Fo(\030)2093 4962 y Fw(\026)-46 b Fr(\031)2136 4974 y Fo(\030)2172 4962 y Fw(\()2214 4941 y(\026)2204 4962 y Fr(L)2261 4974 y Fo(\030)2298 4962 y Fr( )2352 4974 y Fo(\030)2388 4962 y Fw(\))2420 4895 y Fm(\003)2458 4962 y Fw(\026)h Fr(v)2495 4974 y Fo(\030)2531 4962 y Fr(:)678 b Fw(\(8.37\))456 5107 y(By)36 b(standard)f(argumen)n(ts)h (the)h(deriv)-5 b(ativ)n(e)35 b Fr(@)1949 5119 y Fo(\030)1986 5107 y Fr( )2040 5119 y Fo(\030)2113 5107 y Fw(is)h(de\014ned)h(b)n(y)f (the)h(solution)f(of)43 b(\(8.37\))o(,)456 5212 y(whose)27 b(existence)g(is)g(guaran)n(teed)f(from)h(the)h(existence)g(of)2361 5191 y(\026)2351 5212 y Fr(L)2408 5177 y Fp(\000)p Fs(1)2408 5237 y Fo(\030)2497 5212 y Fw(,)g(see)f(\(8.12\))g(and)g(\(8.13\))o(,)p eop %%Page: 40 40 40 39 bop 456 251 a Fs(40)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)555 450 y Fw(In)31 b(order)e(to)h(pro)n(v)n(e)e(the)j(second)e (inequalit)n(y)h(in)h(\(8.31\))e(w)n(e)h(need)g(to)g(b)r(ound)h(the)g (v)-5 b(arious)456 550 y(terms)24 b(on)h(the)h(righ)n(t)e(hand)h(side)h (of)31 b(\(8.37\))o(.)36 b(W)-7 b(e)26 b(start)f(with)g(the)h(term)f Fr(@)2780 562 y Fo(\030)2816 550 y Fr(b)2852 562 y Fo(\030)2888 550 y Fw(.)37 b(Let)25 b Fr(R)3157 562 y Fo(\030)3219 550 y Fw(b)r(e)g(the)456 649 y(function)j(de\014ned)g(as:)551 803 y Fr(R)614 815 y Fo(\030)651 803 y Fw(\()p Fr(x)p Fw(\))84 b(=)993 711 y Fm(n)1049 803 y Fr(m)1122 769 y Fs(0)1122 823 y Fo(\030)1159 803 y Fw(\()p Fr(x)p Fw(\))19 b Fq(\000)f Fw([)e(\026)-58 b Fr(m)p Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))h Fq(\000)f Fr(ae)1906 769 y Fp(\000)p Fo(\013)p Fs(\()p Fo(\030)r Fs(+)p Fo(x)p Fs(\))2178 803 y Fw(])2201 711 y Fm(o)2270 803 y Fz(1)2318 818 y Fo(x)p Fp(2)p Fs([)p Fp(\000)p Fs(1)p Fo(;)p Fs(0])846 985 y Fw(=)993 893 y Fm(n)1049 985 y Fw([)e(\026)-58 b Fr(m)p Fw(\()p Fr(\030)23 b Fw(+)18 b Fr(x)p Fw(\))h(+)f Fr(ae)1583 951 y Fp(\000)p Fo(\013)p Fs(\()p Fo(\030)r Fs(+)p Fo(x)p Fs(\))1854 985 y Fw(])h Fq(\000)f Fw([)e(\026)-58 b Fr(m)p Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))h(+)f Fr(ae)2513 951 y Fp(\000)p Fo(\013)p Fs(\()p Fo(\030)r Fp(\000)p Fo(x)p Fs(\))2785 985 y Fw(])2808 893 y Fm(o)2877 985 y Fz(1)2925 1000 y Fo(x)p Fp(2)p Fs([)p Fp(\000)p Fs(1)p Fo(;)p Fs(0])3187 985 y Fr(:)k Fw(\(8.38\))456 1155 y(F)-7 b(urthermore:)504 1308 y Fr(@)548 1320 y Fo(\030)584 1308 y Fr(R)647 1320 y Fo(\030)683 1308 y Fw(\()p Fr(x)p Fw(\))25 b(=)906 1216 y Fm(n)977 1308 y Fw(\026)-57 b Fr(m)1035 1274 y Fp(0)1058 1308 y Fw(\()p Fr(\030)23 b Fw(+)18 b Fr(x)p Fw(\))h Fq(\000)f Fr(a\013e)1549 1274 y Fp(\000)p Fo(\013)p Fs(\()p Fo(\030)r Fs(+)p Fo(x)p Fs(\))1839 1308 y Fq(\000)34 b Fw(\026)-58 b Fr(m)1995 1274 y Fp(0)2018 1308 y Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))h(+)f Fr(a\013e)2509 1274 y Fp(\000)p Fo(\013)p Fs(\()p Fo(\030)r Fp(\000)p Fo(x)p Fs(\))2782 1216 y Fm(o)2851 1308 y Fz(1)2899 1323 y Fo(x)p Fp(2)p Fs([)p Fp(\000)p Fs(1)p Fo(;)p Fs(0])3161 1308 y Fr(:)48 b Fw(\(8.39\))456 1483 y(Then,)27 b(b)n(y)i(\(2.9\),)1406 1601 y Fq(k)p Fr(R)1511 1613 y Fo(\030)1546 1601 y Fq(k)1588 1613 y Fp(1)1677 1601 y Fw(+)18 b Fq(k)p Fr(@)1846 1613 y Fo(\030)1882 1601 y Fr(R)1945 1613 y Fo(\030)1981 1601 y Fq(k)2023 1613 y Fp(1)2116 1601 y Fq(\024)23 b Fr(C)6 b(e)2308 1567 y Fp(\000)p Fo(\013)2403 1575 y Fl(0)2435 1567 y Fo(\030)2471 1601 y Fr(:)738 b Fw(\(8.40\))456 1733 y(Since)944 1847 y Fr(m)1017 1813 y Fs(0)1017 1868 y Fo(\030)1054 1847 y Fw(\()p Fr(x)p Fw(\))24 b(=)38 b(\026)-58 b Fr(m)p Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))h Fq(\000)f Fr(ae)1787 1813 y Fp(\000)p Fo(\013)p Fs(\()p Fo(x)p Fs(+)p Fo(\030)r Fs(\))2077 1847 y Fw(+)g Fr(R)2223 1859 y Fo(\030)2260 1847 y Fw(\()p Fr(x)p Fw(\))194 b(if)56 b Fr(x)24 b(>)e Fq(\000)p Fw(1)p Fr(;)456 1982 y Fw(b)n(y)j(T)-7 b(a)n(ylor)23 b(expansion)h(and)i (using)f(that)g Fr(L)1832 1994 y Fs(\026)-46 b Fo(m)1897 1982 y Fw(\026)-57 b Fr(m)1955 1952 y Fp(0)2001 1982 y Fw(=)23 b(0,)i(after)g(some)g(simple)g(computations)g(w)n(e)456 2082 y(easily)h(get:)822 2234 y Fr(@)866 2246 y Fo(\030)903 2234 y Fr(b)939 2246 y Fo(\030)975 2234 y Fw(\()p Fr(x)p Fw(\))84 b(=)e Fq(\000)14 b Fr(a\013L)1563 2246 y Fs(\026)-46 b Fo(m)1613 2234 y Fr(e)1652 2200 y Fp(\000)p Fo(\013)p Fs(\()p Fp(\001)p Fs(+)p Fo(\030)r Fs(\))1905 2234 y Fw(\()p Fr(x)p Fw(\))20 b(+)e Fr(L)2189 2246 y Fs(\026)-46 b Fo(m)2238 2234 y Fr(@)2282 2246 y Fo(\030)2319 2234 y Fr(R)2382 2246 y Fo(\030)2418 2234 y Fw(\()p Fr(x)p Fw(\))1317 2369 y(+)p Fr(\014)1433 2334 y Fs(2)1484 2369 y Fw(tanh)1651 2331 y Fp(00)1693 2369 y Fw(\()p Fr(\020)6 b Fw(\)[)p Fq(\000)p Fr(ae)1970 2334 y Fp(\000)p Fo(\013)p Fs(\()p Fo(x)p Fs(+)p Fo(\030)r Fs(\))2260 2369 y Fw(+)19 b Fr(R)2407 2381 y Fo(\030)2443 2369 y Fw(\()p Fr(x)p Fw(\)]\()p Fr(J)28 b Fq(\003)18 b Fr(@)2787 2381 y Fo(\030)2823 2369 y Fr(m)2896 2334 y Fs(0)2896 2389 y Fo(\030)2933 2369 y Fw(\)\()p Fr(x)p Fw(\))156 b(\(8.41\))456 2521 y(where)32 b Fr(\020)38 b Fw(=)31 b Fr(\020)6 b Fw(\()p Fr(x)p Fw(\))34 b(is)f(a)f(n)n(um)n(b)r(er)h(in)f(the)i(in)n(terv)-5 b(al)32 b(with)h(end-p)r(oin)n(ts)f Fr(\014)t Fw(\()p Fr(J)f Fq(\003)37 b Fw(\026)-58 b Fr(m)p Fw(\)\()p Fr(\030)27 b Fq(\000)21 b Fr(x)p Fw(\))34 b(and)456 2623 y Fr(\014)t Fw(\()p Fr(J)c Fq(\003)36 b Fw(\026)-57 b Fr(m)p Fw(\)\()p Fr(\030)26 b Fq(\000)21 b Fr(x)p Fw(\))h Fq(\000)f Fr(a)p Fw(\()p Fr(J)29 b Fq(\003)21 b Fr(e)1403 2593 y Fp(\000)p Fo(\013)p Fs(\()p Fp(\001)p Fs(+)p Fo(\030)r Fs(\))1657 2623 y Fw(\)\()p Fr(x)p Fw(\))i(+)e(\()p Fr(J)30 b Fq(\003)20 b Fr(R)2142 2635 y Fo(\030)2179 2623 y Fw(\)\()p Fr(x)p Fw(\).)52 b(By)31 b(using)i(\(2.8\))f(and)g(\(2.9\),)h(w)n(e)456 2723 y(ha)n(v)n(e,)26 b(for)h(all)g Fr(x)d Fq(\025)f Fw(0,)971 2918 y Fq(j)p Fr(L)1064 2930 y Fs(\026)-46 b Fo(m)1114 2918 y Fr(e)1153 2883 y Fp(\000)p Fo(\013)p Fs(\()p Fp(\001)p Fs(+)p Fo(\030)r Fs(\))1406 2918 y Fw(\()p Fr(x)p Fw(\))p Fq(j)84 b Fw(=)f Fr(e)1811 2883 y Fp(\000)p Fo(\013)p Fs(\()p Fo(\030)r Fs(+)p Fo(x)p Fs(\))2082 2797 y Fm(\014)2082 2847 y(\014)2082 2897 y(\014)2082 2947 y(\014)2110 2918 y Fw(1)18 b Fq(\000)2263 2861 y Fw(\()p Fr(p)2350 2873 y Fs(\026)-46 b Fo(m)2400 2861 y Fr(J)27 b Fq(\003)33 b Fw(\026)-57 b Fr(m)p Fw(\)\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))p 2263 2898 629 4 v 2447 2975 a(1)g Fq(\000)g Fr(m)2663 2946 y Fs(2)2663 3000 y Fo(\014)2901 2797 y Fm(\014)2901 2847 y(\014)2901 2897 y(\014)2901 2947 y(\014)3232 2918 y Fw(\(8.42\))1624 3122 y Fq(\024)83 b Fr(C)6 b(e)1876 3088 y Fp(\000)p Fo(\013)p Fs(\()p Fo(\030)r Fs(+)p Fo(x)p Fs(\))2148 3122 y Fr(e)2187 3088 y Fp(\000)p Fo(\013)p Fp(j)p Fo(\030)r Fp(\000)p Fo(x)p Fp(j)2470 3122 y Fq(\024)22 b Fr(C)6 b(e)2661 3088 y Fp(\000)p Fs(3)p Fo(\013\030)r(=)p Fs(2)2893 3122 y Fr(:)316 b Fw(\(8.43\))456 3271 y(By)36 b(\(8.40\))e Fq(k)p Fr(L)954 3283 y Fs(\026)-46 b Fo(m)1003 3271 y Fr(@)1047 3283 y Fo(\030)1084 3271 y Fr(R)1147 3283 y Fo(\030)1183 3271 y Fq(k)1225 3283 y Fp(1)1330 3271 y Fq(\024)36 b Fr(C)6 b(e)1535 3241 y Fp(\000)p Fo(\013)1630 3249 y Fl(0)1662 3241 y Fo(\030)1698 3271 y Fw(.)60 b(Then,)37 b(since)e Fq(k)14 b Fw(tanh)2463 3234 y Fp(00)2519 3271 y Fq(k)2561 3283 y Fp(1)2666 3271 y Fq(\024)36 b Fw(2,)g(w)n(e)f(ha)n (v)n(e,)h(for)f(all)456 3371 y Fr(x)23 b Fq(\025)g Fw(0,)720 3452 y Fm(\014)720 3502 y(\014)747 3523 y Fr(\014)798 3489 y Fs(2)850 3523 y Fw(tanh)1016 3486 y Fp(00)1058 3523 y Fw(\()p Fr(\020)6 b Fw(\)[)p Fq(\000)p Fr(ae)1335 3489 y Fp(\000)p Fo(\013)p Fs(\()p Fo(x)p Fs(+)p Fo(\030)r Fs(\))1626 3523 y Fw(+)18 b Fr(R)1772 3535 y Fo(\030)1808 3523 y Fw(\()p Fr(x)p Fw(\)]\()p Fr(J)28 b Fq(\003)18 b Fr(@)2152 3535 y Fo(\030)2188 3523 y Fr(m)2261 3489 y Fs(0)2261 3543 y Fo(\030)2299 3523 y Fw(\)\()p Fr(x)p Fw(\))2442 3452 y Fm(\014)2442 3502 y(\014)1253 3683 y Fq(\024)23 b Fr(C)1420 3591 y Fm(\020)1469 3683 y Fr(e)1508 3649 y Fp(\000)p Fo(\013)p Fs(\()p Fo(\030)r Fs(+)p Fo(x)p Fs(\))1798 3683 y Fw(+)18 b Fr(e)1920 3649 y Fp(\000)p Fo(\013)2015 3657 y Fl(0)2047 3649 y Fo(\030)2084 3591 y Fm(\021)2147 3683 y Fr(e)2186 3649 y Fp(\000)p Fo(\013)p