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(jump)e(pro)s(cesses,)h(in)f(a)i(more)f(general)h(framew)m(ork)f(than)g (ours,)0 1699 y Fe(i.e.)j Fo(without)e(assuming)g(rev)m(ersibilit)m(y)e (w.r.t.)43 b(a)32 b(Gibbs)d(measure.)43 b(In)31 b([Pre],)g(and,)g(more) h(recen)m(tly)-8 b(,)32 b(in)e([FF)m(G],)0 1826 y(man)m(y)g(of)h(their) e(general)i(prop)s(erties)e(are)h(studied.)227 1954 y(In)39 b(this)f(pap)s(er)g(w)m(e)i(are)f(in)m(terested)h(in)e(the)h(approac)m (h)h(to)g(the)f(in)m(v)-5 b(arian)m(t)38 b(measure)i(in)d(the)j Fn(L)3592 1921 y Fr(2)3672 1954 y Fo(sense,)0 2081 y(and,)e(in)d (particular,)h(w)m(e)h(sho)m(w)f(that)h(for)g(a)f(p)s(ositiv)m(e,)h (\014nite)f(range,)i(pair)d(p)s(oten)m(tial,)j(if)e Fn(z)t Fo(,)i Fn(\014)j Fo(are)c(suc)m(h)f(that)0 2208 y(there)28 b(is)f(a)h(con)m(v)m(ergen)m(t)j(cluster)c(expansion)g(\(see)i (condition)d(\(CE\))i(in)f(Section)g(2\),)j(then)d(the)i(generator)g (of)f(the)0 2335 y(pro)s(cess)h Fn(L)378 2291 y Fj(\021)378 2362 y Fr(\003)461 2335 y Fo(has)g(a)h(sp)s(ectral)f(gap)g(whic)m(h)g 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y(\(1\))46 b(The)32 b(sp)s(ectral)f(gap)h(of)h(the)f(generator)h(is)e(strictly)g(p)s (ositiv)m(e)g(uniformly)e(in)i(the)h(v)m(olume)g(and)f(b)s(oundary)227 3507 y(condition.)66 3635 y(\(2\))46 b(The)32 b(logarithmic)e(Sob)s (olev)h(constan)m(t)j(is)d(b)s(ounded)f(uniformly)f(in)h(the)i(v)m (olume)g(and)g(b)s(oundary)e(condi-)227 3762 y(tion.)66 3890 y(\(3\))46 b(The)40 b(co)m(v)-5 b(ariance)42 b(w.r.t.)72 b(the)41 b(Gibbs)e(measure)h(of)h(t)m(w)m(o)h(lo)s(cal)e(functions)f (deca)m(ys)j(exp)s(onen)m(tially)d(fast)227 4017 y(in)e(the)h(distance) g(of)g(the)g(\\supp)s(orts")f(of)i(the)f(functions,)h(uniformly)c(in)h (the)j(v)m(olume)f(and)f(b)s(oundary)227 4144 y(condition.)0 4297 y(W)-8 b(e)32 b(observ)m(e)g(that)f(for)g(the)g(system)g(w)m(e)h (consider)e(in)g(this)f(pap)s(er,)i(there)g(is)f(no)h(hop)s(e)f(of)h (pro)m(ving)g(\(in)f(general\))0 4424 y(a)k(logarithmic)f(Sob)s(olev)g (inequalit)m(y)f(\(LSI\).)i(Ev)m(en)g(w)m(orse)g(suc)m(h)f(an)h (inequalit)m(y)e(fails)h(ev)m(en)h(for)g(a)g(\014xed)f(\014nite)0 4551 y(v)m(olume.)74 b(Consider,)42 b(indeed,)h(the)f(trivial)d(case)k (of)e Fn(H)51 b Fo(=)43 b(0.)75 b(Then)40 b(the)i(distribution)37 b(of)42 b(the)g(n)m(um)m(b)s(er)e(of)0 4678 y(particles)33 b(in)g(a)i(v)m(olume)f(\003)g(is)f(P)m(oissonian)g(with)g(mean)h Fn(z)t Fm(j)p Fo(\003)p Fm(j)p Fo(.)53 b(It)34 b(is)g(easy)g(to)h(v)m (erify)f(\(see)h([Le1],)h(Section)e(5.1\))0 4805 y(that)43 b(the)g(P)m(oissonian)e(distribution)e(do)s(es)j(not)h(satisfy)f(a)g (LSI.)g(It)h(is)e(still)g(p)s(ossible,)i(though,)i(that)e(under)0 4933 y(stronger)32 b(conditions)f(on)h(the)h(p)s(oten)m(tial)f(whic)m (h)f(do)h(not)h(include)d(the)i(case)i Fn(H)h Fo(=)28 b(0)33 b(\(e.g.)48 b(sup)s(erstabilit)m(y)-8 b(,)30 b(see)0 5060 y([Ru]\))h(a)f(LSI)g(is)f(indeed)g(satis\014ed.)0 5213 y(Our)35 b(results)g(are)i(presen)m(ted)f(in)g(Section)g(2,)i (while)d(most)i(pro)s(ofs)e(are)i(p)s(ostp)s(oned)e(to)i(Section)f(3.) 59 b(Section)37 b(4)0 5340 y(con)m(tains)c(a)h(partial)e(con)m(v)m (erse)i(of)f(our)g(main)f(result,)h Fe(i.e.)f Fo(that)h(the)h(uniform)d (p)s(ositivit)m(y)g(of)i(the)g(sp)s(ectral)g(gap)0 5558 y Fh(30)p Ff(=mag)r(g)r(io=)p Fh(2000)d Fr([2])1317 b Fo(1.1)p eop %%Page: 3 3 3 2 bop 0 91 a Fo(implies)28 b(the)j(exp)s(onen)m(tial)f(deca)m(y)i(of) f(the)h(co)m(v)-5 b(ariance)32 b(of)f(t)m(w)m(o)h(lo)s(cal)e (functions.)41 b(Finally)-8 b(,)30 b(Section)h(5)g(is)f(a)h(brief)0 219 y(discussion)d(on)i(the)h(p)s(ossibilit)m(y)26 b(of)31 b(ha)m(ving)f(a)h(logarithmic)e(Sob)s(olev)g(inequalit)m(y)-8 b(.)0 473 y Fp(Ac)m(kno)m(wledgemen)m(ts.)40 b Fo(W)-8 b(e)28 b(thank)g(F)-8 b(abio)28 b(Martinelli)d(for)i(man)m(y)h(helpful) c(insigh)m(ts)j(and)f(for)i(his)e(\\semigrou-)0 600 y(pless")k(pro)s (of)f(of)i(the)f(\\2{blo)s(c)m(ks")i(Lemma)e(3.1.)0 854 y Fp(2.)47 b(Notation)34 b(and)h(results)0 1016 y Fe(The)f(Gibbs)g(me) -5 b(asur)g(es)p Fo(.)46 b(Let)33 b Fm(B)s Fo(\()p Fk(R)1188 983 y Fj(d)1236 1016 y Fo(\))f(b)s(e)g(the)f(Borel)h Fn(\033)s Fo({algebra)h(on)e Fk(R)2437 983 y Fj(d)2485 1016 y Fo(;)i(w)m(e)f(denote)g(b)m(y)g Fm(B)3160 1030 y Fj(b)3196 1016 y Fo(\()p Fk(R)3291 983 y Fj(d)3340 1016 y Fo(\))c Fm(\032)f(B)s Fo(\()p Fk(R)3658 983 y Fj(d)3707 1016 y Fo(\))32 b(the)0 1144 y(collection)27 b(of)h(all)f(b)s(ounded)f(Borel)i(sets.)40 b(F)-8 b(or)28 b Fn(A)e Fm(2)f(B)1846 1158 y Fj(b)1882 1144 y Fo(\()p Fk(R)1977 1111 y Fj(d)2025 1144 y Fo(\),)k Fm(j)p Fn(A)p Fm(j)g Fo(indicates)d(the)i(Leb)s(esgue)g(measure)f(of)h Fn(A)p Fo(.)40 b(Let)0 1272 y Fm(R)77 1239 y Fj(d)145 1272 y Fm(\032)25 b(B)301 1286 y Fj(b)337 1272 y Fo(\()p Fk(R)432 1239 y Fj(d)480 1272 y Fo(\))h(b)s(e)f(the)h(set)h(of)e(all)g Fe(r)-5 b(e)g(ctangles)35 b Fo(\(cartesian)26 b(pro)s(ducts)e(of)i (\014nite)f(closed)g(in)m(terv)-5 b(als\).)39 b(W)-8 b(e)27 b(consider,)0 1399 y(as)k Fe(c)-5 b(on\014gur)g(ation)34 b(sp)-5 b(ac)g(e)7 b Fo(,)32 b(the)f(set)g(\012)f(of)g(all)f(lo)s (cally)g(\014nite)h(subsets)f(of)i Fk(R)2546 1366 y Fj(d)2594 1399 y Fo(,)f Fe(i.e.)625 1646 y Fo(\012)25 b(:=)g Fm(f)p Fn(!)j Fm(\032)d Fk(R)1123 1608 y Fj(d)1197 1646 y Fo(:)g(card\()p Fn(!)f Fm(\\)19 b Fn(A)p