Fp(j)p Fo(\030)r Fp(\000)p Fo(x)p Fp(j)2469 3683 y Fq(\024)23 b Fr(C)2636 3591 y Fm(\020)2686 3683 y Fr(e)2725 3649 y Fp(\000)p Fs(3)p Fo(\013\030)r(=)p Fs(2)2974 3683 y Fw(+)18 b Fr(e)3096 3649 y Fp(\000)p Fo(\013)3191 3657 y Fl(0)3224 3649 y Fo(\030)3260 3591 y Fm(\021)3323 3683 y Fr(:)456 3858 y Fw(By)28 b(\(8.42\))f(and)g(the)h(ab)r(o)n(v)n(e)f (estimates)g(it)h(follo)n(ws)e(that:)1291 4032 y Fq(k)p Fr(@)1377 4044 y Fo(\030)1413 4032 y Fr(b)1449 4044 y Fo(\030)1485 4032 y Fw(\()p Fr(x)p Fw(\))p Fq(k)1638 4044 y Fp(1)1732 4032 y Fq(\024)d Fr(C)1899 3940 y Fm(\020)1948 4032 y Fr(e)1987 3997 y Fp(\000)p Fs(3)p Fo(\013\030)r(=)p Fs(2)2237 4032 y Fw(+)18 b Fr(e)2359 3997 y Fp(\000)p Fo(\013)2454 4005 y Fl(0)2486 3997 y Fo(\030)2522 3940 y Fm(\021)2586 4032 y Fr(:)456 4206 y Fw(F)-7 b(rom)27 b(\(2.16\))g(and)g(the)h(fact)g(that)g Fr(@)1606 4218 y Fo(\030)1649 4206 y Fw(\026)-49 b Fr(p)1684 4218 y Fo(\030)1748 4206 y Fw(is)27 b(uniformly)h(b)r(ounded,)g(w)n(e)f (conclude)g(that:)1201 4383 y Fq(k)p Fr(@)1287 4395 y Fo(\030)1323 4383 y Fw(\()p Fr(b)1391 4395 y Fo(\030)1446 4383 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)1619 4395 y Fo(\030)1655 4383 y Fw(\))p Fq(k)1729 4395 y Fp(1)1822 4383 y Fq(\024)23 b Fr(C)1989 4291 y Fm(\020)2038 4383 y Fr(e)2077 4349 y Fp(\000)p Fs(3)p Fo(\013\030)r(=)p Fs(2)2327 4383 y Fw(+)18 b Fr(e)2449 4349 y Fp(\000)p Fo(\013)2544 4357 y Fl(0)2576 4349 y Fo(\030)2612 4291 y Fm(\021)2676 4383 y Fr(:)533 b Fw(\(8.44\))555 4557 y(T)-7 b(o)33 b(b)r(ound)h(the)g(second)f(term)g(on)h(the)f(righ)n(t)g (hand)h(side)f(of)40 b(\(8.37\))33 b(w)n(e)g(use)g(again)f(that)456 4657 y Fr(@)500 4669 y Fo(\030)543 4657 y Fw(\026)-49 b Fr(p)578 4669 y Fo(\030)642 4657 y Fw(is)27 b(uniformly)h(b)r(ounded) g(and)f(the)h(b)r(ound)g(\(8.33\))o(,)g(getting:)1278 4813 y Fq(k)p Fr(@)1364 4825 y Fo(\030)1407 4813 y Fw(\026)-49 b Fr(p)1442 4825 y Fo(\030)1478 4813 y Fr(J)26 b Fq(\003)18 b Fr( )1664 4825 y Fo(\030)1701 4813 y Fq(k)1743 4825 y Fp(1)1835 4813 y Fq(\024)23 b Fr(C)2002 4746 y Fm(\000)2040 4813 y Fr(e)2079 4779 y Fp(\000)p Fs(2)p Fo(\013\030)2262 4813 y Fw(+)18 b Fr(e)2384 4779 y Fp(\000)p Fo(\013)2479 4787 y Fl(0)2511 4779 y Fo(\030)2547 4746 y Fm(\001)2599 4813 y Fr(:)610 b Fw(\(8.45\))456 4963 y(The)28 b(third)g(term)g(on)g (the)g(righ)n(t)f(hand)h(side)g(of)35 b(\(8.37\))27 b(is)h(b)r(ounded)g (b)n(y)g(using)h(\(8.24\))o(,)f(\(2.16\))456 5062 y(and)f(\(8.14\))o(,) h(\(8.15\))o(,)g(obtaining)1295 5216 y Fq(k)t Fw(\026)-46 b Fr(\031)1384 5228 y Fo(\030)1420 5216 y Fw(\()p Fr(b)1488 5228 y Fo(\030)1543 5216 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)1716 5228 y Fo(\030)1752 5216 y Fw(\))1801 5195 y(\026)1784 5216 y Fr(T)1833 5228 y Fo(\030)1869 5216 y Fw(\()p Fr(@)1945 5228 y Fo(\030)1985 5216 y Fw(\026)k Fr(v)2022 5228 y Fo(\030)2058 5216 y Fw(\))p Fq(k)2132 5228 y Fp(1)2226 5216 y Fq(\024)22 b Fr(C)6 b(e)2417 5181 y Fp(\000)p Fs(2)p Fo(\013\030)2582 5216 y Fr(:)627 b Fw(\(8.46\))p eop %%Page: 41 41 41 40 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(41)555 450 y Fw(Finally)-7 b(,)35 b(for)d(the)i(last)f (terms)f(on)h(the)h(righ)n(t)e(hand)h(side)g(of)40 b(\(8.37\))32 b(w)n(e)h(apply)g(the)g(same)456 550 y(argumen)n(t)26 b(as)h(in)h(equations)f(\(4.11\))o({\(4.12\))f(to)i(whic)n(h)f(w)n(e)g (refer)g(for)g(details.)37 b(Then:)578 760 y(\026)-46 b Fr(\031)621 772 y Fo(\030)657 760 y Fw(\()p Fr(@)733 772 y Fo(\030)777 760 y Fw(\026)d Fr(p)812 772 y Fo(\030)848 760 y Fr(J)27 b Fq(\003)18 b Fr( )1035 772 y Fo(\030)1071 760 y Fw(\))h(+)f Fr(@)1249 772 y Fo(\030)1290 760 y Fw(\026)-47 b Fr(\031)1332 772 y Fo(\030)1369 760 y Fw(\()1411 739 y(\026)1401 760 y Fr(L)1458 772 y Fo(\030)1494 760 y Fr( )1548 772 y Fo(\030)1585 760 y Fw(\))23 b(=)1731 738 y(\026)1727 760 y Fr(\025)1775 772 y Fo(\030)1826 647 y Fm(Z)1909 668 y Fp(1)1872 836 y Fs(0)1979 760 y Fr(dx)14 b( )2137 772 y Fo(\030)2174 760 y Fw(\()p Fr(x)p Fw(\))2299 618 y Fm( )2376 704 y Fr(@)2420 716 y Fo(\030)2460 704 y Fw(\026)-46 b Fr(v)2496 716 y Fo(\030)p 2376 741 158 4 v 2422 817 a Fw(\026)d Fr(p)2457 829 y Fo(\030)2561 760 y Fq(\000)2654 704 y Fr(@)2698 716 y Fo(\030)2742 704 y Fw(\026)g Fr(p)2777 716 y Fo(\030)p 2654 741 159 4 v 2701 817 a Fw(\026)g Fr(p)2736 789 y Fs(2)2736 842 y Fo(\030)2826 760 y Fw(\026)k Fr(v)2863 772 y Fo(\030)2899 618 y Fm(!)2979 760 y Fw(\()p Fr(x)p Fw(\))p Fr(:)119 b Fw(\(8.47\))456 971 y(Therefore,)26 b(b)n(y)i(\(8.33\),)904 1111 y Fq(k)t Fw(\026)-46 b Fr(\031)993 1123 y Fo(\030)1030 1111 y Fw(\()p Fr(@)1106 1123 y Fo(\030)1149 1111 y Fw(\026)d Fr(p)1184 1123 y Fo(\030)1220 1111 y Fr(J)27 b Fq(\003)18 b Fr( )1407 1123 y Fo(\030)1443 1111 y Fw(\))h(+)f Fr(@)1621 1123 y Fo(\030)1658 1111 y Fr(\031)1705 1123 y Fo(\030)1741 1111 y Fw(\()1783 1090 y(\026)1773 1111 y Fr(L)1830 1123 y Fo(\030)1866 1111 y Fr( )1920 1123 y Fo(\030)1957 1111 y Fw(\))p Fq(k)2031 1123 y Fp(1)2124 1111 y Fq(\024)23 b Fr(C)2280 1089 y Fw(\026)2277 1111 y Fr(\025)2325 1123 y Fo(\030)2375 1043 y Fm(\000)2414 1111 y Fr(e)2453 1076 y Fp(\000)p Fs(2)p Fo(\013\030)2635 1111 y Fw(+)18 b Fr(e)2757 1076 y Fp(\000)p Fo(\013)2852 1084 y Fl(0)2884 1076 y Fo(\030)2921 1043 y Fm(\001)2972 1111 y Fr(:)237 b Fw(\(8.48\))456 1247 y(F)-7 b(rom)29 b(\(8.12\))o(,)h(\(8.13\))o(,)h (\(8.37\))o(,)f(and)g(the)g(estimates)f(\(8.44\))o(,)i(\(8.45\))o(,)f (\(8.46\))o(,)g(and)g(\(8.48\))f(w)n(e)456 1346 y(\014nally)e(obtain:) 1338 1467 y Fq(k)p Fr(@)1424 1479 y Fo(\030)1460 1467 y Fr( )1514 1479 y Fo(\030)1550 1467 y Fq(k)1592 1479 y Fp(1)1685 1467 y Fq(\024)c Fr(C)1852 1375 y Fm(\020)1901 1467 y Fr(e)1940 1433 y Fp(\000)p Fs(3)p Fo(\013\030)r(=)p Fs(2)2190 1467 y Fw(+)18 b Fr(e)2312 1433 y Fp(\000)p Fo(\013)2407 1441 y Fl(0)2439 1433 y Fo(\030)2476 1375 y Fm(\021)2539 1467 y Fr(:)670 b Fw(\(8.49\))456 1611 y(Setting)23 b Fr(\016)774 1623 y Fs(5)834 1611 y Fw(=)f(min)q Fq(f)p Fr(\013=)p Fw(2;)14 b Fr(\013)1329 1623 y Fs(0)1373 1611 y Fq(\000)8 b Fr(\013)p Fq(g)p Fw(,)24 b(the)f(inequalities)f (\(8.31\))g(follo)n(w)g(from)g(\(8.33\))g(and)g(\(8.49\))o(.)3380 1710 y Fg(\003)456 1828 y Fz(Pro)s(of)38 b(of)f(Theorem)g(2.2.)52 b Fw(Equations)31 b(\(2.18\))i(and)g(\(2.19\))f(follo)n(w)g(from)h(the) g(de\014nition)456 1928 y(\(8.29\))22 b(of)h Fr(n)831 1940 y Fo(\030)891 1928 y Fw(and)g(the)g(estimates)g(\(8.31\))o(.)36 b(Recalling)22 b(the)i(de\014nition)f(\(8.20\))o(,)h(w)n(e)f(next)h (write:)762 2068 y Fr(f)9 b Fw(\()p Fr(n)894 2080 y Fo(\030)930 2068 y Fw(\))24 b(=)e(tanh)p Fq(f)p Fr(\014)1346 2000 y Fm(\002)1381 2068 y Fr(J)k Fq(\003)18 b Fw(\()p Fr(m)1618 2033 y Fs(0)1618 2088 y Fo(\030)1674 2068 y Fw(+)g Fr( )1811 2080 y Fo(\030)1847 2068 y Fw(\))h(+)f Fr(h)2029 2000 y Fm(\003)2063 2068 y Fq(g)g(\000)g Fw(tanh)p Fq(f)p Fr(\014)t(J)27 b Fq(\003)18 b Fr(m)2671 2033 y Fs(0)2671 2088 y Fo(\030)2708 2068 y Fq(g)g Fw(+)g Fr(b)2887 2080 y Fo(\030)2941 2068 y Fq(\000)g Fr( )3078 2080 y Fo(\030)3115 2068 y Fr(;)456 2206 y Fw(from)27 b(whic)n(h,)g(b)n(y)h(T)-7 b(a)n(ylor)26 b(expansion)g(to)i(second)f(order)f(and)h(using)i (\(8.28\))o(,)769 2342 y Fr(f)9 b Fw(\()p Fr(n)901 2354 y Fo(\030)937 2342 y Fw(\))23 b(=)1089 2321 y(\026)1080 2342 y Fr(L)1137 2354 y Fo(\030)1172 2342 y Fr( )1226 2354 y Fo(\030)1281 2342 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)1454 2354 y Fo(\030)1509 2342 y Fw(+)18 b Fr(b)1628 2354 y Fo(\030)1682 2342 y Fw(+)g Fr(R)q Fw(\()p Fr( )1915 2354 y Fo(\030)1952 2342 y Fr(;)c(h)p Fw(\))23 b(=)k(\026)-47 b Fr(\031)2226 2354 y Fo(\030)2263 2342 y Fw(\()p Fr(b)2331 2354 y Fo(\030)2386 2342 y Fw(+)18 b Fr(h)7 b Fw(\026)-49 b Fr(p)2559 2354 y Fo(\030)2595 2342 y Fw(\))s(\026)k Fr(v)2667 2354 y Fo(\030)2722 2342 y Fw(+)18 b Fr(R)q Fw(\()p Fr( )2955 2354 y Fo(\030)2991 2342 y Fr(;)c(h)p Fw(\))p Fr(;)456 2478 y Fw(where)1409 2579 y Fq(j)p Fr(R)q Fw(\()p Fr( )1582 2591 y Fo(\030)1618 2579 y Fr(;)g(h)p Fw(\))p Fq(j)23 b(\024)g Fr(C)1948 2512 y Fm(\000)1986 2579 y Fq(k)p Fr( )2082 2591 y Fo(\030)2118 2579 y Fq(k)2160 2544 y Fs(2)2160 2599 y Fp(1)2248 2579 y Fw(+)18 b Fr(h)2379 2544 y