Fo(\))26 b Fn(<)f Fm(1)30 b Fo(for)g(all)g(b)s(ounded)e(subsets)i Fn(A)g Fo(of)38 b Fk(R)3181 1608 y Fj(d)3230 1646 y Fm(g)0 1892 y Fo(where)24 b(card)o(\()p Fn(A)p Fo(\))h(stands)f(for)g(the)g(cardinalit)m(y)e(of)i (the)h(set)f Fn(A)p Fo(.)39 b(W)-8 b(e)25 b(endo)m(w)f(\012)g(with)e (the)j Fn(\033)s Fm(\000)p Fo(algebra)f Fm(F)33 b Fo(generated)0 2020 y(b)m(y)g(the)g(coun)m(ting)g(v)-5 b(ariables)32 b Fn(N)1115 2034 y Fj(A)1204 2020 y Fo(:)e Fn(!)j Fm(!)d Fo(card)o(\()p Fn(!)c Fm(\\)21 b Fn(A)p Fo(\),)35 b(where)d Fn(A)e Fm(2)g(B)2518 2034 y Fj(b)2554 2020 y Fo(\()p Fk(R)2649 1987 y Fj(d)2698 2020 y Fo(\).)49 b(Giv)m(en)33 b Fn(!)s(;)15 b(\021)33 b Fm(2)d Fo(\012)i(w)m(e)i(let)f Fn(!)s Fm(4)p Fn(\021)0 2148 y Fo(b)s(e)d(the)h(symmetric)f (di\013erence)g(of)g Fn(!)k Fo(and)c Fn(\021)s Fo(,)h Fe(i.e.)h Fn(!)s Fm(4)p Fn(\021)d Fo(:=)c(\()p Fn(!)f Fm([)c Fn(\021)s Fo(\))p Fm(n)p Fo(\()p Fn(!)25 b Fm(\\)20 b Fn(\021)s Fo(\).)42 b(F)-8 b(or)31 b(\003)26 b Fm(2)f(B)3171 2162 y Fj(b)3207 2148 y Fo(\()p Fk(R)3302 2115 y Fj(d)3351 2148 y Fo(\),)31 b(w)m(e)g(consider)0 2275 y(also)f(the)h(\014nite)e(v) m(olume)h(con\014guration)g(space)1420 2522 y(\012)1486 2536 y Fr(\003)1565 2522 y Fo(:=)25 b Fm(f)p Fn(!)k Fm(\032)24 b Fo(\003)i(:)f Fn(!)34 b Fo(is)29 b(\014nite)o Fm(g)0 2768 y Fo(with)i Fn(\033)s Fm(\000)p Fo(algebra)i Fm(F)721 2782 y Fr(\003)808 2768 y Fo(generated)g(b)m(y)g(the)f(functions)f Fn(N)1978 2782 y Fj(A)2038 2768 y Fo(,)i(suc)m(h)f(that)h Fn(A)g Fo(is)e(a)i(Borel)f(subset)g(of)h(\003.)47 b(W)-8 b(e)34 b(write)0 2896 y Fn(f)h Fm(2)24 b(F)230 2910 y Fj(A)320 2896 y Fo(to)31 b(indicate)f(that)h(the)f Fn(f)40 b Fo(is)29 b Fm(F)1369 2910 y Fj(A)1429 2896 y Fm(\000)p Fo(measurable.)40 b(The)29 b(function)g Fn(f)40 b Fo(is)30 b(said)f(to)i(b)s(e)f Fe(lo)-5 b(c)g(al)41 b Fo(if)30 b(there)g(exists)0 3023 y Fn(A)25 b Fm(2)g(B)239 3037 y Fj(b)276 3023 y Fo(\()p Fk(R)371 2990 y Fj(d)419 3023 y Fo(\))31 b(suc)m(h)f(that)h Fn(f)j Fm(2)25 b(F)1117 3037 y Fj(A)1177 3023 y Fo(.)0 3186 y(F)-8 b(or)38 b Fn(x;)15 b(y)41 b Fm(2)c Fk(R)506 3153 y Fj(d)592 3186 y Fo(w)m(e)h(denote)g(b)m(y)f Fn(d)p Fo(\()p Fn(x;)15 b(y)s Fo(\))39 b(the)f(Euclidean)e(distance,)j(while)d Fm(j)p Fn(x)p Fm(j)i Fo(stands)f(for)g Fn(d)p Fo(\()p Fn(x;)15 b Fo(0\).)65 b(Let)38 b Fn(')g Fo(:)0 3313 y Fk(R)60 3280 y Fj(d)137 3313 y Fm(7!)29 b Fk(R)42 b Fo(b)s(e)32 b(a)h(measurable)f(ev)m(en)h(function;)g Fn(')g Fo(is)f(called)g(a)h Fe(p)-5 b(air)36 b(p)-5 b(otential)9 b Fo(.)50 b(W)-8 b(e)34 b(assume)f(that)g Fn(')g Fo(has)g Fe(\014nite)0 3441 y(r)-5 b(ange)38 b Fn(r)s Fo(,)30 b Fe(i.e.)i Fo(that)f Fn(')p Fo(\()p Fn(x)p Fo(\))26 b(=)f(0)31 b(if)f Fm(j)p Fn(x)p Fm(j)25 b Fn(>)g(r)s Fo(.)40 b(Giv)m(en)31 b Fn(A)25 b Fm(\032)g Fk(R)2014 3408 y Fj(d)2092 3441 y Fo(w)m(e)31 b(let)1406 3665 y(\026)1382 3688 y 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Fo(:=)d(\()p Fn(!)c Fm(\\)c Fo(\003\))h Fm([)e Fo(\()p Fn(\021)27 b Fm(\\)21 b Fo(\003)3238 4550 y Fj(c)3276 4583 y Fo(\),)34 b(\003)3433 4550 y Fj(c)3504 4583 y Fo(stands)f(for)0 4711 y(the)40 b(complemen)m(t)f(of)h(\003,)i(and)c Fn(\021)43 b Fo(is)38 b(called)h(the)h Fe(b)-5 b(oundary)42 b(c)-5 b(ondition)p Fo(.)70 b(W)-8 b(e)40 b(denote)g(with)e Fn(Q)3343 4725 y Fr(\003)3437 4711 y Fo(the)h(P)m(oisson)0 4838 y(p)s(oin)m(t)27 b(pro)s(cess)g(on)h(\003)g(with)e(in)m(tensit)m(y)h(1,)i(and)e(w)m(e)h (de\014ne)g Fn(Q)2074 4795 y Fj(\021)2074 4865 y Fr(\003)2153 4838 y Fo(:=)d Fn(Q)2346 4852 y Fr(\003)2415 4838 y Fm(\002)15 b Fn(\016)2541 4852 y Fr(\003)2591 4834 y Ff(c)2625 4852 y Fj(;\021)2691 4838 y Fo(,)29 b(where)e Fn(\016)3045 4852 y Fr(\003)3095 4834 y Ff(c)3129 4852 y Fj(;\021)3223 4838 y Fo(is)f(the)i(probabilit)m(y)0 4966 y(measure)c(on)h(\(\012)570 4980 y Fr(\003)620 4961 y Ff(c)658 4966 y Fn(;)15 b Fm(F)763 4980 y Fr(\003)813 4961 y Ff(c)852 4966 y Fo(\))25 b(whic)m(h)e(giv)m (es)i(mass)f(1)h(to)g(the)g(con\014guration)f Fn(\021)s Fo(.)39 b(F)-8 b(or)25 b(\003)g Fm(2)g(B)2974 4980 y Fj(b)3011 4966 y Fo(\()p Fk(R)3106 4933 y Fj(d)3154 4966 y Fo(\),)h(the)f(\014nite)e(v)m(olume)0 5093 y Fe(Gibbs)33 b(me)-5 b(asur)g(e)38 b Fo(in)29 b(\003)i(at)g(in)m(v)m(erse)f(temp)s (erature)g Fn(\014)5 b Fo(,)31 b(activit)m(y)g Fn(z)j Fo(and)c(b)s(oundary)e(condition)h Fn(\021)34 b Fo(is)c(giv)m(en)g(b)m (y)901 5340 y Fn(\026)956 5296 y Fj(\021)956 5367 y Fr(\003)1010 5340 y Fo(\()p Fn(d!)s Fo(\))c(:=)f Fn(e)1376 5302 y Fi(j)p Fr(\003)p Fi(j)1489 5340 y Fo(\()p Fn(Z)1593 5296 y Fj(\021)1586 5367 y Fr(\003)1641 5340 y Fo(\))1676 5302 y Fi(\000)p Fr(1)1789 5340 y Fn(z)1835 5302 y Fj(N)1893 5311 y Fh(\003)1939 5302 y Fr(\()p Fj(!)r Fr(\))2078 5340 y Fo(exp)2217 5266 y Fg(\002)2255 5340 y Fm(\000)p Fn(\014)5 b(H)2465 5296 y Fj(\021)2458 5367 y Fr(\003)2512 5340 y Fo(\()p Fn(!)s Fo(\))2642 5266 y Fg(\003)2695 5340 y Fn(Q)2767 5296 y Fj(\021)2767 5367 y Fr(\003)2822 5340 y Fo(\()p Fn(d!)s Fo(\))714 b(\(2)p Fn(:)p Fo(1\))0 5558 y Fh(30)p Ff(=mag)r(g)r(io=)p Fh(2000)30 b Fr([3])1317 b Fo(2.1)p eop %%Page: 4 4 4 3 bop 0 91 a Fo(where)32 b Fn(Z)334 47 y Fj(\021)327 118 y Fr(\003)414 91 y Fo(is)g(the)g(appropriate)g(normalization)f (factor)j(\(w)m(e)f(omit)f(for)h(simplicit)m(y)c(the)k(dep)s(endence)f (of)g(these)0 218 y(quan)m(tities)25 b(on)h Fn(z)k Fo(and)25 b Fn(\014)5 b Fo(\).)40 b(W)-8 b(e)27 b(denote)f(with)f Fn(\026)1635 175 y Fj(\021)1635 245 y Fr(\003)1689 218 y Fo(\()p Fn(f)10 b Fo(\))26 b(the)g(exp)s(ectation)g(of)g Fn(f)35 b Fo(with)25 b(resp)s(ect)h(to)g Fn(\026)3327 175 y Fj(\021)3327 245 y Fr(\003)3381 218 y Fo(,)h(while)d Fn(\026)3721 232 y Fr(\003)3775 218 y Fo(\()p Fn(f)10 b Fo(\))0 346 y(denotes)31 b(the)f(function)f Fn(!)g Fm(!)c Fn(\026)1098 313 y Fj(!)1098 371 y Fr(\003)1152 346 y Fo(\()p Fn(f)10 b Fo(\).)40 b(Explicitly)-8 b(,)28 b(for)j(all)e(measurable)g(funcions)g Fn(f)40 b Fo(on)30 b(\012)3150 360 y Fr(\003)3204 346 y Fo(,)g(w)m(e)h(ha)m(v)m(e)1061 672 y Fn(\026)1116 629 y Fj(\021)1116 699 y Fr(\003)1170 672 y Fo(\()p Fn(f)10 b Fo(\))25 b(=)g(\()p Fn(Z)1520 629 y Fj(\021)1513 699 y Fr(\003)1568 672 y Fo(\))1603 635 y Fi(\000)p Fr(1)1764 559 y Fi(1)1734 586 y Fg(X)1731 778 y Fj(n)p Fr(=0)1894 611 y Fn(z)1940 578 y Fj(n)p 1894 651 96 4 v 1902 735 a Fn(n)p Fo(!)2031 549 y Fg(Z)2082 755 y Fr(\003)2132 737 y Ff(n)2196 672 y Fn(e)2238 635 y Fi(\000)p Fj(\014)s(H)2404 604 y Ff(\021)2399 658 y Fh(\003)2445 635 y Fr(\()p Fj(x)p Fr(\))2548 672 y Fn(f)10 b Fo(\()p Fn(x)p Fo(\))15 b Fn(dx)0 1012 y Fo(where)29 b(w)m(e)h(ha)m(v)m(e)h(iden)m(ti\014ed)c(the)j(functions)e(on)h(\012) 1737 1026 y Fr(\003)1821 1012 y Fo(with)f(the)i(symmetric)e(functions)g (on)3141 944 y Fg(S)3217 966 y Fi(1)3217 1039 y Fj(n)p Fr(=0)3373 1012 y Fo(\003)3436 979 y Fj(n)3486 1012 y Fo(.)40 b(F)-8 b(or)31 b(a)e(set)0 1139 y Fn(X)37 b Fm(2)29 b(F)42 b Fo(w)m(e)33 b(set)g Fn(\026)644 1153 y Fr(\003)699 1139 y Fo(\()p Fn(X)7 b Fo(\))30 b(:=)f Fn(\026)1061 1153 y Fr(\003)1115 1139 y Fo(\(1)-25 b(I)1204 1153 y Fj(X)1273 1139 y Fo(\),)34 b(where)e(1)-25 b(I)1685 1153 y Fj(X)1787 1139 y Fo(is)32 b(the)h(c)m(haracteristic)h(function)d(on)i Fn(X)7 b Fo(.)48 b(W)-8 b(e)34 b(write)e Fn(\026)p Fo(\()p Fn(f)5 b(;)15 b(g)s Fo(\))0 1267 y(to)31 b(denote)g(the)g(co)m(v)-5 b(ariance)31 b(\(with)f(resp)s(ect)g(to)h Fn(\026)p Fo(\))g(of)f Fn(f)40 b Fo(and)30 b Fn(g)s Fo(.)41 b(The)30 b(family)f(of)h(measures) h(\(2.1\))h(satis\014es)e(the)0 1394 y(DLR)g(compatibilit)m(y)f (conditions)721 1637 y Fn(\026)776 1651 y Fr(\003)830 1637 y Fo(\()p Fn(\026)920 1651 y Fj(V)983 1637 y Fo(\()p Fn(X)7 b Fo(\)\))27 b(=)e Fn(\026)1348 1651 y Fr(\003)1402 1637 y Fo(\()p Fn(X)7 b Fo(\))183 b Fm(8)15 b Fn(X)32 b Fm(2)25 b(F)192 b(8)15 b Fn(V)5 b(;)15 b Fo(\003)25 b Fm(2)g(B)2651 1651 y Fj(b)2687 1637 y Fo(\()p Fk(R)2782 1599 y Fj(d)2831 1637 y Fo(\))p Fn(;)30 b(V)46 b Fm(\032)25 b Fo(\003)534 b(\(2)p Fn(:)p Fo(2\))0 1940 y Fe(The)33 b(dynamics.)42 b Fo(F)-8 b(or)31 b(a)f(giv)m(en)h(function)e Fn(f)39 b Fo(on)31 b(\012)f(w)m(e)h(let)219 2183 y Fn(D)297 2145 y Fi(\000)294 2205 y Fj(x)358 2183 y Fn(f)10 b Fo(\()p Fn(!)s Fo(\))25 b(:=)g Fn(f)10 b Fo(\()p Fn(!)s Fm(nf)p Fn(x)p Fm(g)p Fo(\))21 b Fm(\000)f Fn(f)10 b Fo(\()p Fn(!)s Fo(\))182 b Fn(D)1618 2145 y Fr(+)1615 2205 y Fj(x)1678 2183 y Fn(f)10 b Fo(\()p Fn(!)s Fo(\))25 b(:=)h Fn(f)10 b Fo(\()p Fn(!)22 b Fm([)e(f)p Fn(x)p Fm(g)p Fo(\))h Fm(\000)f Fn(f)10 b Fo(\()p Fn(!)s Fo(\))182 b Fn(!)28 b Fm(2)d Fo(\012)p Fn(;)45 b(x)26 b Fm(2)e Fk(R)3446 2145 y Fj(d)3713 2183 y Fo(\(2)p Fn(:)p Fo(3\))0 2425 y(where)30 b(it)g(is)f(understo)s(o)s(d)g(that)i Fn(D)1196 2392 y Fi(\000)1193 2447 y Fj(x)1257 2425 y Fn(f)10 b Fo(\()p Fm(;)p Fo(\))26 b(=)e(0.)42 b(F)-8 b(or)31 b(simplicit)m(y)c(w)m(e)k(use)f(the)h(notation)1087 2667 y(\()p Fn(D)1200 2628 y Fi(\000)1197 2694 y Fr(\003)1262 2667 y Fn(f)e Fm(\001)21 b Fn(D)1460 2628 y Fi(\000)1457 2694 y Fr(\003)1521 2667 y Fn(g)s Fo(\)\()p Fn(!)s 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Fg(R)1508 5242 y Fr(\003)1577 5210 y Fn(dx)15 b(\026)1746 5166 y Fj(\021)1746 5236 y Fr(\003)1801 5210 y Fo(\(\()p Fn(D)1949 5177 y Fr(+)1946 5232 y Fj(x)2010 5210 y Fn(f)10 b Fo(\)\()p Fn(D)2213 5177 y Fr(+)2210 5232 y Fj(x)2273 5210 y Fn(g)s Fo(\)\))p Fc(,)32 b(for)e(all)f Fn(f)5 b(;)15 b(g)29 b Fm(2)c(D)3029 5224 y Fr(0)3069 5210 y Fo(\()p Fn(L)3166 5166 y Fj(\021)3166 5236 y Fr(\003)3221 5210 y Fo(\))66 5340 y Fc(\(3\))46 b Fn(L)289 5296 y Fj(\021)289 5367 y Fr(\003)374 5340 y Fc(is)29 b(symmetric)h(on)g Fm(D)1104 5354 y Fr(0)1145 5340 y Fo(\()p Fn(L)1242 5296 y Fj(\021)1242 5367 y Fr(\003)1297 5340 y Fo(\))0 5558 y Fh(30)p Ff(=mag)r(g)r(io=)p Fh(2000)g Fr([4])1317 b Fo(2.2)p eop %%Page: 5 5 5 4 bop 66 91 a Fc(\(4\))46 b Fm(E)283 47 y Fj(\021)275 118 y Fr(\003)362 91 y Fc(is)32 b(closable)h(and)f(its)g(closure)h(is)f (asso)s(ciated)h(with)f(a)h(self{adjoin)m(t)f(extension)h(of)g Fn(L)3288 47 y Fj(\021)3288 118 y Fr(\003)3342 91 y Fc(.)48 b(W)-8 b(e)34 b(man)m(tain)227 218 y(the)i(same)f(sym)m(b)s(ols)f Fn(L)1035 174 y Fj(\021)1035 245 y Fr(\003)1124 218 y Fc(and)g Fm(E)1361 174 y Fj(\021)1353 245 y Fr(\003)1442 218 y Fc(to)i(denote)g(these)f(extensions,)h(and)f(denote)g(b)m(y)g Fm(D)s Fo(\()p Fn(L)3342 174 y Fj(\021)3342 245 y Fr(\003)3396 218 y Fo(\))p Fc(,)i Fm(D)s Fo(\()p Fm(E)3657 174 y Fj(\021)3649 245 y Fr(\003)3703 218 y Fo(\))f Fc(the)227 345 y(resp)s(ectiv)m(e)31 b(domains)66 473 y(\(5\))46 b Fn(P)298 430 y Fr(\003)p Fj(;\021)285 496 y(t)448 473 y Fo(:=)34 b Fn(e)620 440 y