Fs(2)2416 2512 y Fm(\001)2468 2579 y Fr(:)456 2697 y Fw(F)-7 b(rom)27 b(\(8.24\))o(,)h(\(8.31\))o(,)g(and)f(\(2.16\)) g(w)n(e)g(get)g(\(for)h Fr(\030)f Fq(2)c Fw(\000)2214 2709 y Fo(`)2242 2717 y Fl(0)2274 2709 y Fo(;\024)2337 2697 y Fw(\):)720 2840 y Fq(k)p Fr(f)9 b Fw(\()p Fr(n)894 2852 y Fo(\030)929 2840 y Fw(\))19 b Fq(\000)f Fr(V)h Fw(\()p Fr(\030)t Fw(\))p Fr(@)1278 2852 y Fo(\030)1315 2840 y Fr(n)1365 2852 y Fo(\030)1401 2840 y Fq(k)1443 2852 y Fp(1)1596 2840 y Fq(\024)83 b Fr(C)6 b(e)1848 2806 y Fp(\000)p Fs(2)p Fo(\013\030)2012 2840 y Fq(k)s Fw(\026)-45 b Fr(v)2094 2852 y Fo(\030)2149 2840 y Fq(\000)2232 2784 y(p)p 2301 2784 51 4 v 56 x Fr(\026)14 b(@)2409 2852 y Fo(\030)2445 2840 y Fr(n)2495 2852 y Fo(\030)2531 2840 y Fq(k)2573 2852 y Fp(1)1744 2993 y Fw(+)g Fr(C)1901 2900 y Fm(\020)1951 2993 y Fr(e)1990 2958 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)2225 2966 y Fl(3)2257 2958 y Fs(\))p Fo(\030)2338 2993 y Fw(+)k Fr(e)2460 2958 y Fp(\000)p Fs(2\()p Fo(\013)p Fs(+)p Fo(\016)2695 2966 y Fl(5)2727 2958 y Fs(\))p Fo(\030)2808 2993 y Fw(+)g Fr(e)2930 2958 y Fp(\000)p Fs(4)p Fo(\013\030)3094 2900 y Fm(\021)3157 2993 y Fr(:)456 3153 y Fw(On)27 b(the)h(other)f(hand,)h (b)n(y)g(\(8.6\),)f(\(8.7\),)h(and)f(the)h(second)f(inequalit)n(y)g(in) h(\(8.31\))o(,)958 3314 y Fq(k)s Fw(\026)-45 b Fr(v)1040 3326 y Fo(\030)1095 3314 y Fq(\000)1178 3258 y(p)p 1247 3258 V 56 x Fr(\026)14 b(@)1355 3326 y Fo(\030)1391 3314 y Fr(n)1441 3326 y Fo(\030)1477 3314 y Fq(k)1519 3326 y Fp(1)1612 3314 y Fq(\024)23 b Fr(C)1779 3222 y Fm(\020)1829 3314 y Fr(e)1868 3280 y Fs(3)p Fo(\013\030)r(=)p Fs(2)2047 3314 y Fr(\030)2087 3280 y Fs(4)2143 3314 y Fw(+)18 b Fr(e)2265 3280 y Fp(\000)p Fo(\013)2360 3255 y Fj(0)2382 3280 y Fo(\030)2437 3314 y Fw(+)g Fr(e)2559 3280 y Fp(\000)p Fs(\()p Fo(\013)p Fs(+)p Fo(\016)2761 3288 y Fl(5)2793 3280 y Fs(\))p Fo(\030)2855 3222 y Fm(\021)2919 3314 y Fr(:)456 3475 y Fw(F)-7 b(rom)34 b(the)i(ab)r(o)n(v)n(e)e(b)r(ounds)h (the)h(inequalit)n(y)f(\(2.20\))g(easily)f(follo)n(ws)g(with)i(a)f (suitable)g Fr(\016)3307 3487 y Fs(0)3380 3475 y Fr(>)456 3574 y Fw(0.)2859 b Fg(\003)555 3692 y Fw(F)-7 b(or)28 b(the)h(pro)r(of)f(of)g(Prop)r(osition)f(2.3)h(w)n(e)g(ha)n(v)n(e)f(to) i(sho)n(w)e(that)i Fr(n)2600 3704 y Fo(\030)2655 3692 y Fw(+)19 b Fr(h)24 b Fq(2)h Fr(G)p Fw(\()p Fr(c)3024 3662 y Fp(\003)3063 3692 y Fr(;)14 b(\030)t(;)g(\016)3217 3662 y Fp(\003)3255 3692 y Fw(\).)40 b(T)-7 b(o)456 3792 y(this)25 b(end)g(w)n(e)f(need)i(more)e(re\014ned)g(estimates)h(on)g Fr( )2084 3804 y Fo(\030)2120 3792 y Fw(,)h(and)e(this)h(is)g(done)g (in)g(the)g(next)g(lemma.)456 3891 y(W)-7 b(e)28 b(\014rst)f(in)n(tro)r (duce)g(some)g(notation.)37 b(W)-7 b(e)28 b(de\014ne)1422 4080 y(\000)1474 4092 y Fo(\030)1554 4033 y Fr(:)1533 4080 y Fw(=)1621 3967 y Fm(Z)1704 3988 y Fo(\030)1736 3963 y Fl(2)1667 4156 y Fs(0)1773 4080 y Fr(dt)14 b(e)1907 4031 y Fs(\026)1899 4046 y Fo(L)1945 4055 y Fi(\030)1977 4046 y Fo(t)2022 4059 y Fw(\026)2006 4080 y Fr(T)2055 4092 y Fo(\030)2091 4080 y Fw(\()p Fr(b)2159 4092 y Fo(\030)2213 4080 y Fw(+)19 b Fr(h)7 b Fw(\026)-49 b Fr(p)2387 4092 y Fo(\030)2422 4080 y Fw(\))p Fr(:)456 4263 y Fw(Then,)27 b(from)h(\(8.11\))o(,)g(\(8.30\))o(,)g(and)f(\(8.32\))o(,)1207 4414 y Fq(k)p Fr( )1303 4426 y Fo(\030)1357 4414 y Fq(\000)18 b Fw(\000)1492 4426 y Fo(\030)1528 4414 y Fq(k)1570 4426 y Fp(1)1663 4414 y Fq(\024)23 b Fr(C)6 b(e)1855 4380 y Fp(\000)p Fo(d)1942 4388 y Fj(\000)1991 4380 y Fo(\030)2023 4355 y Fl(2)2073 4347 y Fm(\000)2111 4414 y Fr(e)2150 4380 y Fp(\000)p Fs(2)p Fo(\013\030)2333 4414 y Fw(+)18 b Fr(e)2455 4380 y Fp(\000)p Fo(\013)2550 4388 y Fl(0)2582 4380 y Fo(\030)2618 4347 y Fm(\001)2670 4414 y Fr(:)539 b Fw(\(8.50\))456 4550 y(Recalling)28 b(\(8.21\))f(and)g(\(8.22\))o(,)h (w)n(e)f(write:)934 4686 y(\000)986 4698 y Fo(\030)1105 4686 y Fw(=)82 b Fr(P)1305 4698 y Fo(\030)1361 4686 y Fw(+)18 b Fr(Q)1510 4698 y Fo(\030)1546 4686 y Fr(;)932 4887 y(P)985 4899 y Fo(\030)1126 4840 y Fr(:)1105 4887 y Fw(=)1252 4774 y Fm(Z)1336 4794 y Fo(\030)1368 4769 y Fl(2)1299 4962 y Fs(0)1404 4887 y Fr(dt)c(e)1538 4838 y Fs(\026)1530 4853 y Fo(L)1576 4862 y Fi(\030)1608 4853 y Fo(t)1653 4866 y Fw(\026)1637 4887 y Fr(T)1686 4899 y Fo(\030)1722 4887 y Fw(\()p Fr(b)1790 4899 y Fo(\030)1845 4887 y Fw(+)k Fr(k)1971 4899 y Fo(\030)2007 4887 y Fw(\))p Fr(;)181 b(k)2286 4899 y Fo(\030)2322 4887 y Fw(\()p Fr(x)p Fw(\))2478 4840 y Fr(:)2457 4887 y Fw(=)23 b Fr(e)2584 4853 y Fp(\000)p Fs(2)p Fo(\013\030)2748 4887 y Fr(k)2794 4853 y Fs(0)2791 4907 y Fo(\030)2831 4887 y Fw(\()p Fr(x)p Fw(\))p Fr(;)920 5140 y(Q)986 5152 y Fo(\030)1126 5093 y Fr(:)1105 5140 y Fw(=)82 b Fq(\000)1331 5027 y Fm(Z)1414 5047 y Fo(\030)1446 5022 y Fl(2)1377 5216 y Fs(0)1483 5140 y Fr(dt)14 b(e)1617 5091 y Fs(\026)1609 5106 y Fo(L)1655 5115 y Fi(\030)1686 5106 y Fo(t)1732 5119 y Fw(\026)1716 5140 y Fr(T)1765 5152 y Fo(\030)1801 5140 y Fw(\()p Fr(k)1876 5152 y Fo(\030)1931 5140 y Fq(\000)k Fr(h)7 b Fw(\026)-49 b Fr(p)2104 5152 y Fo(\030)2140 5140 y Fw(\))p Fr(:)1037 b Fw(\(8.51\))p eop %%Page: 42 42 42 41 bop 456 251 a Fs(42)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fz(Lemma)29 b(8.4.)40 b Fk(F)-6 b(or)30 b(e)l(ach)h Fr(\013)1379 420 y Fp(\003)1440 450 y Fr(<)23 b(\013)30 b Fk(ther)l(e)g(exists)f Fr(C)2104 462 y Fp(\003)2166 450 y Fr(>)23 b Fw(0)29 b Fk(so)h(that)632 610 y Fq(j)p Fr(P)708 622 y Fo(\030)744 610 y Fw(\()p Fr(x)p Fw(\))p Fq(j)24 b(\024)f Fr(C)1049 622 y Fp(\003)1101 518 y Fm(h)1141 610 y Fr(e)1180 576 y Fp(\000)p Fo(\013)1275 551 y Fj(\003)1309 576 y Fs(\()p Fo(\030)r Fs(+)p Fo(x)p Fs(\))1504 610 y Fw(+)18 b Fr(e)1626 576 y Fp(\000)p Fs(2)p Fo(\013\030)1804 518 y Fm(\020)1854 610 y Fr(e)1893 576 y Fp(\000)p Fo(\013)1988 551 y Fj(\003)2022 576 y Fo(\030)r(=)p Fs(4)2126 610 y Fz(1)2173 631 y Fp(j)p Fo(x)p Fp(\000)p Fo(\030)r Fp(j\024)2387 587 y(p)p 2441 587 33 3 v 2441 631 a Fo(\030)2496 610 y Fw(+)g Fz(1)2627 631 y Fp(j)p Fo(x)p Fp(\000)p Fo(\030)r Fp(j)p Fo(>)2841 587 y Fp(p)p 2894 587 V 2894 631 a Fo(\030)2931 518 y Fm(\021)o(i)3033 610 y Fr(:)176 b Fw(\(8.52\))456 774 y Fk(F)-6 b(urthermor)l(e:)1619 882 y Fq(j)p Fr(Q)1708 894 y Fo(\030)1744 882 y Fw(\()p Fr(x)p Fw(\))p Fq(j)24 b(\024)f Fr(C)6 b(e)2094 848 y Fp(\000)p Fs(2)p Fo(\013\030)2258 882 y Fr(:)951 b Fw(\(8.53\))456 1058 y Fz(Pro)s(of.)61 b Fw(The)34 b(pro)r(of)f(of)g(this)h(lemma)f(is)g(similar)g(to)g(the)h (pro)r(of)f(of)g(the)h(second)f(inequalit)n(y)456 1158 y(in)28 b([7,)g(Eq.)g(\(6.1\)],)g(therefore)f(w)n(e)h(only)g(giv)n(e)f (an)h(outline)g(of)g(the)h(argumen)n(t,)e(b)n(y)h(referring)f(to)456 1257 y([7)o(])i(for)f(more)f(details.)40 b(W)-7 b(e)29 b(\014rst)f(pro)n(v)n(e)f(\(8.53\))o(.)39 b(By)30 b(\(8.11\))o(,)f (\(8.23\))o(,)g(\(8.30\))o(,)g(and)f(\(2.16\))g(w)n(e)456 1357 y(ha)n(v)n(e:)966 1518 y Fq(k)p Fr(Q)1074 1530 y Fo(\030)1110 1518 y Fq(k)1152 1530 y Fp(1)1244 1518 y Fq(\024)23 b Fr(C)1411 1405 y Fm(Z)1494 1426 y Fo(\030)1526 1401 y Fl(2)1457 1594 y Fs(0)1563 1518 y Fr(dt)14 b(e)1689 1484 y Fp(\000)p Fo(d)1776 1492 y Fj(\000)1824 1484 y Fo(t)1853 1518 y Fw(\()p Fq(k)p Fr(h)7 b Fw(\026)-49 b Fr(p)2017 1530 y Fo(\030)2053 1518 y Fq(k)2095 1530 y Fp(1)2184 1518 y Fw(+)18 b Fq(k)p Fr(k)2352 1530 y Fo(\030)2388 1518 y Fq(k)2430 1530 y Fp(1)2499 1518 y Fw(\))24 b Fq(\024)e Fr(C)6 b(e)2746 1484 y Fp(\000)p Fs(2)p Fo(\013\030)2911 1518 y Fr(:)298 b Fw(\(8.54\))555 1697 y(Since)788 1676 y(\026)772 1697 y Fr(T)821 1709 y Fo(\030)857 1697 y Fr( )26 b Fw(=)d Fr( )e Fq(\000)i Fw(\026)-47 b Fr(\031)1230 1709 y Fo(\030)1267 1697 y Fw(\()p Fr( )s Fw(\))s(\026)i Fr(v)1428 1709 y Fo(\030)1493 1697 y Fw(and)27 b Fq(k)s Fw(\026)-45 b Fr(v)1736 1709 y Fo(\030)1772 1697 y Fq(k)1814 1709 y Fp(1)1907 1697 y Fq(\024)23 b Fr(C)6 b Fw(,)28 b(b)n(y)g(\(8.21\))f(and)h(\(8.25\))o (,)497 1917 y Fq(j)p Fr(P)573 1929 y Fo(\030)610 1917 y Fw(\()p Fr(x)p Fw(\))p Fq(j)c(\024)f Fr(C)6 b(\030)961 1882 y Fs(2)999 1917 y Fr(e)1038 1882 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)1273 1890 y Fl(4)1305 1882 y Fs(\))p