Fj(tL)697 409 y Ff(\021)697 464 y Fh(\003)784 473 y Fc(is)h(a)h(p)s(ositiv)m(e)f(preserving)f(con)m(traction)j(semigroup)d (\(th)m(us)i(a)g(Mark)m(o)m(v)h(semigroup\))e(on)227 601 y Fn(L)289 568 y Fj(p)331 601 y Fo(\()p Fn(\026)421 557 y Fj(\021)421 627 y Fr(\003)476 601 y Fo(\))30 b Fc(for)g(all)g Fn(p)25 b Fm(2)f Fo([1)p Fn(;)15 b Fm(1)p Fo(])p Fc(.)0 834 y Fe(R)-5 b(emark)p Fo(.)63 b(A)38 b(more)f(explicit)f(construction)h(for)g(these)h(pro)s(cesses)f(can)h 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Fo(\))26 b(:=)f Fn(f)10 b Fo(\()p Fn(x)1108 4246 y Fr(1)1148 4232 y Fn(;)15 b(:)g(:)g(:)i(;)k Fo(^)-51 b Fn(x)1402 4246 y Fj(i)1433 4232 y Fn(;)15 b(:)g(:)g(:)h(;)f(x)1686 4246 y Fj(n)1736 4232 y Fo(\))e Fm(\000)g Fn(f)d Fo(\()p Fn(x)2010 4246 y Fr(1)2050 4232 y Fn(;)15 b(:)g(:)g(:)i(;)e(x)2304 4246 y Fj(n)2353 4232 y Fo(\),)28 b(where)k(^)-51 b Fn(x)2752 4246 y Fj(i)2809 4232 y Fo(denotes)27 b(that)h(the)e(v)-5 b(ariable)26 b Fn(x)3870 4246 y Fj(i)0 4359 y Fo(is)j(omitted.)41 b(By)31 b(telescopic)f(sums)g(w)m(e)h(ha)m(v)m(e)414 4654 y Fn(f)10 b Fo(\()p Fn(x)556 4668 y Fr(1)596 4654 y Fn(;)15 b(:)g(:)g(:)i(;)e(x)850 4668 y Fj(n)900 4654 y Fo(\))20 b Fm(\000)g Fn(f)10 b Fo(\()p Fn(y)1181 4668 y Fr(1)1221 4654 y Fn(;)15 b(:)g(:)g(:)i(;)e(y)1468 4668 y Fj(m)1537 4654 y Fo(\))25 b(=)g Fm(\000)1823 4540 y Fj(n)1780 4568 y Fg(X)1779 4763 y Fj(k)r Fr(=1)1927 4654 y Fn(D)2005 4615 y Fi(\000)2002 4681 y Fj(k)2066 4654 y Fn(f)10 b Fo(\()p Fn(x)2208 4668 y Fr(1)2248 4654 y Fn(;)15 b(:)g(:)g(:)i(;)e(x)2502 4668 y Fj(k)2547 4654 y Fo(\))20 b(+)2728 4540 y Fj(m)2695 4568 y Fg(X)2693 4763 y Fj(h)p Fr(=1)2843 4654 y Fn(D)2921 4615 y Fi(\000)2918 4681 y Fj(h)2982 4654 y Fn(f)10 b Fo(\()p Fn(y)3117 4668 y Fr(1)3157 4654 y Fn(;)15 b(:)g(:)g(:)i(;)e(y)3404 4668 y Fj(h)3451 4654 y Fo(\))0 4963 y(whence,)30 b(b)m(y)h(Sc)m(h)m(w)m (arz)g(inequalit)m(y)-8 b(,)182 5196 y(1)p 182 5237 46 4 v 182 5320 a(2)253 5258 y([)q Fn(f)10 b Fo(\()p Fn(x)421 5272 y Fr(1)461 5258 y Fn(;)15 b(:)g(:)g(:)i(;)e(x)715 5272 y Fj(n)764 5258 y Fo(\))21 b Fm(\000)f Fn(f)10 b Fo(\()p Fn(y)1046 5272 y Fr(1)1086 5258 y Fn(;)15 b(:)g(:)g(:)i(;)e(y) 1333 5272 y Fj(m)1402 5258 y Fo(\)])1462 5212 y Fr(2)1528 5258 y Fm(\024)25 b Fn(n)1738 5144 y Fj(n)1694 5172 y Fg(X)1694 5367 y Fj(k)r Fr(=1)1841 5184 y Fg(\002)1879 5258 y Fn(D)1957 5219 y Fi(\000)1954 5285 y Fj(k)2018 5258 y Fn(f)10 b Fo(\()p Fn(x)2160 5272 y Fr(1)2201 5258 y Fn(;)15 b(:)g(:)g(:)h(;)f(x)2454 5272 y Fj(k)2499 5258 y Fo(\))2534 5184 y Fg(\003)2573 5202 y Fr(2)2634 5258 y Fo(+)20 b Fn(m)2855 5144 y Fj(m)2821 5172 y Fg(X)2820 5367 y Fj(h)p Fr(=1)2969 5184 y Fg(\002)3007 5258 y Fn(D)3085 5219 y Fi(\000)3082 5285 y Fj(h)3146 5258 y Fn(f)10 b Fo(\()p Fn(y)3281 5272 y Fr(1)3321 5258 y Fn(;)15 b(:)g(:)g(:)i(;)e(y) 3568 5272 y Fj(h)3615 5258 y Fo(\))3650 5184 y Fg(\003)3688 5202 y Fr(2)0 5558 y Fh(30)p Ff(=mag)r(g)r(io=)p Fh(2000)30 b Fr([8])1317 b Fo(3.2)p eop %%Page: 9 9 9 8 bop 0 91 a Fo(whic)m(h,)29 b(plugged)h(in)m(to)g(\(3.3\),)j(yields) 181 330 y Fn(\026)236 287 y Fj(\021)236 357 y Fr(\003)290 330 y Fo(\()p Fn(f)5 b(;)15 b(f)10 b Fo(\))25 b Fm(\024)g Fo(2\()p Fn(Z)775 287 y Fj(\021)768 357 y Fr(\003)823 330 y Fo(\))858 293 y Fi(\000)p Fr(1)971 229 y Fg(\020)1026 330 y Fo(1)20 b Fm(\000)g Fn(e)1224 293 y Fi(\000)p Fj(z)s Fi(j)p Fr(\003)p Fi(j)1416 229 y Fg(\021)1518 217 y Fi(1)1488 244 y Fg(X)1486 436 y Fj(n)p Fr(=1)1681 217 y Fj(n)1638 244 y Fg(X)1638 440 y Fj(k)r Fr(=1)1902 269 y Fn(z)1948 236 y Fj(n)p 1796 309 308 4 v 1796 393 a Fo(\()p Fn(n)g Fm(\000)g Fo(1\)!)2129 206 y Fg(Z)2180 413 y Fr(\003)2230 395 y Ff(n)2279 330 y Fn(dx)15 b(e)2435 293 y Fi(\000)p Fj(\014)s(H)2601 262 y Ff(\021)2596 316 y Fh(\003)2642 293 y Fr(\()p Fj(x)p Fr(\))2760 257 y Fg(\002)2798 330 y Fn(D)2876 291 y Fi(\000)2873 358 y Fj(k)2937 330 y Fn(f)10 b Fo(\()p Fn(x)3079 344 y Fr(1)3119 330 y Fn(;)15 b(:)g(:)g(:)i(;)e(x)3373 344 y Fj(k)3418 330 y Fo(\))3453 257 y Fg(\003)3491 274 y Fr(2)3713 330 y Fo(\(3)p Fn(:)p Fo(4\))0 578 y(in)29 b(whic)m(h)g(w)m(e)i(used)788 795 y(\()p Fn(Z)892 751 y Fj(\021)885 821 y Fr(\003)940 795 y Fo(\))975 757 y Fi(\000)p Fr(1)1130 681 y Fi(1)1101 708 y Fg(X)1088 901 y Fj(m)p Fr(=1)1271 733 y Fn(z)1317 700 y Fj(m)p 1271 774 116 4 v 1276 857 a Fn(m)p Fo(!)1412 671 y Fg(Z)1463 878 y Fr(\003)1513 859 y Ff(m)1577 795 y Fn(dy)19 b(e)1730 757 y Fi(\000)p Fj(\014)s(H)1896 726 y Ff(\021)1891 781 y Fh(\003)1937 757 y Fr(\()p Fj(y)r Fr(\))2063 795 y Fo(=)25 b(1)20 b Fm(\000)g Fo(\()p Fn(Z)2419 751 y Fj(\021)2412 821 y Fr(\003)2467 795 y Fo(\))2502 757 y Fi(\000)p Fr(1)2625 795 y Fm(\024)25 b Fo(1)c Fm(\000)f Fn(e)2920 757 y Fi(\000)p Fj(z)s Fi(j)p Fr(\003)p Fi(j)0 1060 y Fo(The)31 b(inequalit)m(y)e(holds)h(b)s(ecause)i Fn(')27 b Fm(\025)f Fo(0)32 b(so)g(that)g Fn(e)1800 1027 y Fi(\000)p Fj(\014)s(H)1966 996 y Ff(\021)1961 1050 y Fh(\003)2007 1027 y Fr(\()p Fj(y)r Fr(\))2134 1060 y Fm(\024)27 b Fo(1.)44 b(By)31 b(the)h(same)f(reason,)h(for)g Fn(k)e Fm(\024)c Fn(n)31 b Fo(w)m(e)h(ha)m(v)m(e)0 1186 y Fn(H)83 1143 y Fj(\021)76 1213 y Fr(\003)130 1186 y Fo(\()p Fn(x)217 1200 y Fr(1)258 1186 y Fn(;)15 b(:)g(:)g(:)i(;)e(x)512 1200 y Fj(n)561 1186 y Fo(\))26 b Fm(\025)f Fn(H)801 1143 y Fj(\021)794 