Fo(\030)1386 1917 y Fw(+)1469 1804 y Fm(Z)1552 1824 y Fo(\030)1584 1799 y Fl(2)1515 1992 y Fs(0)1620 1917 y Fr(dt)1707 1804 y Fm(Z)1790 1824 y Fs(+)p Fp(1)1754 1992 y Fs(0)1912 1917 y Fr(dy)16 b(e)2059 1868 y Fs(\026)2051 1883 y Fo(L)2097 1892 y Fi(\030)2129 1882 y Fo(t)2159 1917 y Fw(\()p Fr(x;)e(y)s Fw(\))2365 1824 y Fm(\020)2415 1917 y Fr(e)2454 1882 y Fp(\000)p Fo(\013)2549 1890 y Fl(0)2581 1882 y Fo(\030)2617 1917 y Fz(1)2665 1932 y Fo(y)r Fp(2)p Fs([0)p Fo(;)p Fs(1])2891 1917 y Fw(+)k Fr(e)3013 1882 y Fp(\000)p Fs(2)p Fo(\013)p Fs(\()p Fo(\030)r Fs(+)p Fo(y)r Fs(\))3316 1824 y Fm(\021)3380 1917 y Fr(:)3232 2063 y Fw(\(8.55\))456 2162 y(An)34 b(estimate)g(of)g(the)g(last)g(in)n (tegral)f(in)h(\(8.55\))g(can)f(b)r(e)i(found)f(in)g([7])g(but,)i (since)e(it)h(is)e(not)456 2262 y(written)e(in)g(the)h(form)f(w)n(e)g (need,)h(w)n(e)f(giv)n(e)f(the)h(argumen)n(t)f(b)r(elo)n(w.)48 b(F)-7 b(rom)30 b(\(8.5\))h(and)g(\(8.19\))456 2362 y(w)n(e)c(get,)g (for)g(an)n(y)g Fr(x)d Fq(\025)e Fw(0,)953 2461 y Fm(Z)1036 2482 y Fo(\030)1068 2456 y Fl(2)999 2650 y Fs(0)1105 2574 y Fr(dt)1192 2461 y Fm(Z)1275 2482 y Fs(+)p Fp(1)1238 2650 y Fs(0)1396 2574 y Fr(dy)17 b(e)1544 2525 y Fs(\026)1536 2540 y Fo(L)1582 2549 y Fi(\030)1614 2540 y Fo(t)1643 2574 y Fw(\()p Fr(x;)d(y)s Fw(\))p Fr(e)1874 2540 y Fp(\000)p Fo(\013)1969 2548 y Fl(0)2002 2540 y Fo(\030)2038 2574 y Fz(1)2086 2589 y Fo(y)r Fp(2)p Fs([0)p Fo(;)p Fs(1])2317 2574 y Fq(\024)22 b Fr(C)6 b(\030)2509 2540 y Fs(2)2547 2574 y Fr(e)2586 2540 y Fp(\000)p Fs(\()p Fo(\013)2707 2548 y Fl(0)2739 2540 y Fo(\030)r Fs(+)p Fo(\020)t(x)p Fs(\))2924 2574 y Fr(:)285 b Fw(\(8.56\))456 2764 y(Again)27 b(b)n(y)h(\(8.19\))o(,)689 2868 y Fm(Z)772 2888 y Fo(\030)804 2863 y Fl(2)735 3056 y Fs(0)840 2981 y Fr(dt)927 2868 y Fm(Z)1010 2888 y Fs(+)p Fp(1)974 3056 y Fs(0)1132 2981 y Fr(dy)16 b(e)1279 2932 y Fs(\026)1271 2947 y Fo(L)1317 2956 y Fi(\030)1349 2947 y Fo(t)1378 2981 y Fw(\()p Fr(x;)e(y)s Fw(\))p Fr(e)1609 2947 y Fp(\000)p Fs(2)p Fo(\013)p Fs(\()p Fo(\030)r Fs(+)p Fo(y)r Fs(\))1936 2981 y Fq(\024)23 b Fr(C)6 b(\030)2129 2947 y Fs(2)2166 2981 y Fr(e)2205 2947 y Fp(\000)p Fs(2)p Fo(\013\030)2383 2868 y Fm(Z)2466 2888 y Fs(+)p Fp(1)2429 3056 y Fs(0)2587 2981 y Fr(dy)17 b(e)2727 2947 y Fp(\000)p Fo(\020)t Fp(j)p Fo(x)p Fp(\000)p Fo(y)r Fp(j)2982 2981 y Fr(e)3021 2947 y Fp(\000)p Fs(2)p Fo(\013y)3188 2981 y Fr(:)456 3178 y Fw(The)33 b(in)n(tegral)g(on)g (the)h(r.h.s.)g(is)f(uniformly)h(b)r(ounded;)j(moreo)n(v)n(er,)c(if)h Fq(j)p Fr(\030)27 b Fq(\000)22 b Fr(x)p Fq(j)34 b(\024)3117 3115 y(p)p 3186 3115 41 4 v 63 x Fr(\030)k Fw(then,)456 3277 y(since)27 b Fq(j)p Fr(x)19 b Fq(\000)f Fr(y)s Fq(j)23 b(\025)f(j)p Fr(y)g Fq(\000)c Fr(\030)t Fq(j)g(\000)g(j)p Fr(x)h Fq(\000)f Fr(\030)t Fq(j)p Fw(,)565 3365 y Fm(Z)648 3386 y Fs(+)p Fp(1)611 3554 y Fs(0)769 3478 y Fr(dy)f(e)909 3444 y Fp(\000)p Fo(\020)t Fp(j)p Fo(x)p Fp(\000)p Fo(y)r Fp(j)1163 3478 y Fr(e)1202 3444 y Fp(\000)p Fs(2)p Fo(\013y)1453 3478 y Fw(=)1600 3365 y Fm(Z)1683 3386 y Fs(+)p Fp(1)1646 3554 y Fs(0)1805 3478 y Fr(dy)f(e)1944 3444 y Fp(\000)p Fo(\020)t Fp(j)p Fo(x)p Fp(\000)p Fo(y)r Fp(j)2199 3478 y Fr(e)2238 3444 y Fp(\000)p Fs(2)p Fo(\013y)2419 3411 y Fm(\000)2457 3478 y Fz(1)2505 3493 y Fp(j)p Fo(y)r Fp(\000)p Fo(\030)r Fp(j\024)p Fo(\030)r(=)p Fs(2)2838 3478 y Fw(+)i Fz(1)2969 3493 y Fp(j)p Fo(y)r Fp(\000)p Fo(\030)r Fp(j)p Fo(>\030)r(=)p Fs(2)3284 3411 y Fm(\001)1453 3683 y Fq(\024)82 b Fr(C)1679 3591 y Fm(\020)1729 3683 y Fr(e)1768 3649 y Fp(\000)p Fs(3)p Fo(\013=)p Fs(2)1986 3683 y Fw(+)18 b Fr(e)2108 3649 y Fo(\020)t Fs(\()2168 3605 y Fp(p)p 2222 3605 33 3 v 2222 3649 a Fo(\030)r Fp(\000)p Fo(\030)r(=)p Fs(2\))2435 3591 y Fm(\021)2499 3683 y Fr(:)456 3851 y Fw(Collecting)26 b(all)g(the)h(ab)r(o)n(v)n(e)e (estimates)h(w)n(e)g(obtain)g(\(8.52\))g(b)n(y)g(c)n(ho)r(osing)f Fr(\013)2821 3821 y Fp(\003)2883 3851 y Fr(<)d(\020)33 b Fw(\(recall)26 b(that)456 3951 y(in)h(\(8.19\))g(the)h(parameter)e Fr(\020)k Fq(2)23 b Fw(\(0)p Fr(;)14 b(\013)p Fw(\))28 b(can)g(b)r(e)g(\014xed)f(arbitrarily)f(close)h(to)g Fr(\013)p Fw(\).)406 b Fg(\003)456 4072 y Fz(Pro)s(of)35 b(of)g(Prop)s(osition)f(2.3)p Fw(.)46 b(F)-7 b(rom)30 b(\(2.16\))o(,)i(\(8.50\))o(,)g(\(8.52\))o(,)g(and)e(\(8.53\))g(w)n(e)h (easily)f(get)456 4172 y(that)20 b Fr( )682 4184 y Fo(\030)722 4172 y Fw(+)t Fr(h)f Fw(satis\014es)h(the)g(b)r(ounds)g(on)g(the)h (righ)n(t)e(hand)h(side)g(of)27 b(\(8.2\))20 b(for)f(a)h(suitable)g (constan)n(t)456 4271 y Fr(c)492 4241 y Fp(\003)562 4271 y Fw(if)33 b Fr(\030)j(>)31 b(`)846 4283 y Fs(0)915 4271 y Fw(is)h(c)n(hosen)g(large)f(enough.)51 b(Instead)33 b(the)f(estimate)h(\(8.3\))f(is)h(not)f(immediate.)456 4371 y(By)c(\(8.50\))o(,)g(\(8.51\))o(,)g(and)f(\(8.52\))g(there)h(is)f Fr(\016)1847 4383 y Fs(6)1907 4371 y Fr(>)c Fw(0)k(so)g(that)571 4444 y Fm(\014)571 4493 y(\014)571 4543 y(\014)571 4593 y(\014)612 4451 y(Z)659 4640 y Fp(j)p Fo(x)p Fp(\000)p Fo(\030)r Fp(j\024)873 4596 y(p)p 926 4596 V 926 4640 a Fo(\030)963 4564 y Fr(dx)h Fw([)p Fr( )1158 4576 y Fo(\030)1194 4564 y Fw(\()p Fr(x)p Fw(\))20 b(+)e Fr(h)g Fq(\000)g Fr(Q)1623 4576 y Fo(\030)1659 4564 y Fw(\()p Fr(x)p Fw(\)])31 b(\026)-58 b Fr(m)1881 4530 y Fp(0)1904 4564 y Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))2157 4530 y Fs(2)2211 4564 y Fw(\026)-58 b Fr(m)p Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))2521 4444 y Fm(\014)2521 4493 y(\014)2521 4543 y(\014)2521 4593 y(\014)2572 4564 y Fq(\024)23 b Fr(C)6 b(e)2764 4530 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)2999 4538 y Fl(6)3032 4530 y Fs(\))p Fo(\030)3094 4564 y Fr(:)115 b Fw(\(8.57\))456 4774 y(\(note)30 b(that)h(since)g(\()16 b(\026)-58 b Fr(m)1170 4744 y Fp(0)1193 4774 y Fw(\))1225 4744 y Fs(2)1278 4774 y Fw(\026)h Fr(m)30 b Fw(is)h(an)n(tisymmetric)f(the)h(constan)n (t)e(+)p Fr(h)i Fw(do)r(es)f(not)g(con)n(tribute)h(to)456 4873 y(the)f(in)n(tegral\).)43 b(The)31 b(con)n(tribution)e(of)h Fr(Q)1787 4885 y Fo(\030)1853 4873 y Fw(to)g(the)h(ab)r(o)n(v)n(e)d(in) n(tegral)h(has)h(b)r(een)g(estimated)g(in)456 4973 y([7)o(,)e(Sect.)g (8].)36 b(More)27 b(precisely)-7 b(,)27 b(setting)1133 5159 y Fr(M)1214 5171 y Fs(2)1295 5112 y Fr(:)1274 5159 y Fw(=)1362 5046 y Fm(Z)1408 5234 y Fp(j)p Fo(x)p Fp(\000)p Fo(\030)r Fp(j\024)1622 5190 y(p)p 1675 5190 V 1675 5234 a Fo(\030)1712 5159 y Fr(dx)14 b(Q)1882 5171 y Fo(\030)1918 5159 y Fw(\()p Fr(x)p Fw(\))j(\026)-59 b Fr(m)2102 5124 y Fp(0)2126 5159 y Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))2379 5124 y Fs(2)2433 5159 y Fw(\026)-58 b Fr(m)p Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))p Fr(;)p eop %%Page: 43 43 43 42 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(43)456 450 y Fw(it)28 b(is)f(pro)n(v)n(ed)f(that)i(there)f (is)h Fr(\016)1405 462 y Fs(7)1470 450 y Fw(suc)n(h)f(that)h Fq(j)p Fr(M)1941 462 y Fs(2)1996 450 y Fq(\000)18 b Fr(M)2160 462 y Fs(3)2197 450 y Fq(j)23 b(\024)g Fr(C)6 b(e)2435 420 y Fp(\000)p Fs(\(2)p Fo(\013)p Fs(+)p Fo(\016)2670 428 y Fl(7)2702 420 y Fs(\))p Fo(\030)2765 450 y Fw(,)27 b(where)559 676 y Fr(M)640 688 y Fs(3)721 628 y Fr(:)700 676 y Fw(=)c Fq(\000)867 563 y Fm(Z)913 751 y Fp(j)p Fo(x)p Fp(\000)p Fo(\030)r Fp(j\024)1127 707 y(p)p 1180 707 33 3 v 1180 751 a Fo(\030)1216 676 y Fr(dx)31 b Fw(\026)-59 b Fr(m)1393 641 y Fp(0)1417 676 y Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))1670 641 y Fs(2)1724 676 y Fw(\026)-58 b Fr(m)p Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))2048 563 y Fm(Z)2131 583 y Fo(\030)2163 558 y Fl(2)2095 751 y Fs(0)2200 676 y Fr(dt)2287 563 y Fm(Z)2370 676 y Fr(dy)f(G)2536 688 y Fo(\030)r(;t)2617 676 y Fw(\()p Fr(x;)d(y)s Fw(\))g(\()q Fr(h)7 b Fw(\026)-49 b Fr(p)2946 688 y Fo(\030)3000 676 y Fq(\000)18 b Fr(k)3126 688 y Fo(\030)3163 676 y Fw(\))c(\()p Fr(y)s Fw(\))p Fr(;)456 888 y Fw(with)27 b Fr(G)709 900 y Fo(\030)r(;t)791 888 y Fw(\()p Fr(x;)14 b(y)s Fw(\))27 b(explicitly)g(giv)n(en)f(in)i ([7)o(,)g(Eq.)e(\(8.10\)])g(and)h(satisfying)f Fr(G)2804 900 y Fo(\030)r(;t)2886 888 y Fw(\()p Fr(\030)21 b Fq(\000)c Fr(x;)d(y)21 