1213 y Fr(\003)848 1186 y Fo(\()p Fn(x)935 1200 y Fr(1)976 1186 y Fn(;)15 b(:)g(:)g(:)i(;)e(x)1230 1200 y Fj(k)1275 1186 y Fo(\);)31 b(therefore,)g(b)m(y)f(using)f (\(3.4\),)435 1430 y Fn(\026)490 1386 y Fj(\021)490 1456 y Fr(\003)544 1430 y Fo(\()p Fn(f)5 b(;)15 b(f)10 b Fo(\))25 b Fm(\024)g Fo(2)15 b(\()p Fn(Z)1044 1386 y Fj(\021)1037 1456 y Fr(\003)1092 1430 y Fo(\))1127 1392 y Fi(\000)p Fr(1)1225 1356 y Fg(\000)1267 1430 y Fo(1)20 b Fm(\000)g Fn(e)1465 1392 y Fi(\000)p Fj(z)s Fi(j)p Fr(\003)p Fi(j)1657 1356 y Fg(\001)1746 1316 y Fi(1)1717 1343 y Fg(X)1714 1536 y Fj(n)p Fr(=1)1910 1316 y Fj(n)1867 1343 y Fg(X)1866 1539 y Fj(k)r Fr(=1)2025 1368 y Fn(z)2071 1335 y Fj(n)2120 1368 y Fm(j)p Fo(\003)p Fm(j)2233 1335 y Fj(n)p Fi(\000)p Fj(k)p 2025 1409 356 4 v 2049 1492 a Fo(\()p Fn(n)g Fm(\000)g Fo(1\)!)2407 1306 y Fg(Z)2457 1513 y Fr(\003)2507 1494 y Ff(k)2551 1430 y Fn(dx)15 b(e)2707 1392 y Fi(\000)p Fj(\014)s(H)2873 1361 y Ff(\021)2868 1416 y Fh(\003)2915 1392 y Fr(\()p Fj(x)p Fr(\))3032 1356 y Fg(\002)3070 1430 y Fn(D)3148 1391 y Fi(\000)3145 1457 y Fj(k)3209 1430 y Fn(f)10 b Fo(\()p Fn(x)p Fo(\))3386 1356 y Fg(\003)3424 1374 y Fr(2)784 1762 y Fo(=)25 b(2)15 b(\()p Fn(Z)1044 1718 y Fj(\021)1037 1789 y Fr(\003)1092 1762 y Fo(\))1127 1724 y Fi(\000)p Fr(1)1225 1688 y Fg(\000)1267 1762 y Fo(1)20 b Fm(\000)g Fn(e)1465 1724 y Fi(\000)p Fj(z)s Fi(j)p Fr(\003)p Fi(j)1657 1688 y Fg(\001)1744 1648 y Fi(1)1715 1676 y Fg(X)1714 1871 y Fj(k)r Fr(=1)1978 1700 y Fn(z)2024 1667 y Fj(k)p 1872 1741 303 4 v 1872 1824 a Fo(\()p Fn(k)k Fm(\000)c Fo(1\)!)2201 1638 y Fg(Z)2252 1845 y Fr(\003)2302 1827 y Ff(k)2346 1762 y Fn(dx)15 b(e)2502 1724 y Fi(\000)p Fj(\014)s(H)2668 1694 y Ff(\021)2663 1748 y Fh(\003)2709 1724 y Fr(\()p Fj(x)p Fr(\))2827 1688 y Fg(\002)2865 1762 y Fn(D)2943 1723 y Fi(\000)2940 1789 y Fj(k)3004 1762 y Fn(f)10 b Fo(\()p Fn(x)p Fo(\))3181 1688 y Fg(\003)3219 1706 y Fr(2)1256 2094 y Fm(\002)1382 1981 y Fi(1)1352 2008 y Fg(X)1347 2203 y Fj(n)p Fr(=)p Fj(k)1514 2033 y Fn(z)1560 2000 y Fj(n)p Fi(\000)p Fj(k)1707 2033 y Fm(j)p Fo(\003)p Fm(j)1820 2000 y Fj(n)p Fi(\000)p Fj(k)p 1514 2073 453 4 v 1585 2157 a Fo(\()p Fn(n)20 b Fm(\000)g Fn(k)s Fo(\)!)1989 2033 y(\()p Fn(n)g Fm(\000)g Fn(k)s Fo(\)!\()p Fn(k)25 b Fm(\000)19 b Fo(1\)!)p 1989 2073 615 4 v 2143 2157 a(\()p Fn(n)h Fm(\000)g Fo(1\)!)784 2426 y Fm(\024)25 b Fo(2)p Fn(e)967 2389 y Fj(z)s Fi(j)p Fr(\003)p Fi(j)1102 2353 y Fg(\000)1144 2426 y Fo(1)c Fm(\000)f Fn(e)1343 2389 y Fi(\000)p Fj(z)s Fi(j)p Fr(\003)p Fi(j)1535 2353 y Fg(\001)1576 2426 y Fo(\()p Fn(Z)1680 2383 y Fj(\021)1673 2453 y Fr(\003)1728 2426 y Fo(\))1763 2389 y Fi(\000)p Fr(1)1906 2313 y Fi(1)1877 2340 y Fg(X)1876 2536 y Fj(k)r Fr(=1)2140 2365 y Fn(z)2186 2332 y Fj(k)p 2034 2405 303 4 v 2034 2489 a Fo(\()p Fn(k)k Fm(\000)c Fo(1\)!)2363 2303 y Fg(Z)2414 2509 y Fr(\003)2464 2491 y Ff(k)2508 2426 y Fn(dx)15 b(e)2664 2389 y Fi(\000)p Fj(\014)s(H)2830 2358 y Ff(\021)2825 2412 y Fh(\003)2871 2389 y Fr(\()p Fj(x)p Fr(\))2989 2353 y Fg(\002)3027 2426 y Fn(D)3105 2387 y Fi(\000)3102 2454 y Fj(k)3166 2426 y Fn(f)10 b Fo(\()p Fn(x)p Fo(\))3343 2353 y Fg(\003)3381 2371 y Fr(2)784 2775 y Fo(=)25 b(2)p Fn(e)967 2738 y Fj(z)s Fi(j)p Fr(\003)p Fi(j)1102 2702 y Fg(\000)1144 2775 y Fo(1)c Fm(\000)f Fn(e)1343 2738 y Fi(\000)p Fj(z)s 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Fo(9\))0 4800 y(The)30 b(second)g(term)h(can)f(b)s(e)g(written)g(as)260 5035 y Fn(\026)315 4992 y Fj(\021)315 5062 y Fr(\003)369 5035 y Fo(\()p Fn(f)24 b(\026)528 5049 y Fj(A)587 5035 y Fn(f)10 b Fo(\))25 b(=)g Fm(h)p Fo(\()p Fn(f)31 b Fm(\000)19 b Fn(\026)1089 5049 y Fj(B)1152 5035 y Fn(f)10 b Fo(\))p Fn(;)15 b(\026)1337 5049 y Fj(A)1396 5035 y Fn(f)10 b Fm(i)20 b Fo(+)g Fm(h)p Fn(\026)1687 5049 y Fj(B)1749 5035 y Fn(f)5 b(;)15 b(\026)1894 5049 y Fj(A)1953 5035 y Fn(f)10 b Fm(i)25 b Fo(=)g Fm(h)p Fo(\()p Fn(f)30 b Fm(\000)20 b Fn(\026)2455 5049 y Fj(B)2517 5035 y Fn(f)10 b Fo(\))p Fn(;)15 b(\026)2702 5049 y Fj(A)2761 5035 y Fn(f)10 b Fm(i)20 b Fo(+)g Fm(h)p Fn(f)5 b(;)15 b(\026)3142 5049 y Fj(B)3204 5035 y Fn(\026)3259 5049 y Fj(A)3318 5035 y Fn(f)10 b Fm(i)285 5190 y(\024)25 b(k)p Fn(f)30 b Fm(\000)20 b Fn(\026)647 5204 y Fj(B)709 5190 y Fn(f)10 b Fm(k)809 5204 y Fr(2)850 5190 y Fm(k)p Fn(\026)950 5204 y Fj(A)1009 5190 y Fn(f)g Fm(k)1109 5204 y Fr(2)1170 5190 y Fo(+)20 b Fm(k)p Fn(f)10 b Fm(k)1406 5204 y Fr(2)1447 5190 y Fm(k)p Fn(\026)1547 5204 y Fj(B)1609 5190 y Fn(\026)1664 5204 y Fj(A)1723 5190 y Fn(f)g Fm(k)1823 5204 y Fr(2)1889 5190 y Fm(\024)25 b Fo([)16 b Fm(k)p Fn(f)30 b Fm(\000)19 b Fn(\026)2291 5204 y Fj(B)2354 5190 y Fn(f)10 b Fm(k)2454 5204 y Fr(2)2515 5190 y Fo(+)19 b Fn(")p Fm(k)p Fn(f)10 b Fm(k)2792 5204 y Fr(2)2849 5190 y Fo(])15 b Fm(k)p Fn(\026)2989 5204 y Fj(A)3049 5190 y Fn(f)10 b Fm(k)3149 5204 y Fr(2)285 5345 y Fo(=)25 b([)15 b Fm(k)p Fn(f)30 b Fm(\000)20 b Fn(\026)687 5359 y Fj(B)749 5345 y Fn(f)10 b Fm(k)849 5359 y Fr(2)910 5345 y Fo(+)20 b Fn(")p Fm(k)p Fn(f)10 b Fm(k)1188 5359 y Fr(2)1245 5345 y Fo(])15 b Fn(\026)1340 5301 y Fj(\021)1340 5371 y Fr(\003)1394 5345 y Fo(\()p Fn(f)25 b(\026)1554 5359 y Fj(A)1613 5345 y Fn(f)10 b Fo(\))1703 5307 y Fr(1)p Fj(=)p Fr(2)3668 5190 y Fo(\(3)p Fn(:)p Fo(10\))0 5558 y Fh(30)p Ff(=mag)r(g)r(io=)p Fh(2000)30 b Fr([10])1281 b Fo(3.4)p eop %%Page: 11 11 11 10 bop 0 91 a Fo(where)29 b(the)g(second)g(and)g(the)g(last)g (equalities)f(follo)m(w)h(from)f(the)i(DLR)f(conditions)f(\(2.2\),)j (while)d(in)f(the)j(second)0 218 y(inequalit)m(y)f(w)m(e)i(ha)m(v)m(e)g (used)f(\(3.7\))i(with)d Fn(p)c Fo(=)g(2.)41 b(F)-8 b(rom)31 b(\(3.10\))i(w)m(e)d(get)856 441 y Fn(\026)911 398 y Fj(\021)911 468 y Fr(\003)966 441 y Fo(\()p Fn(f)24 b(\026)1125 455 y Fj(A)1184 441 y Fn(f)10 b Fo(\))25 b Fm(\024)g(k)p Fn(f)30 b Fm(\000)20 b Fn(\026)1661 455 y Fj(B)1723 441 y Fn(f)10 b Fm(k)1823 404 y Fr(2)1823 464 y(2)1884 441 y Fo(+)20 b Fn(")2017 404 y Fr(2)2058 441 y Fm(k)p Fn(f)10 b Fm(k)2203 404 y Fr(2)2203 464 y(2)2264 441 y Fo(+)20 b(2)p Fn(")p Fm(k)p Fn(f)10 b Fm(k)2587 455 y Fr(2)2629 441 y Fm(k)p Fn(f)30 b Fm(\000)20 b Fn(\026)2895 455 y Fj(B)2957 441 y Fm(k)3002 455 y Fr(2)1299 596 y Fm(\024)25 b Fn(\026)1450 552 y Fj(\021)1450 623 y Fr(\003)1504 596 y Fo(\()p Fn(\026)1594 610 y Fj(B)1657 596 y Fo(\()p Fn(f)5 b(;)15 b(f)10 b Fo(\)\))20 b(+)g Fm(k)p Fn(f)10 b Fm(k)2163 558 y Fr(2)2163 619 y(2)2219 596 y Fo(\(2)p Fn(")22 b Fo(+)d Fn(")2495 558 y Fr(2)2537 596 y Fo(\))0 818 y(whic)m(h,)29 b(together)j(with)d(\(3.9\),)k(implies)27 b(\(3.8\))93 b Fb(\003)0 996 y Fo(In)24 b(order)f(to)j(pro)s(ceed)e (with)f(the)h(pro)s(of)g(of)g(Theorem)h(2.2)g(w)m(e)g(go)g(bac)m(k)g (to)g(the)g(geometry)g(of)g Fn(A)f Fo(and)g Fn(B)29 b Fo(describ)s(ed)0 1123 y(b)s(efore)j(\(3.6\))j(and)d(w)m(e)h(w)m(an)m (t)g(to)g(sho)m(w)g(that)g(Corollary)e(2.5)j(implies)c(that)j (inequalit)m(y)e(\(3.7\))j(holds)d(with,)i(sa)m(y)-8 b(,)0 1250 y Fn(p)31 b Fo(=)g Fm(1)p Fo(,)k(for)f(all)f(b)s(oundary)f (conditions.)50 b(In)33 b(fact,)j(if)d Fn(g)i Fm(2)c(F)2145 1264 y Fj(A)2200 1245 y Ff(c)2238 1250 y Fo(,)k(and)f Fn(L)g Fo(is)f(large)h(enough)f(so)i(that)f(Corollary)0 1377 y(2.5)d(can)g(b)s(e)f(applied,)e(w)m(e)j(ha)m(v)m(e,)h(b)m(y)e (\(2.2\))653 1599 y Fm(k)p Fn(\026)753 1626 y Fj(B)816 1599 y Fn(g)23 b Fm(\000)d Fn(\026)1028 1555 y Fj(\021)1028 1626 y Fr(\003)1082 1599 y Fn(g)s Fm(k)1173 1613 y Fi(1)1277 1599 y Fo(=)25 b Fm(k)p Fn(\026)1473 1626 y Fj(B)1536 1599 y Fn(g)e Fm(\000)d Fn(\026)1748 1555 y Fj(\021)1748 1626 y Fr(\003)1802 1599 y Fn(\026)1857 1626 y Fj(B)1920 1599 y Fn(g)s Fm(k)2011 1613 y Fi(1)2114 1599 y Fm(\024)62 b Fo(sup)2210 1675 y Fj(!)r(;\034)7 b Fi(2)p Fr(\012)2437 1599 y Fm(j)p Fn(\026)2517 1561 y Fj(!)2517 1621 y(B)2579 1599 y Fn(g)24 b Fm(\000)c Fn(\026)2792 1561 y Fj(\034)2792 1621 y(B)2854 1599 y Fn(g)s Fm(j)1277 1846 y(\024)25 b Fn(\013)15 b Fm(k)p Fn(g)s Fm(k)1582 1860 y Fr(1)1640 1846 y Fn(e)1682 1808 y Fi(\000)p Fj(md)p Fr(\(\003)p Fi(n)p Fj(A;)p Fr(\003)p Fi(n)p Fj(B)s Fr(\))2236 1846 y Fm(\024)25 b Fn(\013)16 b Fm(k)p Fn(g)s Fm(k)2542 1860 y Fi(1)2636 1846 y Fn(e)2678 1808 y Fi(\000)p Fj(md)p Fr(\(\003)p Fi(n)p Fj(A;)p Fr(\003)p Fi(n)p Fj(B)s Fr(\))3222 1846 y Fn(:)3668 1722 y Fo(\(3)p Fn(:)p Fo(11\))0 2071 y(The)30 b(same)h(b)s(ound)e(applies)g(to)i(the)g(quan)m(tit)m(y)g Fm(k)p Fn(\026)1732 2085 y Fj(A)1792 2071 y Fn(g)24 b Fm(\000)c 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b(\(3.6\))h(and)e(\(3.12\))i(w)m(e)f(get)613 3936 y Fn(\026)668 3893 y Fj(\021)668 3963 y Fr(\003)722 3936 y Fo(\()p Fn(f)5 b(;)15 b(f)10 b Fo(\))25 b Fm(\024)1058 3863 y Fg(\000)1099 3936 y Fo(1)c(+)f Fn(c)1295 3950 y Fr(1)1336 3936 y Fn(e)1378 3899 y Fi(\000)p Fj(c)1467 3908 y Fh(2)1503 3899 y Fj(m)1568 3845 y Fi(p)p 1627 3845 50 4 v 1627 3899 a Fj(L)1682 3863 y Fg(\001)1739 3936 y Fo(\()p Fn(G)1845 3950 y Fj(A)1925 3936 y Fm(_)g Fn(G)2077 3950 y Fj(B)2140 3936 y Fo(\))2205 3863 y Fg(\002)2258 3936 y Fm(E)2314 3893 y Fj(\021)2306 3963 y Fr(\003)2361 3936 y Fo(\()p Fn(f)10 b Fo(\))20 b(+)g Fn(\026)2652 3893 y Fj(\021)2652 3963 y Fr(\003)2706 3936 y Fo(\()p Fm(j)p Fn(D)2844 3898 y Fi(\000)2841 3963 y Fj(A)p Fi(\\)p Fj(B)3008 3936 y Fo(\()p Fn(f)10 b Fo(\))p Fm(j)3158 3899 y Fr(2)3199 3936 y Fo(\))3249 3863 y Fg(\003)3668 3936 y Fo(\(3)p Fn(:)p Fo(13\))0 4149 y(A)m(t)37 b(this)d(p)s(oin)m(t)h (one)h(ma)m(y)g(b)s(e)f(tempted)h(to)h(discourage,)g(b)s(ecause)e(if)g 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Fe(aver)-5 b(age)43 b(over)g(the)g(lo)-5 b(c)g(ation)45 b(of)e(the)g(overlap)p Fo(.)0 4838 y(Consider)30 b(in)g(fact)j(a)f(sequence)g(of)f(pairs)g Fm(f)p Fn(A)1566 4852 y Fj(i)1597 4838 y Fn(;)15 b(B)1706 4852 y Fj(i)1736 4838 y Fm(g)1781 4805 y Fj(s)1781 4862 y(i)p Fr(=1)1904 4838 y Fo(,)32 b(where,)g(for)f(instance,)h Fn(s)27 b Fo(:=)g Fm(b)p Fn(L)3067 4805 y Fr(1)p Fj(=)p Fr(3)3182 4838 y Fm(c)p Fo(,)32 b(where)f Fm(b\001c)i Fo(is)e(the)0 4965 y(in)m(tegral)f(part.)41 b(By)30 b(a)m(v)m(eraging)i (\(3.13\))h(o)m(v)m(er)e Fn(i)g Fo(w)m(e)g(obtain)255 5260 y Fn(\026)310 5216 y Fj(\021)310 5287 y Fr(\003)364 5260 y Fo(\()p Fn(f)5 b(;)15 b(f)10 b Fo(\))25 b Fm(\024)700 5186 y Fg(\000)742 5260 y Fo(1)c(+)f Fn(c)938 5274 y Fr(1)979 5260 y Fn(e)1021 5222 y Fi(\000)p Fj(c)1110 5231 y Fh(2)1146 5222 y Fj(m)1211 5168 y Fi(p)p 1270 5168 V 1270 5222 a Fj(L)1324 5186 y Fg(\001)1396 5260 y Fo(sup)1452 5335 y Fj(i)1533 5260 y Fo(\()p Fn(G)1639 5274 y Fj(A)1694 5283 y Ff(i)1748 5260 y Fm(_)g Fn(G)1900 5274 y