b Fq(\000)16 b Fr(\030)t Fw(\))24 b(=)456 988 y Fr(G)521 1000 y Fo(\030)r(;t)602 988 y Fw(\()p Fr(\030)16 b Fw(+)c Fr(x;)i(y)h Fw(+)d Fr(\030)t Fw(\).)35 b(It)25 b(is)f(\014nally)g(sho)n(wn)g(that)g Fr(M)2031 1000 y Fs(3)2093 988 y Fw(is)g(b)r(ounded)h(b)n(y)f(the)g(r.h.s.)g(of) 31 b(\(8.3\):)k(here)456 1088 y(w)n(e)29 b(only)h(remark)f(that,)i (since)f(\()16 b(\026)-58 b Fr(m)1569 1058 y Fp(0)1592 1088 y Fw(\))1624 1058 y Fs(2)1677 1088 y Fw(\026)h Fr(m)30 b Fw(is)g(an)n(tisymmetric,)g(only)g(the)g(o)r(dd)h(part)e(\(w.r.t.)45 b Fr(\030)t Fw(\))456 1187 y(of)30 b Fr(h)7 b Fw(\026)-49 b Fr(p)643 1199 y Fo(\030)700 1187 y Fq(\000)20 b Fr(k)828 1199 y Fo(\030)896 1187 y Fw(con)n(tributes)30 b(to)h(the)g(in)n (tegral,)g(and)f(since)38 b(\026)-49 b Fr(p)2325 1199 y Fo(\030)2392 1187 y Fw(is)31 b(ev)n(en,)g(only)g(the)g(o)r(dd)g(part) g(of)456 1287 y Fr(k)499 1299 y Fo(\030)565 1287 y Fw(surviv)n(es.)44 b(The)30 b(estimate)g(of)g(this)g(last)g(term)g(uses)g(the)h(fact)f (that)h Fr(J)38 b Fw(is)30 b(non)g(increasing,)456 1387 y(w)n(e)d(refer)g(to)g([7])g(for)h(the)g(details.)555 1486 y(W)-7 b(e)23 b(ha)n(v)n(e)f(th)n(us)h(pro)n(v)n(ed)e(that)i(for)f (an)n(y)g Fr(\024)h(>)f Fw(0)h(there)f(exists)h Fr(`)2462 1498 y Fs(1)2521 1486 y Fr(>)g(`)2644 1498 y Fs(0)2704 1486 y Fw(and)f(suitable)h(p)r(ositiv)n(e)456 1586 y(constan)n(ts)34 b Fr(c)867 1556 y Fp(\003)941 1586 y Fw(and)h Fr(\016)1150 1556 y Fp(\003)1224 1586 y Fw(suc)n(h)g(that)g Fr(n)1656 1598 y Fo(\030)1716 1586 y Fw(+)24 b Fr(h)36 b Fq(2)g Fr(G)p Fw(\()p Fr(c)2113 1556 y Fp(\003)2152 1586 y Fr(;)14 b(\030)t(;)g(\016)2306 1556 y Fp(\003)2344 1586 y Fw(\))36 b(for)f(all)g Fr(\030)40 b Fq(2)c Fw(\000)2889 1598 y Fo(`)2917 1606 y Fl(1)2949 1598 y Fo(;\024)3012 1586 y Fw(.)61 b(Therefore)456 1685 y(Prop)r(osition)27 b(2.3)i(as)g(w)n (ell)g(as)f(the)i(estimates)f(\(2.33\))g(and)g(\(2.34\))g(follo)n(w)f (from)h(the)h(sp)r(ectral)456 1785 y(analysis)c(in)i Fr(G)p Fw(\()p Fr(c)998 1755 y Fp(\003)1036 1785 y Fr(;)14 b(\030)t(;)g(\016)1190 1755 y Fp(\003)1229 1785 y Fw(\).)2096 b Fg(\003)456 1911 y Fz(Pro)s(of)28 b(of)h(Lemma)e(2.4)p Fw(.)35 b(The)25 b(pro)r(of)f(of)h(of)g(inequalities)g(\(2.35\))f (\(2.36\))g(and)h(\(2.37\))f(can)h(b)r(e)456 2010 y(done)30 b(follo)n(wing)f(line)i(b)n(y)f(line)h(that)g(one)f(of)37 b(\(8.14\))o(,)32 b(\(8.15\))o(,)f(and)g(\(8.16\))f(giv)n(en)f(in)i ([6,)g(Sect.)456 2110 y(11].)53 b(The)34 b(argumen)n(ts)e(giv)n(en)h (there)g(apply)g(also)g(in)g(our)g(case)g(\(where)g Fr(m)2863 2080 y Fs(0)2863 2133 y Fo(\030)2934 2110 y Fw(is)g(replaced)g(b)n(y) 456 2221 y Fr(n)506 2233 y Fo(\030)542 2221 y Fw(\):)39 b(the)29 b(k)n(ey)e(to)r(ol)i(is)f(Lemma)g(8.3,)g(whic)n(h)h(giv)n(es)e (su\016cien)n(tly)h(strict)h(b)r(ounds)f(on)g Fr(n)3172 2233 y Fo(\030)3228 2221 y Fq(\000)18 b Fr(m)3384 2191 y Fs(0)3384 2244 y Fo(\030)3421 2221 y Fw(.)456 2321 y(In)37 b(order)g(to)g(mak)n(e)g(shorter)f(the)i(pap)r(er)g(w)n(e)f (omit)h(the)g(details.)67 b(Finally)-7 b(,)40 b(recalling)c(that)456 2421 y Fr(v)496 2433 y Fo(\030)555 2421 y Fw(=)23 b Fr(p)685 2433 y Fo(\030)721 2421 y Fr(v)764 2391 y Fp(\003)761 2445 y Fo(\030)830 2421 y Fw(and)k(that)h Fq(k)p Fr(@)1257 2433 y Fo(\030)1293 2421 y Fr(p)1335 2433 y Fo(\030)1371 2421 y Fq(k)1413 2433 y Fp(1)1506 2421 y Fq(\024)23 b Fr(C)6 b Fw(,)28 b(\(2.38\))f(follo)n(ws)f(from)i(\(2.26\))f(and)g (\(2.36\))o(.)555 2526 y(W)-7 b(e)34 b(next)g(pro)n(v)n(e)f(\(2.40\))o ({\(2.42\))n(.)56 b(By)33 b(the)i(second)e(inequalit)n(y)g(in)h (\(8.31\))f(and)h(recalling)456 2626 y(the)28 b(de\014nitions)f (\(2.27\))g(and)h(\(8.1\))f(w)n(e)g(ha)n(v)n(e:)1434 2782 y Fq(k)1476 2726 y(p)p 1544 2726 51 4 v 1544 2782 a Fr(\026)14 b(@)1652 2794 y Fo(\030)1689 2782 y Fr(n)1739 2794 y Fo(\030)1793 2782 y Fq(\000)34 b Fw(~)-58 b Fr(m)1949 2748 y Fp(0)1949 2802 y Fo(\030)1985 2782 y Fq(k)2027 2794 y Fp(1)2120 2782 y Fq(\024)23 b Fr(C)6 b(e)2312 2748 y Fp(\000)p Fo(\013\030)2443 2782 y Fr(:)766 b Fw(\(8.58\))456 2936 y(On)25 b(the)i(other)e(hand,)i(since)f Fr(n)1431 2948 y Fo(\030)1490 2936 y Fq(2)d Fr(G)p Fw(\()p Fr(c)1701 2906 y Fp(\003)1740 2936 y Fr(;)14 b(\030)t(;)g(\016)1894 2906 y Fp(\003)1932 2936 y Fw(\),)27 b(the)f(inequalities)g(\(8.6\))g (and)g(\(8.7\))g(apply)g(to)456 3036 y Fr(v)496 3048 y Fo(n)537 3057 y Fi(\030)597 3036 y Fw(=)c Fr(v)724 3048 y Fo(\030)761 3036 y Fw(,)28 b(from)f(whic)n(h)g(it)h(follo)n(ws)f (that)1500 3208 y Fq(k)p Fr(v)1582 3220 y Fo(\030)1637 3208 y Fq(\000)34 b Fw(~)-58 b Fr(m)1793 3174 y Fp(0)1793 3229 y Fo(\030)1829 3208 y Fq(k)1871 3220 y Fp(1)1964 3208 y Fq(\024)23 b Fr(C)6 b(e)2156 3174 y Fp(\000)p Fo(\013)2251 3149 y Fj(0)2273 3174 y Fo(\030)r(=)p Fs(2)2377 3208 y Fr(:)456 3363 y Fw(Then,)27 b(since)h Fr(\031)946 3375 y Fo(\030)983 3363 y Fw(\()p Fr(v)1055 3375 y Fo(\030)1091 3363 y Fw(\))c(=)e(1,)28 b(from)f(the)h(previous)e(estimates)i(and)f (\(2.31\))g(w)n(e)g(get)1398 3459 y Fm(\014)1398 3508 y(\014)1425 3473 y Fq(p)p 1494 3473 V 56 x Fr(\026)14 b(\031)1605 3541 y Fo(\030)1642 3529 y Fw(\()p Fr(@)1718 3541 y Fo(\030)1755 3529 y Fr(n)1805 3541 y Fo(\030)1841 3529 y Fw(\))19 b Fq(\000)f Fw(1)2017 3459 y Fm(\014)2017 3508 y(\014)2067 3529 y Fq(\024)k Fr(C)6 b(e)2258 3495 y Fp(\000)p Fo(\013)2353 3470 y Fj(0)2376 3495 y Fo(\030)r(=)p Fs(2)2479 3529 y Fr(:)456 3681 y Fw(Collecting)27 b(together)f(the)j (ab)r(o)n(v)n(e)d(estimates,)h(\(2.40\))g(and)h(\(2.41\))f(follo)n(w.) 36 b(W)-7 b(e)28 b(are)f(left)h(with)456 3781 y(the)g(pro)r(of)f(of)34 b(\(2.42\))o(.)j(By)28 b(\(2.26\))o(,)g(\(2.35\))o(,)g(\(2.36\))o(,)g (\(8.31\))o(,)g(and)f(\(8.58\))g(w)n(e)g(ha)n(v)n(e:)854 3870 y Fm(\014)854 3920 y(\014)854 3969 y(\014)854 4019 y(\014)882 3990 y Fr(@)926 4002 y Fo(\030)962 3990 y Fr(\031)1009 4002 y Fo(\030)1046 3990 y Fw(\()p Fr(@)1122 4002 y Fo(\030)1159 3990 y Fr(n)1209 4002 y Fo(\030)1245 3990 y Fw(\))19 b Fq(\000)1379 3934 y(p)p 1448 3934 V 56 x Fr(\026)1512 3877 y Fm(Z)1595 3898 y Fs(+)p Fp(1)1558 4066 y Fs(0)1716 3990 y Fr(dx)1846 3934 y Fw(\026)-57 b Fr(m)1904 3904 y Fp(00)1946 3934 y Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))e(\026)-58 b Fr(m)2272 3904 y Fp(0)2296 3934 y Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))p 1831 3971 719 4 v 2011 4047 a Fr(p)2066 4059 y Fs(\026)-46 b Fo(m)2116 4047 y Fw(\()p Fr(\030)22 b Fq(\000)c Fr(x)p Fw(\))2559 3870 y Fm(\014)2559 3920 y(\014)2559 3969 y(\014)2559 4019 y(\014)2610 3990 y Fq(\024)23 b Fr(C)6 b(e)2802 3956 y Fp(\000)p Fo(\013)2897 3931 y Fj(0)2919 3956 y Fo(\030)r(=)p Fs(2)3023 3990 y Fr(:)456 4187 y Fw(But,)28 b(since)43 b(\026)-58 b Fr(m)28 b Fw(is)f(o)r(dd)h(and)f Fr(p)1411 4199 y Fs(\026)-46 b Fo(m)1489 4187 y Fw(ev)n(en,)1190 4269 y Fm(Z)1273 4382 y Fr(dx)1403 4326 y Fw(\026)-57 b Fr(m)1461 4296 y Fp(00)1503 4326 y Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))e(\026)-58 b Fr(m)1829 4296 y Fp(0)1853 4326 y Fw(\()p Fr(\030)23 b Fq(\000)18 b Fr(x)p Fw(\))p 1388 4363 V 1568 4439 a Fr(p)1623 4451 y Fs(\026)-46 b Fo(m)1673 4439 y Fw(\()p Fr(\030)22 b Fq(\000)c Fr(x)p Fw(\))2139 4382 y(=)23 b(0)166 b Fq(8)14 b Fr(\030)26 b(>)c Fw(0)p Fr(:)456 4583 y Fw(Then)27 b(b)n(y)i(\(2.9\))e(and)h(the)g (previous)e(estimate)i(\(2.42\))e(follo)n(ws.)917 b Fg(\003)456 4708 y Fz(Pro)s(of)36 b(of)g(Prop)s(osition)f(2.11)p Fw(.)48 b(The)31 b(critical)g(droplet)g Fr(q)i Fw(=)c Fr(n)2566 4713 y Fs(\026)2559 4728 y Fo(\030)2617 4708 y Fw(+)34 b(\026)-56 b Fr(')32 b Fw(b)r(elongs)f(to)h(the)f(set)456 4817 y Fr(G)p Fw(\()p Fr(c)589 4787 y Fp(\003)627 4817 y Fr(;)673 4795 y Fw(\026)664 4817 y Fr(\030)t(;)14 b(\016)781 4787 y Fp(\003)819 4817 y Fw(\))33 b(for)f Fr(h)g Fw(small.)51 b(Therefore)31 b(items)i(1\),)g(3\),)h(and)e(4\))g(of)g(Prop)r(osition) f(2.11)g(follo)n(w)456 4917 y(from)19 b(the)h(sp)r(ectral)g(analysis)e (in)i(the)g(set)g Fr(G)p Fw(\()p Fr(c)1867 4887 y Fp(\003)1906 4917 y Fr(;)1951 4895 y Fw(\026)1943 4917 y Fr(\030)t(;)14 b(\016)2060 4887 y Fp(\003)2098 4917 y Fw(\).)35 b(Let)20 b(us)f(pro)n(v)n(e)f(item)j(2\).)34 b(The)20 b(existence)456 5016 y(and)27 b(uniqueness)h(of)f