Fj(B)1955 5283 y Ff(i)1989 5260 y Fo(\))2039 5159 y Fg(h)2097 5260 y Fm(E)2153 5216 y Fj(\021)2145 5287 y Fr(\003)2200 5260 y Fo(\()p Fn(f)10 b Fo(\))20 b(+)2447 5198 y(1)p 2447 5239 46 4 v 2448 5322 a Fn(s)2503 5260 y(\026)2558 5216 y Fj(\021)2558 5287 y Fr(\003)2612 5186 y Fg(\000)2702 5146 y Fj(s)2654 5174 y Fg(X)2661 5367 y Fj(i)p Fr(=1)2800 5260 y Fm(j)p Fn(D)2903 5221 y Fi(\000)2900 5287 y Fj(A)2955 5296 y Ff(i)2984 5287 y Fi(\\)p Fj(B)3088 5296 y Ff(i)3122 5260 y Fo(\()p Fn(f)10 b Fo(\))p Fm(j)3272 5222 y Fr(2)3313 5186 y Fg(\001)3369 5159 y(i)3668 5260 y Fo(\(3)p Fn(:)p Fo(14\))0 5558 y Fh(30)p Ff(=mag)r(g)r(io=)p Fh(2000)30 b Fr([11])1281 b Fo(3.5)p eop %%Page: 12 12 12 11 bop 0 91 a Fo(If)27 b(the)h(sets)g Fn(A)485 105 y Fj(i)515 91 y Fo(,)h Fn(B)638 105 y Fj(i)696 91 y Fo(are)f(c)m(hosen) g(in)e(suc)m(h)h(a)h(w)m(a)m(y)h(that)f Fn(A)1955 105 y Fj(i)2000 91 y Fm(\\)15 b Fn(B)2145 105 y Fj(i)2189 91 y Fm(\\)g Fn(A)2333 105 y Fj(j)2386 91 y Fm(\\)g Fn(B)2531 105 y Fj(j)2594 91 y Fo(=)25 b Fm(;)j Fo(for)f(all)f Fn(i)g Fm(6)p Fo(=)f Fn(j)33 b Fo(then)27 b(there)h(exists)0 217 y Fn(L)62 231 y Fr(1)128 217 y Fo(=)d Fn(L)286 231 y Fr(1)327 217 y Fo(\()p Fn(r)m(;)15 b(z)t(;)g(\014)5 b Fo(\))27 b Fm(\025)e Fn(L)802 231 y Fr(0)873 217 y Fo(suc)m(h)30 b(that)h(for)f(all)g Fn(L)25 b Fm(\025)g Fn(L)1786 231 y Fr(1)173 451 y Fn(G)244 465 y Fr(\003)324 451 y Fm(\024)420 377 y Fg(\000)461 451 y Fo(1)c(+)f Fn(c)657 465 y Fr(1)698 451 y Fn(e)740 413 y Fi(\000)p Fj(c)829 422 y Fh(2)865 413 y Fj(m)930 359 y Fi(p)p 989 359 50 4 v 989 413 a Fj(L)1044 377 y Fg(\001)1100 350 y(\020)1155 451 y Fo(1)g(+)1322 389 y(1)p 1322 430 46 4 v 1323 513 a Fn(s)1379 350 y Fg(\021)1463 451 y Fo(sup)1519 526 y Fj(i)1600 451 y Fo(\()p Fn(G)1706 465 y Fj(A)1761 474 y Ff(i)1815 451 y Fm(_)g Fn(G)1967 465 y Fj(B)2022 474 y Ff(i)2056 451 y Fo(\))25 b Fm(\024)2212 323 y Fg(\022)2279 451 y Fo(1)c(+)2552 389 y(2)p 2447 430 257 4 v 2447 515 a Fm(b)p Fn(L)2549 489 y Fr(1)p Fj(=)p Fr(3)2663 515 y Fm(c)2715 323 y Fg(\023)2812 451 y Fo(sup)2867 526 y Fj(i)2964 451 y Fo(\()p Fn(G)3070 465 y Fj(A)3125 474 y Ff(i)3179 451 y Fm(_)f Fn(G)3331 465 y Fj(B)3386 474 y Ff(i)3419 451 y Fo(\))15 b Fn(:)174 b Fo(\(3)p Fn(:)p Fo(15\))0 680 y(In)28 b(order)f(to)i(conclude)f(the)g(pro)s(of)g(of)g (Theorem)g(2.2)i(all)d(is)g(left)h(to)h(do)g(is)e(to)i(organize)g(the)f (geometric)h(iterativ)m(e)0 806 y(construction.)70 b(Let)41 b Fn(l)786 820 y Fj(k)872 806 y Fo(:=)h(\(3)p Fn(=)p Fo(2\))1215 773 y Fj(k)r(=d)1337 806 y Fo(,)h(and)d(let)g Fm(R)1810 773 y Fj(d)1810 832 y(k)1895 806 y Fo(b)s(e)f(the)i(set)g(of) f(all)f(rectangles)i(in)e Fm(R)3224 773 y Fj(d)3307 806 y Fo(whic)m(h,)j(mo)s(dulo)0 933 y(translations)29 b(and)h(p)s(erm)m (utations)f(of)i(the)f(co)s(ordinates,)h(are)f(con)m(tained)h(in)1279 1102 y([0)p Fn(;)15 b(l)1416 1116 y Fj(k)r Fr(+1)1554 1102 y Fo(])21 b Fm(\002)e Fo([0)p Fn(;)c(l)1827 1116 y Fj(k)r Fr(+2)1965 1102 y Fo(])21 b Fm(\002)f(\001)15 b(\001)g(\001)21 b(\002)f Fo([0)p Fn(;)15 b(l)2456 1116 y Fj(k)r Fr(+)p Fj(d)2596 1102 y Fo(])0 1271 y(Let)29 b(also)g Fn(G)413 1285 y Fj(k)483 1271 y Fo(:=)c(sup)741 1293 y Fj(V)16 b Fi(2R)911 1271 y Ff(d)911 1317 y(k)971 1271 y Fn(G)1042 1285 y Fj(V)1105 1271 y Fo(.)40 b(The)28 b(idea)g(b)s(ehind)e(this)i(construction)g(is)f(that)j(eac)m(h)f (rectangle)h(in)d Fm(R)3595 1238 y Fj(d)3595 1296 y(k)3640 1271 y Fm(nR)3762 1238 y Fj(d)3762 1296 y(k)r Fi(\000)p Fr(1)0 1397 y Fo(can)37 b(b)s(e)e(obtained)h(as)h(a)f(\\sligh)m(tly)f (o)m(v)m(erlapping)h(union")f(of)h(t)m(w)m(o)i(rectangles)f(in)e Fm(R)2956 1364 y Fj(d)2956 1423 y(k)r Fi(\000)p Fr(1)3094 1397 y Fo(.)58 b(More)38 b(precisely)c(w)m(e)0 1524 y(ha)m(v)m(e:)0 1691 y Fp(Prop)s(osition)43 b(3.2.)53 b Fc(F)-8 b(or)37 b(all)e Fo(\003)g Fm(2)g(R)1359 1658 y Fj(d)1359 1717 y(k)1404 1691 y Fm(nR)1526 1658 y Fj(d)1526 1717 y(k)r Fi(\000)p Fr(1)1700 1691 y Fc(there)i(exists)f(a)h(\014nite)e(sequence) i Fm(f)p Fn(A)3012 1705 y Fj(i)3042 1691 y Fn(;)15 b(B)3151 1705 y Fj(i)3182 1691 y Fm(g)3227 1650 y Fj(s)3261 1660 y Ff(k)3227 1716 y Fj(i)p Fr(=1)3350 1691 y Fc(,)38 b(where)e Fn(s)3725 1705 y Fj(k)3804 1691 y Fo(:=)0 1856 y Fm(b)p Fn(l)69 1808 y Fr(1)p Fj(=)p Fr(3)67 1883 y Fj(k)184 1856 y Fm(c)p Fc(,)31 b(suc)m(h)f(that)66 2004 y(\(1\))46 b Fo(\003)26 b(=)f Fn(A)480 2018 y Fj(i)530 2004 y Fm([)20 b Fn(B)680 2018 y Fj(i)741 2004 y Fc(and)29 b Fn(A)985 2018 y Fj(i)1016 2004 y Fn(;)15 b(B)1125 2018 y Fj(i)1181 2004 y Fm(2)25 b(R)1344 1971 y Fj(d)1344 2030 y(k)r Fi(\000)p Fr(1)1481 2004 y Fc(,)31 b(for)f(all)f Fn(i)d Fo(=)f(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(s)2245 2018 y Fj(k)66 2131 y Fc(\(2\))46 b Fn(d)p Fo(\(\003)p Fm(n)p Fn(A)485 2145 y Fj(i)517 2131 y Fn(;)15 b Fo(\003)p Fm(n)p Fn(B)734 2145 y Fj(i)765 2131 y Fo(\))26 b Fm(\025)933 2095 y Fr(1)p 933 2110 37 4 v 933 2163 a(8)980 2061 y Fm(p)p 1056 2061 72 4 v 70 x Fn(l)1083 2145 y Fj(k)1127 2131 y Fc(,)31 b(for)f(all)f Fn(i)d Fo(=)f(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(s) 1891 2145 y Fj(k)66 2259 y Fc(\(3\))46 b Fn(A)295 2273 y Fj(i)346 2259 y Fm(\\)20 b Fn(B)496 2273 y Fj(i)546 2259 y Fm(\\)g Fn(A)695 2273 y Fj(j)754 2259 y Fm(\\)f Fn(B)903 2273 y Fj(j)967 2259 y Fo(=)25 b Fm(;)31 