Fr(\015)1173 5028 y Fo(v)1236 5016 y Fr(>)c(\015)32 b Fw(solving)c(\(2.62\))f(follo)n(ws)f (from)i([6)o(,)g(Lemma)f(3.1].)37 b(The)27 b(pro)r(of)h(of)456 5116 y(\(2.63\))d(is)h(not)h(giv)n(en)e(in)i([6)o(],)g(but)g(it)g(can)f (b)r(e)g(done)g(in)h(the)g(same)e(w)n(a)n(y)g(as)h(the)h(pro)r(of)f(of) 32 b(\(2.55\))456 5216 y(in)f(Sect.)g(5.)46 b(In)31 b(fact,)g (recalling)f(the)h(de\014nition)g(\(2.56\))f(of)h(the)g(linear)f(op)r (erator)f Fr(L)p Fw(,)i(w)n(e)g(can)p eop %%Page: 44 44 44 43 bop 456 251 a Fs(44)723 b(P)-5 b(.)21 b(BUTT)1522 236 y(\022)1514 251 y(A,)j(A.)e(DE)g(MASI,)f(AND)i(E.)e(R)n(OSA)-5 b(TELLI)456 450 y Fw(rewrite)28 b(the)h(equation)f(for)g Fr(v)k Fw(in)d(\(2.59\))f(as)g Fr(v)g Fw(=)d(\(1)19 b(+)g Fr(\025)p Fw(\))2280 420 y Fp(\000)p Fs(1)2370 450 y Fr(pJ)27 b Fq(\003)18 b Fr(v)s Fw(;)30 b(then,)g(b)n(y)e(arguing)f(as)h (in)456 550 y(Sect.)g(5,)f(w)n(e)g(ha)n(v)n(e:)972 732 y Fr(v)s Fw(\()p Fr(x)20 b Fw(+)e Fr(\030)1233 744 y Fo(q)1270 732 y Fw(\))23 b(=)1413 619 y Fm(Z)1496 639 y Fo(s)1459 807 y(s)p Fp(\000)p Fs(1)1579 732 y Fr(dy)1699 711 y Fw(~)1680 732 y Fr(G)1745 744 y Fo(s)1781 732 y Fw(\()p Fr(x;)14 b(y)s Fw(\))p Fr(v)s Fw(\()p Fr(y)22 b Fw(+)c Fr(\030)2230 744 y Fo(q)2267 732 y Fw(\))p Fr(;)180 b Fq(8)14 b Fr(x)23 b(>)f(s)h Fq(\025)g Fr(`;)456 925 y Fw(with)28 b Fr(\030)681 937 y Fo(q)718 925 y Fw(,)f Fr(`)h Fw(as)f(in)g(\(5.20\))g(and)625 1113 y(~)606 1134 y Fr(G)671 1146 y Fo(s)707 1134 y Fw(\()p Fr(x;)14 b(y)s Fw(\))944 1087 y Fr(:)923 1134 y Fw(=)1040 1031 y Fp(1)1013 1056 y Fm(X)1010 1231 y Fo(n)p Fs(=1)1289 1078 y Fw(1)p 1159 1115 302 4 v 1159 1191 a(\(1)19 b(+)f Fr(\025)p Fw(\))1415 1167 y Fo(n)1484 1021 y Fm(Z)1567 1042 y Fs(+)p Fp(1)1530 1210 y Fo(s)1689 1134 y Fr(dy)1773 1146 y Fs(1)1824 1134 y Fq(\001)c(\001)g(\001)1934 1021 y Fm(Z)2017 1042 y Fs(+)p Fp(1)1980 1210 y Fo(s)2139 1134 y Fr(dy)2223 1146 y Fo(n)p Fp(\000)p Fs(1)2414 1031 y Fo(n)2381 1056 y Fm(Y)2380 1232 y Fo(i)p Fs(=1)2502 1134 y Fr(p)2544 1146 y Fo(\030)2574 1154 y Fi(q)2611 1134 y Fw(\()p Fr(y)2684 1146 y Fo(i)p Fp(\000)p Fs(1)2797 1134 y Fw(\))p Fr(J)8 b Fw(\()p Fr(y)2956 1146 y Fo(i)p Fp(\000)p Fs(1)3087 1134 y Fq(\000)18 b Fr(y)3211 1146 y Fo(i)3238 1134 y Fw(\))p Fr(:)456 1351 y Fw(Finally)24 b(\(2.64\))e(is)h(exactly)f([6)o (,)i(Eq.)f(\(2.23\)],)g(the)g(only)g(di\013erence)g(here)f(is)h(that)g (this)h(estimate)456 1451 y(can)31 b(b)r(e)g(pro)n(v)n(ed)f(also)h(for) f Fr(\020)36 b Fw(=)29 b Fr(\015)1512 1463 y Fo(v)1552 1451 y Fw(,)j(as)f(it)h(can)f(b)r(e)g(c)n(hec)n(k)n(ed)g(b)n(y)g (exploiting)g(the)g(pro)r(of)g(in)h([6)o(,)456 1551 y(Sect.)c(10],)e(w) n(e)i(omit)f(the)h(details.)1847 b Fg(\003)456 1670 y Fz(Pro)s(of)41 b(of)g(Theorem)f(2.5.)61 b Fw(In)36 b(Theorem)f(2.2,)i (Prop)r(osition)d(2.3,)j(and)f(Lemma)g(2.4)f(w)n(e)456 1770 y(ha)n(v)n(e)c(in)n(tro)r(duced)h(functions)h Fr(`)1470 1782 y Fo(i)1529 1770 y Fw(=)e Fr(`)1660 1782 y Fo(i)1687 1770 y Fw(\()p Fr(\024)p Fw(\))i(and)g Fr(c)2035 1782 y Fo(i)2094 1770 y Fw(=)e Fr(c)2226 1782 y Fo(i)2253 1770 y Fw(\()p Fr(\024)p Fw(\))j(for)e Fr(i)f Fw(=)g(0)p Fr(;)14 b Fw(1)p Fr(;)g Fw(2)31 b(\(whic)n(h)i(w)n(e)f(can)456 1870 y(assume)e(non)g(decreasing\).)46 b(In)31 b(the)g(sequel,)g(giv)n (en)i(\026)-45 b Fr(\024)29 b(>)f Fw(0,)j(w)n(e)f(shorthand)2927 1848 y(\026)2921 1870 y Fr(`)2956 1882 y Fo(i)3012 1870 y Fw(=)e Fr(`)3140 1882 y Fo(i)3167 1870 y Fw(\()s(\026)-45 b Fr(\024)p Fw(\))32 b(and)457 1969 y(\026)-43 b Fr(c)492 1981 y Fo(i)542 1969 y Fw(=)23 b Fr(c)666 1981 y Fo(i)693 1969 y Fw(\()s(\026)-45 b Fr(\024)q Fw(\),)28 b Fr(i)22 b Fw(=)h(0)p Fr(;)14 b Fw(1)p Fr(;)g Fw(2.)35 b(F)-7 b(or)27 b Fr(")c(>)g Fw(0)k(and)1789 1947 y(\026)1783 1969 y Fr(`)c(>)1934 1947 y Fw(\026)1928 1969 y Fr(`)1963 1981 y Fs(2)2028 1969 y Fw(to)k(b)r(e)h(\014xed,)g(w)n(e)f(de\014ne:) 1040 2109 y Fr(F)35 b Fw(:)23 b(\000)1230 2114 y Fs(\026)1226 2129 y Fo(`;)s Fs(\026)-36 b Fo(\024)1335 2109 y Fq(\002)18 b(B)1477 2114 y Fs(\026)1473 2129 y Fo(`)o(;)s Fs(\026)-36 b Fo(\024;")1637 2109 y Fq(!)23 b Fn(R)112 b Fw(:)106 b Fr(F)12 b Fw(\()p Fr(\030)t(;)i(m)p Fw(\))24 b(=)e Fr(\031)2475 2121 y Fo(\030)2512 2109 y Fw(\()p Fr(n)2594 2121 y Fo(\030)2649 2109 y Fq(\000)c Fr(m)p Fw(\))p Fr(:)456 2258 y Fw(Clearly)29 b Fr(F)12 b Fw(\()p Fr(\030)t(;)i(n)971 2270 y Fo(\030)1008 2258 y Fw(\))28 b(=)g(0)i(for)g(all)h Fr(\030)h Fq(2)d Fw(\000)1690 2263 y Fs(\026)1686 2278 y Fo(`)o(;)s Fs(\026)-36 b Fo(\024)1776 2258 y Fw(.)46 b(Giv)n(en)31 b Fr(`)2124 2270 y Fs(3)2189 2258 y Fr(>)2287 2236 y Fw(\026)2281 2258 y Fr(`)g Fw(and)f Fr(\024)e(<)j Fw(\026)-45 b Fr(\024)p Fw(,)32 b(w)n(e)e(next)h(apply)f(the)456 2357 y(con)n(traction)c(mapping)h(principle)h(in)f(the)h(\014rst)g (argumen)n(t)e(of)i(the)g(function)540 2501 y Fr(G)23 b Fw(:)g(\000)730 2506 y Fs(\026)726 2521 y Fo(`;)s Fs(\026)-36 b Fo(\024)835 2501 y Fq(\002)18 b(B)977 2506 y Fs(\026)973 2521 y Fo(`;)s Fs(\026)-36 b Fo(\024)o(;")1133 2501 y Fq(\002)18 b Fw(\000)1268 2513 y Fo(`)1296 2521 y Fl(3)1328 2513 y Fo(;\024)1414 2501 y Fq(!)23 b Fn(R)112 b Fw(:)106 b Fr(G)p Fw(\()p Fr(\030)t(;)14 b(m;)g(\030)2139 2467 y Fp(\003)2178 2501 y Fw(\))23 b(=)g Fr(\030)f Fq(\000)c Fw([)p Fr(@)2529 2513 y Fo(\030)2566 2501 y Fr(F)12 b Fw(\()p Fr(\030)2703 2467 y Fp(\003)2741 2501 y Fr(;)i(n)2828 2513 y Fo(\030)2860 2497 y Fj(\003)2899 2501 y Fw(\)])2955 2460 y Fp(\000)p Fs(1)3058 2501 y Fr(F)e Fw(\()p Fr(\030)t(;)i(m)p Fw(\))p Fr(:)456 2643 y Fw(Since)27 b Fr(@)716 2655 y Fo(\030)753 2643 y Fr(F)12 b Fw(\()p Fr(\030)890 2613 y Fp(\003)928 2643 y Fr(;)i(n)1015 2655 y Fo(\030)1047 2639 y Fj(\003)1086 2643 y Fw(\))23 b(=)g Fr(\031)1276 2655 y Fo(\030)1308 2639 y Fj(\003)1347 2643 y Fw(\()p Fr(@)1423 2655 y Fo(\030)1455 2639 y Fj(\003)1495 2643 y Fr(n)1545 2655 y Fo(\030)1577 2639 y Fj(\003)1616 2643 y Fw(\))28 b(then)806 2787 y Fr(@)850 2799 y Fo(\030)887 2787 y Fr(G)23 b Fw(=)g Fr(\031)1110 2799 y Fo(\030)1142 2783 y Fj(\003)1181 2787 y Fw(\()p Fr(@)1257 2799 y Fo(\030)1289 2783 y Fj(\003)1329 2787 y Fr(n)1379 2799 y Fo(\030)1411 2783 y Fj(\003)1449 2787 y Fw(\))1481 2753 y Fp(\000)p Fs(1)1585 2787 y Fw([)p Fr(\031)1655 2799 y Fo(\030)1687 2783 y Fj(\003)1726 2787 y Fw(\()p Fr(@)1802 2799 y Fo(\030)1834 2783 y Fj(\003)1874 2787 y Fr(n)1924 2799 y Fo(\030)1956 2783 y Fj(\003)1994 2787 y Fw(\))c Fq(\000)f Fr(\031)2175 2799 y Fo(\030)2212 2787 y Fw(\()p Fr(@)2288 2799 y Fo(\030)2325 2787 y Fr(n)2375 2799 y Fo(\030)2411 2787 y Fw(\))g(+)g Fr(@)2588 2799 y Fo(\030)2625 2787 y Fr(\031)2672 2799 y Fo(\030)2709 2787 y Fw(\()p Fr(n)2791 2799 y Fo(\030)2845 2787 y Fq(\000)h Fr(m)p Fw(\)])14 b Fr(;)456 2927 y Fw(from)27 b(whic)n(h,)g(b)n(y)h(using)g(\(2.38\))f(and)g(\(2.41\),)835 3067 y Fq(j)p Fr(@)902 3079 y Fo(\030)938 3067 y Fr(G)p Fq(j)d(\024)e Fw(\()1169 3011 y Fq(p)p 1239 3011 51 4 v 1239 3067 a Fr(\026)c Fw(+)i(\026)-44 b Fr(c)1426 3079 y Fs(2)1464 3067 y Fw(\))1496 3000 y Fm(\000)1534 3067 y Fq(j)p Fr(\031)1604 3079 y Fo(\030)1636 3063 y Fj(\003)1675 3067 y Fw(\()p Fr(@)1751 3079 y Fo(\030)1783 3063 y Fj(\003)1823 3067 y Fr(n)1873 3079 y Fo(\030)1905 3063 y Fj(\003)1944 3067 y Fw(\))18 b Fq(\000)g Fr(\031)2124 3079 y Fo(\030)2161 3067 y Fw(\()p Fr(@)2237 3079 y Fo(\030)2274 3067 y Fr(n)2324 3079 y Fo(\030)2360 3067 y Fw(\))p Fq(j)h Fw(+)h(\026)-44 b Fr(c)2553 3079 y Fs(2)2590 3067 y Fq(k)p Fr(m)18 b Fq(\000)g Fr(n)2856 3079 y Fo(\030)2892 3067 y Fq(k)2934 3079 y Fp(1)3004 3000 y Fm(\001)3042 3067 y Fr(:)456 3207 y Fw(But)29 b(\(2.19\))f(implies)h Fq(k)p Fr(n)1238 3219 y Fo(\030)1293 3207 y Fq(\000)19 b Fr(n)1427 3219 y Fo(\030)1459 3203 y Fj(\003)1498 3207 y Fq(k)1540 3219 y Fp(1)1635 3207 y Fq(\024)25 b Fw(\()r(\026)-44 b Fr(c)1793 3219 y Fs(0)1850 3207 y Fw(+)19 b Fq(k)c Fw(\026)-57 b Fr(m)2049 3177 y Fp(0)2072 3207 y Fq(k)2114 3219 y Fp(1)2183 3207 y Fw(\))p Fq(j)p Fr(\030)24 b Fq(\000)19 b Fr(\030)2422 3177 y Fp(\003)2461 3207 y Fq(j)p Fw(,)29 b(hence)g(there)g(is)g(a)f(constan)n(t)456 3306 y Fr(C)h Fw(=)23 b Fr(C)6 b Fw(\()s(\026)-45 b Fr(\024)p Fw(\))28 b(so)f(that:)817 3446 y Fq(j)p Fr(@)884 3458 y Fo(\030)921 3446 y Fr(G)p Fq(j)c(\024)g Fr(C)1185 3379 y Fm(\000)1223 3446 y Fq(j)p Fr(\031)1293 3458 y Fo(\030)1325 3442 y Fj(\003)1364 3446 y Fw(\()p Fr(@)1440 3458 y Fo(\030)1472 3442 y Fj(\003)1512 3446 y Fr(n)1562 3458 y Fo(\030)1594 3442 y Fj(\003)1633 3446 y Fw(\))18 b Fq(\000)g Fr(\031)1813 3458 y Fo(\030)1850 3446 y Fw(\()p Fr(@)1926 3458 y Fo(\030)1963 3446 y Fr(n)2013 3458 y Fo(\030)2049 3446 y Fw(\))p Fq(j)h Fw(+)f Fq(j)p Fr(\030)k Fq(\000)c Fr(\030)2410 3412 y Fp(\003)2449 3446 y Fq(j)g Fw(+)g Fq(k)p Fr(m)g Fq(\000)g Fr(n)2839 3458 y Fo(\030)2871 3442 y Fj(\003)2910 3446 y Fq(k)2952 3458 y Fp(1)3022 3379 y Fm(\001)3060 3446 y Fr(:)456 3594 y Fw(Using)27 b(again)f(\(2.41\),)h(w)n(e)h(can)f(no)n (w)g(\014x)1744 3572 y(\026)1738 3594 y Fr(`)h Fw(so)e(large)h(that)h (\(for)f(all)g Fr(h)c(<)g(e)2758 3564 y Fp(\000)p Fs(2)p Fo(\013`)2914 3572 y Fl(3)2950 3594 y Fw(\))1374 3773 y Fr(C)1439 3702 y Fm(\014)1439 3752 y(\014)1467 3773 y Fr(\031)1514 3785 y Fo(\030)1546 3769 y Fj(\003)1586 3773 y Fw(\()p Fr(@)1662 3785 y Fo(\030)1694 3769 y Fj(\003)1733 3773 y Fr(n)1783 3785 y Fo(\030)1815 3769 y Fj(\003)1854 3773 y Fw(\))c Fq(\000)f Fr(\031)2035 3785 y Fo(\030)2072 3773 y Fw(\()p Fr(@)2148 3785 y Fo(\030)2184 3773 y Fr(n)2234 3785 y Fo(\030)2270 3773 y Fw(\))2302 3702 y Fm(\014)2302 3752 y(\014)2353 3773 y Fr(<)2451 3717 y Fw(1)p 2451 3754 42 4 v 2451 3830 a(8)2502 3773 y Fr(:)456 3941 y Fw(Then,)35 b(if)g Fq(j)p Fr(\030)27 b Fq(\000)22 b Fr(\030)999 3911 y Fp(\003)1037 3941 y Fq(j)34 b Fr(<)g(r)i Fw(and)e Fq(k)p Fr(m)22 b Fq(\000)h Fr(n)1709 3953 y Fo(\030)1741 3937 y Fj(\003)1780 3941 y Fq(k)1822 3953 y Fp(1)1925 3941 y Fr(<)39 b Fw(\026)-47 b Fr(")33 b Fw(with)i Fr(r)r Fw(,)42 b(\026)-48 b Fr(")34 b(>)f Fw(0)h(small)f(enough,)i(w)n(e)f (\014nd)456 4043 y Fq(j)p Fr(@)523 4055 y Fo(\030)559 4043 y Fr(G)p Fq(j)39 b Fr(<)f Fw(1)p Fr(=)p Fw(4)d(\(for)i(all)g Fr(h)h(<)g(e)1472 4013 y Fp(\000)p Fs(2)p Fo(\013`)1628 4021 y Fl(3)1664 4043 y Fw(\).)65 b(Moreo)n(v)n(er,)36 b(if)i(w)n(e)e(assume)g(also)g Fr(`)2907 4055 y Fs(3)2983 4043 y Fr(>)3091 4021 y Fw(\026)3086 4043 y Fr(`)24 b Fw(+)g Fr(r)40 b Fw(and)456 4143 y Fr(\024)22 b(<)k Fw(\026)-45 b Fr(\024)18 b Fq(\000)g Fr(r)r Fw(,)29 b(the)f(function)g Fr(G)p Fw(\()p Fq(\001)p Fr(;)14 b(m;)g(\030)1629 4113 y Fp(\003)1668 4143 y Fw(\))28 b(is)f(de\014ned)h(for)f(all)g Fr(\030)h Fq(2)23 b Fw(\()p Fr(\030)2553 4113 y Fp(\003)2610 4143 y Fq(\000)18 b Fr(r)n(;)c(\030)2805 4113 y Fp(\003)2862 4143 y Fw(+)k Fr(r)r Fw(\))29 b(and)533 4283 y Fq(j)p Fr(G)p Fw(\()p Fr(\030)t(;)14 b(m;)g(\030)880 4248 y Fp(\003)918 4283 y Fw(\))19 b Fq(\000)f Fr(\030)1092 4248 y Fp(\003)1130 4283 y Fq(j)23 b(\024)g(j)p Fr(G)p Fw(\()p Fr(\030)t(;)14 b(m;)g(\030)1611 4248 y Fp(\003)1650 4283 y Fw(\))19 b Fq(\000)f Fr(G)p Fw(\()p Fr(\030)t(;)c(n)2008 4295 y Fo(\030)2040 4278 y Fj(\003)2079 4283 y Fr(;)g(\030)2156 4248 y Fp(\003)2194 4283 y Fw(\))p Fq(j)19 b Fw(+)f Fq(j)p Fr(G)p Fw(\()p Fr(\030)t(;)c(n)2598 4295 y Fo(\030)2630 4278 y Fj(\003)2670 4283 y Fr(;)g(\030)2747 4248 y Fp(\003)2785 4283 y Fw(\))19 b Fq(\000)f Fr(G)p Fw(\()p Fr(\030)3056 4248 y Fp(\003)3095 4283 y Fr(;)c(n)3182 4295 y Fo(\030)3214 4278 y Fj(\003)3252 4283 y Fr(;)g(\030)3329 4248 y Fp(\003)3368 4283 y Fw(\))p Fq(j)1187 4458 y(\024)22 b(j)p Fr(\031)1344 4470 y Fo(\030)1376 4454 y Fj(\003)1416 4458 y Fw(\()p Fr(@)1492 4470 y Fo(\030)1524 4454 y Fj(\003)1563 4458 y Fr(n)1613 4470 y Fo(\030)1645 4454 y Fj(\003)1684 4458 y Fw(\))p Fq(j)1739 4424 y Fp(\000)p Fs(1)1828 4458 y Fq(j)p Fr(\031)1898 4470 y Fo(\030)1935 4458 y Fw(\()p Fr(n)2017 4470 y Fo(\030)2049 4454 y Fj(\003)2107 4458 y Fq(\000)c Fr(m)p Fw(\))p Fq(j)g Fw(+)2429 4402 y Fq(j)p Fr(\030)23 b Fq(\000)18 b Fr(\030)2634 4372 y Fp(\003)2672 4402 y Fq(j)p 2429 4439 267 4 v 2542 4515 a Fw(4)1187 4667 y Fq(\024)24 b Fw(\026)-44 b Fr(c)1310 4679 y Fs(2)1347 4667 y Fw(\()1379 4611 y Fq(p)p 1449 4611 51 4 v 1449 4667 a Fr(\026)18 b Fw(+)i(\026)-44 b Fr(c)1636 4679 y Fs(2)1673 4667 y Fw(\))p Fq(k)p Fr(n)1797 4679 y Fo(\030)1829 4662 y Fj(\003)1887 4667 y Fq(\000)18 b Fr(m)p Fq(k)2085 4679 y Fp(1)2173 4667 y Fw(+)2266 4610 y Fq(j)p Fr(\030)k Fq(\000)c Fr(\030)2470 4580 y Fp(\003)2509 4610 y Fq(j)p 2266 4647 267 4 v 2378 4724 a Fw(4)1187 4849 y Fq(\024)24 b Fw(\026)-44 b Fr(c)1310 4861 y Fs(2)1347 4849 y Fw(\()1379 4793 y Fq(p)p 1449 4793 51 4 v 1449 4849 a Fr(\026)18 b Fw(+)i(\026)-44 b Fr(c)1636 4861 y Fs(2)1673 4849 y Fw(\))p Fr(")19 b Fw(+)1857 4792 y Fr(r)p 1856 4830 42 4 v 1856 4906 a Fw(4)1930 4849 y Fr(<)2029 4792 y(r)p 2028 4830 V 2028 4906 a Fw(2)2079 4849 y Fr(;)456 5016 y Fw(where)f(w)n(e)g(used)g(\(2.31\),)i(\(2.41\))o(,)h(and)d(w)n(e)g (further)h(assumed)k(\026)-47 b Fr(")18 b Fw(so)g(small)g(that)h(4)r (\026)-44 b Fr(c)2959 5028 y Fs(2)2996 5016 y Fw(\()3028 4965 y Fq(p)p 3097 4965 51 4 v 51 x Fr(\026)q Fw(+)r(\026)g Fr(c)3249 5028 y Fs(2)3286 5016 y Fw(\))6 b(\026)-48 b Fr(")23 b(<)456 5116 y(r)r Fw(.)50 b(Then)32 b(w)n(e)f(can)g(apply)h (the)g(con)n(traction)e(mapping)h(principle,)i(b)n(y)f(obtaining)f (there)g(is)h(a)456 5216 y(unique)26 b(solution)f Fr(\030)i Fw(=)c(\004\()p Fr(m;)14 b(\030)1424 5185 y Fp(\003)1462 5216 y Fw(\))26 b(of)g(the)g(equation)f Fr(F)12 b Fw(\()p Fr(\030)t(;)i(m)p Fw(\))24 b(=)e(0)k(in)g(\()p Fr(\030)2716 5185 y Fp(\003)2769 5216 y Fq(\000)14 b Fr(r)n(;)g(\030)2960 5185 y Fp(\003)3013 5216 y Fw(+)g Fr(r)r Fw(\))27 b(for)e(an)n(y)p eop %%Page: 45 45 45 44 bop 781 251 a Fs(SLO)n(W)29 b(MOTION)g(AND)g(MET)-5 b(AST)g(ABILITY)29 b(F)n(OR)g(A)g(NON)g(LOCAL)h(EQUA)-5 b(TION)258 b(45)456 450 y Fr(\030)496 420 y Fp(\003)562 450 y Fq(2)28 b Fw(\000)697 462 y Fo(`)725 470 y Fl(3)757 462 y Fo(;\024)851 450 y Fw(and)i Fr(m)h Fw(suc)n(h)f(that)h Fq(k)p Fr(m)20 b Fq(\000)g Fr(n)1762 462 y Fo(\030)1794 446 y Fj(\003)1833 450 y Fq(k)1875 462 y Fp(1)1972 450 y Fr(<)34 b Fw(\026)-48 b Fr(")p Fw(.)46 b(Moreo)n(v)n(er)27 b Fr(m)h Fq(7!)g Fw(\004\()p Fr(m;)14 b(\030)2989 420 y Fp(\003)3028 450 y Fw(\))31 b(is)f Fr(C)3242 420 y Fs(1)3311 450 y Fw(and)456 550 y Fq(j)p Fw(\004\()p Fr(m;)14 b(\030)716 520 y Fp(\003)754 550 y Fw(\))19 b Fq(\000)f Fr(\030)928 520 y Fp(\003)966 550 y Fq(j)24 b Fr(<)e(r)r(=)p Fw(2.)555 649 y(W)-7 b(e)35 b(\014nally)f(observ)n(e)e(that)j(if)g Fr(`)1570 661 y Fs(3)1641 649 y Fw(is)f(large)f(enough)g(there)h(is)g Fr(")2581 661 y Fs(0)2652 649 y Fq(2)h Fw(\(0)p Fr(;)19 b Fw(\026)-47 b Fr(")o Fw(])35 b(suc)n(h)f(that)g(the)456 749 y(follo)n(wing)28 b(holds.)40 b(If)30 b Fr(\030)1186 719 y Fp(\003)1224 749 y Fw(,)f Fr(\030)1316 719 y Fp(\003\003)1414 749 y Fq(2)d Fw(\000)1547 761 y Fo(`)1575 769 y Fl(3)1607 761 y Fo(;\024)1699 749 y Fw(and)j Fq(k)p Fr(n)1954 761 y Fo(\030)1986 745 y Fj(\003)2043 749 y Fq(\000)19 b Fr(n)2177 761 y Fo(\030)2209 745 y Fj(\003\003)2279 749 y Fq(k)2321 761 y Fp(1)2416 749 y Fr(<)25 b Fw(2)p Fr(")2587 761 y Fs(0)2653 749 y Fw(then)k Fq(j)p Fr(\030)2906 719 y Fp(\003)2964 749 y Fq(\000)19 b Fr(\030)3088 719 y Fp(\003\003)3160 749 y Fq(j)26 b Fr(<)f(r)r(=)p Fw(2.)456 849 y(In)34 b(fact,)i(b)n(y)d(using)i(\(2.9\))f(and)g(\(2.19\))o(,)i (from)e(the)g(condition)g Fq(j)p Fr(\030)2537 818 y Fp(\003)2598 849 y Fq(\000)22 b Fr(\030)2725 818 y Fp(\003\003)2798 849 y Fq(j)34 b(\025)f Fr(r)r(=)p Fw(2)h(it)g(follo)n(ws)456 948 y(there)27 b(is)g Fr(C)j Fw(=)22 b Fr(C)6 b Fw(\()s(\026)-45 b Fr(\024)q Fw(\))28 b(so)f(that)951 1113 y Fq(j)p Fr(n)1024 1125 y Fo(\030)1056 1109 y Fj(\003)1095 1113 y Fw(\(0\))19 b Fq(\000)f Fr(n)1353 1125 y Fo(\030)1385 1109 y Fj(\003\003)1454 1113 y Fw(\(0\))p Fq(j)23 b(\025)1694 1021 y Fm(\020)1744 1113 y Fr(\013ae)1880 1078 y Fp(\000)p Fo(\013`)2003 1086 y Fl(3)2057 1113 y Fq(\000)18 b Fr(C)6 b(e)2244 1078 y Fp(\000)11 b Fs(min)o Fp(f)p Fo(\013)2494 1086 y Fl(0)2527 1078 y Fs(;)p Fo(\013)p Fs(+)p Fo(\016)2670 1086 y Fl(0)2703 1078 y Fp(g)p Fo(`)2765 1086 y Fl(3)2801 1021 y Fm(\021)2875 1057 y Fr(r)p 2874 1094 42 4 v 2874 1170 a Fw(2)2926 1113 y Fr(;)456 1276 y Fw(and,)25 b(for)g Fr(`)798 1288 y Fs(3)861 1276 y Fw(large)f(enough,)h(the)h(r.h.s.)f(is) g(not)h(smaller)e(than)i(2)p Fr(")2500 1288 y Fs(0)2562 1276 y Fw(for)f(an)n(y)f(su\016cien)n(tly)i(small)456 1375 y Fr(")495 1387 y Fs(0)532 1375 y Fw(.)555 1475 y(W)-7 b(e)42 b(no)n(w)e(pro)n(v)n(e)g(Theorem)g(2.5.)77 b(Giv)n(en)41 b Fr(\024)46 b(>)f Fw(0,)f(w)n(e)d(\014x)j(\026)-45 b Fr(\024)46 b(>)f(\024)c Fw(and)g(then)h(w)n(e)f(in-)456 1574 y(tro)r(duce)f(the)g(parameters)f Fr(")1408 1586 y Fs(0)1445 1574 y Fw(,)1517 1553 y(\026)1512 1574 y Fr(`)o Fw(,)44 b(and)c Fr(`)1822 1586 y Fs(3)1899 1574 y Fw(so)g(that)g(all)h(the)f(previous)f(statemen)n(ts)i(hold.)456 1674 y(Then)j(w)n(e)h(de\014ne)f(the)h(map)g(\004)51 b(:)h Fq(B)1683 1686 y Fo(`)1711 1694 y Fl(3)1743 1686 y Fo(;\024;")1853 1694 y Fl(0)1940 1674 y Fq(!)f Fw(\000)2130 1679 y Fs(\026)2126 1694 y Fo(`;)s Fs(\026)-36 b Fo(\024)2261 1674 y Fw(b)n(y)45 b(setting)f(\004\()p Fr(m)p Fw(\))52 b(=)f(\004\()p Fr(m;)14 b(\030)3280 1644 y Fp(\003)3319 1674 y Fw(\))45 b(if)456 1776 y Fq(k)p Fr(m)26 b Fq(\000)h Fr(n)739 1788 y Fo(\030)771 1772 y Fj(\003)810 1776 y Fq(k)852 1788 y Fp(1)966 1776 y Fr(<)44 b(")1114 1788 y Fs(0)1192 1776 y Fw(\(the)d(existence)f(of)g(suc)n(h)g Fr(\030)2093 1746 y Fp(\003)2176 1776 y Fq(2)45 b Fw(\000)2328 1788 y Fo(`)2356 1796 y Fl(3)2388 1788 y Fo(;\024)2491 1776 y Fw(follo)n(ws)40 b(from)g(\(2.43\))o(\).)76 b(W)-7 b(e)456 1876 y(see)35 b(that)i(the)f(de\014nition)g(is)g(w)n(ell)g(p)r (osed:)54 b(assume)35 b Fq(k)p Fr(m)23 b Fq(\000)h Fr(n)2439 1888 y Fo(\030)2471 1872 y Fj(\003\003)2540 1876 y Fq(k)2582 1888 y Fp(1)2689 1876 y Fr(<)37 b(")2830 1888 y Fs(0)2903 1876 y Fw(for)f(some)f(other)456 1975 y Fr(\030)496 1945 y Fp(\003\003)614 1975 y Fq(2)47 b Fw(\000)768 1987 y Fo(`)796 1995 y Fl(3)828 1987 y Fo(;\024)891 1975 y Fw(,)e(hence)d Fq(k)p Fr(n)1296 1987 y Fo(\030)1328 1971 y Fj(\003)1394 1975 y Fq(\000)27 b Fr(n)1536 1987 y Fo(\030)1568 1971 y Fj(\003\003)1638 1975 y Fq(k)1680 1987 y Fp(1)1796 1975 y Fr(<)46 b Fw(2)p Fr(")1988 1987 y Fs(0)2066 1975 y Fw(and)c(then)g Fq(j)p Fr(\030)2508 1945 y Fp(\003)2574 1975 y Fq(\000)27 b Fr(\030)2706 1945 y Fp(\003\003)2779 1975 y Fq(j)46 b(\024)g Fr(r)r(=)p Fw(2)41 b(for)h(what)456 2075 y(stated)37 b(b)r(efore.)68 b(Then)38 b Fq(j)p Fw(\004\()p Fr(m;)14 b(\030)1515 2045 y Fp(\003\003)1588 2075 y Fw(\))25 b Fq(\000)g Fr(\030)1775 2045 y Fp(\003)1814 2075 y Fq(j)40 b(\024)g(j)p Fw(\004\()p Fr(m;)14 b(\030)2242 2045 y Fp(\003\003)2315 2075 y Fw(\))25 b Fq(\000)g Fr(\030)2502 2045 y Fp(\003\003)2574 2075 y Fq(j)h Fw(+)f Fq(j)p Fr(\030)2776 2045 y Fp(\003\003)2873 2075 y Fq(\000)g Fr(\030)3003 2045 y Fp(\003)3042 2075 y Fq(j)40 b Fr(<)g(r)h Fw(and,)456 2175 y(b)n(y)30 b(lo)r(cal)g(uniqueness,)i(\004\()p Fr(m;)14 b(\030)1456 2144 y Fp(\003\003)1529 2175 y Fw(\))28 b(=)g(\004\()p Fr(m;)14 b(\030)1919 2144 y Fp(\003)1958 2175 y Fw(\).)47 b(Moreo)n(v)n(er,)29 b(to)i(pro)n(v)n(e)e(\(2.45\))o(,)j(w)n(e)e (observ)n(e)456 2274 y(that)36 b Fq(k)p Fr(m)23 b Fq(\000)g Fr(n)920 2286 y Fo(\030)957 2274 y Fq(k)999 2286 y Fp(1)1105 2274 y Fq(\024)36 b(k)p Fr(m)24 b Fq(\000)f Fr(n)1483 2286 y Fo(\030)1515 2270 y Fj(\003)1554 2274 y Fq(k)1596 2286 y Fp(1)1690 2274 y Fw(+)g(\()1810 2222 y Fq(p)p 1880 2222 51 4 v 1880 2274 a Fr(\026)h Fw(+)h(\026)-44 b Fr(c)2078 2286 y Fs(2)2115 2274 y Fw(\))p Fq(j)p Fr(\030)29 b Fq(\000)23 b Fr(\030)2363 2244 y Fp(\003)2402 2274 y Fq(j)36 b Fw(and,)h(on)f(the)g(other)f(hand,)j(if)456 2378 y Fr(\030)29 b Fw(=)c(\004\()p Fr(m;)14 b(\030)848 2348 y Fp(\003)886 2378 y Fw(\))30 b(then)f Fq(j)p Fr(\030)24 b Fq(\000)19 b Fr(\030)1345 2348 y Fp(\003)1383 2378 y Fq(j)25 b(\024)g Fw(const)o Fq(j)p Fr(\031)1780 2390 y Fo(\030)1812 2374 y Fj(\003)1852 2378 y Fw(\()p Fr(@)1928 2390 y Fo(\030)1960 2374 y Fj(\003)1999 2378 y Fr(n)2049 2390 y Fo(\030)2081 2374 y Fj(\003)2120 2378 y Fw(\))p Fq(j)2175 2348 y Fp(\000)p Fs(1)2265 2378 y Fq(j)p Fr(\031)2335 2390 y Fo(\030)2367 2374 y Fj(\003)2406 2378 y Fw(\()p Fr(n)2488 2390 y Fo(\030)2520 2374 y Fj(\003)2579 2378 y Fq(\000)18 b Fr(m)p Fw(\))p Fq(j)26 b(\024)f Fr(C)6 b Fq(k)p Fr(m)19 b Fq(\000)g Fr(n)3239 2390 y Fo(\030)3271 2374 y Fj(\003)3309 2378 y Fq(k)3351 2390 y Fp(1)3421 2378 y Fw(,)456 2478 y(for)34 b(some)g Fr(C)42 b Fw(=)35 b Fr(C)6 b Fw(\()s(\026)-45 b Fr(\024)p Fw(\).)60 b(This)35 b(last)f(b)r(ound)i(also)e(pro)n(v)n(e)f(\(2.46\))o(,)k(since)e Fq(k)p Fr(m)23 b Fq(\000)g Fr(n)3055 2490 y Fo(\030)3085 2498 y Fl(0)3121 2478 y Fq(k)3163 2490 y Fp(1)3268 2478 y Fr(<)35 b(")3407 2490 y Fs(0)456 2577 y Fw(implies)h Fr(\030)41 b Fw(=)36 b(\004\()p Fr(m;)14 b(\030)1157 2589 y Fs(0)1195 2577 y Fw(\).)62 b(Finally)36 b(Corollary)d(2.6)i(is)h (an)f(immediate)h(consequence)f(of)h(the)456 2677 y(ab)r(o)n(v)n(e)26 b(reasoning.)456 2856 y Fz(Ac)m(kno)m(wledgmen)m(ts.)34 b Fw(W)-7 b(e)25 b(are)e(indebted)i(to)f(G.)h(F)-7 b(usco,)25 b(E.)f(Olivieri,)g(and)g(E.)g(Presutti)g(for)456 2956 y(discussions)i(and)h(commen)n(ts.)36 b(Tw)n(o)27 b(of)g(us)g(\(P)-7 b(.)27 b(Butt\022)-42 b(a)28 b(and)f(A.)g(De)h(Masi\))f(ac)n(kno)n (wledge)e(the)456 3056 y(v)n(ery)32 b(kind)j(hospitalit)n(y)e(of)h(the) g(IHES.)g(The)g(researc)n(h)e(has)i(b)r(een)g(partially)f(supp)r(orted) h(b)n(y)456 3155 y(MURST,)28 b(CNR,)g(and)f(b)n(y)h(NA)-7 b(TO)28 b(Gran)n(t)f(PST.CLG.976552.)1708 3339 y Fv(References)491 3472 y Fy([1])35 b(P)-6 b(.)28 b(Butt\022)-35 b(a:)41 b(On)29 b(the)g(v)l(alidit)n(y)f(of)g(an)h(Einstein)f(relation)h(in)f (mo)r(dels)f(of)h(in)n(terface)h(dynamics.)f Ft(J.)i(Stat.)601 3555 y(Phys.)c Ff(72)c Fy(\(1993\))k(1401{1406.)491 3638 y([2])35 b(J.)28 b(Carr,)h(R.)g(P)n(ego:)42 b(Metastable)30 b(patterns)g(in)f(solutions)g(of)g Fe(u)2353 3646 y Fd(t)2409 3638 y Fy(=)f Fe(")2525 3615 y Fc(2)2559 3638 y Fe(u)2600 3646 y Fd(xx)2693 3638 y Fb(\000)19 b Fe(f)7 b Fy(\()p Fe(u)p Fy(\).)30 b Ft(Pr)l(o)l(c.)h(R)l(oy.)g(So)l(c.)601 3721 y(Edinbur)l(gh)c(Se)l(ct.)e(A)g Ff(116)e Fy(\(1990\))i(133{160.) 491 3804 y([3])35 b(J.)23 b(Carr,)g(R.)g(P)n(ego:)32 b(In)n(v)l(arian)n(t)25 b(manifolds)d(for)h(metastable)h(patterns)h(in) e Fe(u)2648 3812 y Fd(t)2696 3804 y Fy(=)c Fe(")2803 3781 y Fc(2)2837 3804 y Fe(u)2878 3812 y Fd(xx)2968 3804 y Fb(\000)c Fe(f)7 b Fy(\()p Fe(u)p Fy(\).)24 b Ft(Comm.)601 3887 y(Pur)l(e)i(Appl.)g(Math.)g Ff(42)d Fy(\(1989\))j(523{576.)491 3970 y([4])35 b(M.)21 b(Cassandro,)i(A.)f(Galv)n(es,)h(E.)g(Olivieri,)d (E.)j(V)-6 b(ares:)30 b(Metastable)24 b(b)r(eha)n(vior)g(of)e(sto)r(c)n (hastic)i(dynamics:)601 4053 y(a)f(path)n(wise)i(approac)n(h.)f Ft(J.)i(Stat.)f(Phys.)h Ff(35)d Fy(\(1984\))j(603{634.)491 4136 y([5])35 b(A.)21 b(De)i(Masi,)e(T.)h(Gobron,)g(E.)g(Presutti:)31 b(T)-6 b(ra)n(v)n(eling)22 b(fron)n(ts)g(in)g(non-lo)r(cal)g(ev)n (olution)i(equations.)f Ft(A)n(r)l(ch.)601 4219 y(R)l(ational)k(Me)l (ch.)e(A)n(nal.)h Ff(132)d Fy(\(1995\))j(143{205.)491 4302 y([6])35 b(A.)22 b(De)i(Masi,)e(E.)h(Olivieri,)e(E.)i(Presutti:)31 b(Sp)r(ectral)24 b(prop)r(erties)f(of)g(in)n(tegral)h(op)r(erators)g 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b(Galv)n(es,)j(E.)d(Olivieri,)g(E.)h(V)-6 b(ares:)40 b(Metastabilit)n(y)29 b(for)f(a)g(class)g(of)g(dynamical)f(systems)h (sub)t(ject)h(to)601 699 y(small)21 b(random)i(p)r(erturbations.)h Ft(A)n(nn.)i(Pr)l(ob)l(ab.)h Ff(15)c Fy(\(1987\))i(1288{1305.)456 782 y([14])35 b(D.)29 b(Henry:)44 b(Geometric)30 b(theory)h(of)f (semilinear)e(parab)r(olic)i(equations)i Ft(Springer)f(L)l(e)l(ctur)l (es)h(Notes)f(in)601 865 y(Mathematics)26 b Ff(840)c Fy(Springer-V)-6 b(erlag,)23 b(Berlin)f(New)i(Y)-6 b(ork)24 b(1981.)456 948 y([15])35 b(M.)29 b(Kac,)i(G.)f(Uhlen)n(b)r(ec)n(k,)i (P)-6 b(.C.)29 b(Hemmer:)41 b(On)30 b(the)g(v)l(an)h(der)f(W)-6 b(aals)30 b(theory)h(of)e(v)l(ap)r(or-liquid)g(equi-)601 1031 y(librium.)i(I.)i(Discussion)h(of)g(a)g(one)g(dimensional)f(mo)r (del.)g Ft(J.)i(Math.)h(Phys.)f Ff(4)f Fy(\(1963\))i(216{228.)f(I)r(I.) 601 1114 y(Discussion)23 b(of)g(the)i(distribution)f(functions.)f Ft(J.)j(Math.)g(Phys.)g Ff(4)d Fy(\(1963\))j(229{247.)f(I)r(I)r(I.)f (Discussion)g(of)601 1197 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