b Fc(if)e Fn(i)d Fm(6)p Fo(=)f Fn(j)0 2426 y Fe(Pr)-5 b(o)g(of.)40 b Fo(Let)f(\003)f(:=)h([)p Fn(a)758 2440 y Fr(1)799 2426 y Fn(;)15 b(b)878 2440 y Fr(1)919 2426 y Fo(])26 b Fm(\002)f(\001)15 b(\001)g(\001)27 b(\002)e Fo([)p Fn(a)1367 2440 y Fj(d)1409 2426 y Fn(;)15 b(b)1488 2440 y Fj(d)1531 2426 y Fo(])39 b Fm(2)f(R)1771 2393 y Fj(d)1771 2452 y(k)1816 2426 y Fm(nR)1938 2393 y Fj(d)1938 2452 y(k)r Fi(\000)p Fr(1)2076 2426 y Fo(,)j(W)-8 b(e)39 b(can)g(assume)f Fn(a)2851 2440 y Fj(n)2939 2426 y Fo(=)g(0)h(and)f Fn(b)3356 2440 y Fj(n)3444 2426 y Fm(\024)g Fn(l)3580 2440 y Fj(k)r Fr(+)p Fj(n)3726 2426 y Fo(,)i(for)0 2552 y Fn(n)25 b Fo(=)g(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(d)p Fo(.)41 b(Then)30 b(necessarily)f Fn(b)1266 2566 y Fj(d)1334 2552 y Fn(>)24 b(l)1456 2566 y Fj(k)1501 2552 y Fo(,)31 b(since,)f(otherwise,)g(\003) 25 b Fm(2)g(R)2486 2519 y Fj(d)2486 2578 y(k)r Fi(\000)p Fr(1)2624 2552 y Fo(.)41 b(De\014ne)917 2789 y Fn(A)985 2803 y Fj(i)1041 2789 y Fo(:=)25 b([0)p Fn(;)15 b(b)1311 2803 y Fr(1)1353 2789 y Fo(])20 b Fm(\002)g(\001)15 b(\001)g(\001)22 b(\002)e Fo([0)p Fn(;)15 b(b)1856 2803 y Fj(d)p Fi(\000)p Fr(1)1992 2789 y Fo(])21 b Fm(\002)2128 2688 y Fg(h)2171 2789 y Fo(0)p Fn(;)2283 2728 y(b)2322 2742 y Fj(d)p 2283 2768 82 4 v 2301 2852 a Fo(2)2396 2789 y(+)2498 2728 y(2)p Fn(i)p 2498 2768 77 4 v 2513 2852 a Fo(8)2585 2705 y Fg(p)p 2676 2705 72 4 v 84 x Fn(l)2703 2803 y Fj(k)2748 2688 y Fg(i)916 3040 y Fn(B)985 3054 y Fj(i)1041 3040 y Fo(:=)k([0)p Fn(;)15 b(b)1311 3054 y Fr(1)1353 3040 y Fo(])20 b Fm(\002)g(\001)15 b(\001)g(\001)22 b(\002)e Fo([0)p Fn(;)15 b(b)1856 3054 y Fj(d)p Fi(\000)p Fr(1)1992 3040 y Fo(])21 b Fm(\002)2128 2939 y Fg(h)2182 2978 y Fn(b)2221 2992 y Fj(d)p 2182 3019 82 4 v 2200 3102 a Fo(2)2295 3040 y(+)2397 2978 y(2)p Fn(i)g Fm(\000)f Fo(1)p 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Fj(k)p 2276 3473 118 4 v 2311 3556 a Fo(4)2424 3494 y(+)2526 3432 y(1)p 2526 3473 46 4 v 2526 3556 a(4)2582 3494 y Fn(l)2611 3447 y Fr(5)p Fj(=)p Fr(6)2609 3521 y Fj(k)2751 3494 y Fm(\024)25 b Fn(l)2874 3508 y Fj(k)0 3663 y Fo(whic)m(h,)h (together)i(with)d(the)i(fact)g(that)g Fn(l)1388 3677 y Fj(k)1458 3663 y Fn(<)e(b)1593 3677 y Fj(d)1636 3663 y Fo(,)i(implies)d(that)j Fn(A)2254 3677 y Fj(i)2311 3663 y Fo(and)f Fn(B)2553 3677 y Fj(i)2609 3663 y Fo(are)h(b)s(oth)f (subsets)f(of)i(\003.)40 b(Moreo)m(v)m(er,)0 3789 y(since,)30 b(for)g(all)f Fn(i)d Fo(=)f(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(s)956 3803 y Fj(k)910 3949 y Fn(b)949 3963 y Fj(d)p 910 3990 82 4 v 928 4073 a Fo(2)1023 4011 y(+)1125 3949 y(2)p Fn(i)p 1125 3990 77 4 v 1140 4073 a Fo(8)1212 3926 y Fg(p)p 1303 3926 72 4 v 85 x Fn(l)1330 4025 y Fj(k)1400 4011 y Fm(\024)25 b Fn(l)1523 4025 y Fj(k)1568 4011 y Fn(;)106 b(b)1738 4025 y Fr(1)1804 4011 y Fm(\024)25 b Fn(l)1927 4025 y Fj(k)r Fr(+1)2064 4011 y Fn(;)107 b(:)15 b(:)g(:)h(;)106 b(b)2487 4025 y Fj(d)p Fi(\000)p Fr(1)2648 4011 y Fm(\024)25 b Fn(l)2771 4025 y Fj(k)r Fi(\000)p Fr(1+2)0 4229 y Fo(w)m(e)34 b(\014nd)d(that)j Fn(A)589 4243 y Fj(i)653 4229 y Fo(b)s(elongs)e(to)h Fm(R)1172 4196 y Fj(d)1172 4255 y(k)r Fi(\000)p Fr(1)1310 4229 y Fo(.)49 b(The)33 b(sets)h Fn(B)1824 4243 y Fj(i)1854 4229 y Fo('s)f(also)g(b)s(elong)f(to)i Fm(R)2617 4196 y Fj(d)2617 4255 y(k)r Fi(\000)p Fr(1)2755 4229 y Fo(,)g(since)f(they)g (are)g(smaller)f(than)0 4356 y(the)f Fn(A)225 4370 y Fj(i)255 4356 y Fo('s.)91 b Fb(\003)0 4503 y Fo(Let)32 b(then)f Fn(k)419 4517 y Fr(0)491 4503 y Fo(b)s(e)f(the)i(smallest)e (in)m(teger)i(suc)m(h)e(that)i Fn(l)1855 4517 y Fj(k)1927 4503 y Fm(\025)26 b Fn(L)2086 4517 y Fr(1)2127 4503 y Fo(.)43 b(F)-8 b(rom)31 b(\(3.15\))j(and)c(Prop)s(osition)g(3.2)i(w)m (e)f(obtain)0 4629 y(that)g(for)f(all)f Fn(k)g(>)c(k)681 4643 y Fr(0)703 4872 y Fn(G)774 4886 y Fj(k)844 4872 y Fm(\024)g Fo(\(1)c(+)f(2\(3)p Fn(=)p Fo(2\))1382 4835 y Fi(\000)p Fj(k)r(=)p Fr(\(3)p Fj(d)p Fr(\))1654 4872 y Fo(\))p Fn(G)1760 4886 y Fj(k)r Fi(\000)p Fr(1)1924 4872 y Fm(\024)25 b Fn(G)2091 4886 y Fj(k)2130 4895 y Fh(0)2280 4758 y Fi(1)2259 4786 y Fg(Y)2186 4981 y Fj(k)r Fr(=)p Fj(k)2322 4990 y Fh(0)2356 4981 y Fr(+1)2448 4872 y Fo(\(1)c(+)f(2\(3)p Fn(=)p Fo(2\))2890 4835 y Fi(\000)p Fj(k)r(=)p Fr(\(3)p Fj(d)p Fr(\))3162 4872 y Fo(\))844 5148 y Fm(\024)25 b Fn(G)1011 5162 y Fj(k)1050 5171 y Fh(0)1106 5148 y Fo(exp)o([2\(1)d Fm(\000)e Fo(\(2)p Fn(=)p Fo(3\))1712 5111 y Fr(1)p Fj(=)p Fr(\(3)p Fj(d)p Fr(\))1923 5148 y Fo(\))1958 5111 y Fi(\000)p Fr(1)2056 5148 y Fo(])0 5322 y(whic)m(h,)29 b(together)j(with)d(Prop)s(osition)g (2.6,)i(yields)e(Theorem)h(2.2)92 b Fb(\003)0 5558 y Fh(30)p Ff(=mag)r(g)r(io=)p Fh(2000)30 b Fr([12])1281 b Fo(4.1)p eop %%Page: 13 13 13 12 bop 0 91 a Fp(4.)47 b(Sp)s(ectral)35 b(gap)g Fo(=)-15 b Fm(\))34 b Fp(deca)m(y)h(of)g(correlations)0 245 y Fo(In)d(this)f(Section)h(w)m(e)h(pro)m(v)m(e)g(a)g(partial)e(con)m(v)m (erse)j(to)f(Theorem)f(2.2.)48 b(More)33 b(precisely)-8 b(,)32 b(assuming)f(\014nite)h(range)0 372 y(p)s(ositiv)m(e)h(pair)f(p) s(oten)m(tial,)j(w)m(e)f(get)h(the)f(exp)s(onen)m(tial)f(deca)m(y)i(of) f(correlation)f(in)g(a)h(v)m(olume)g(\003)f(with)g(b)s(oundary)0 499 y(condition)39 b Fn(\021)44 b Fo(pro)m(vided)39 b(our)h(Glaub)s (er{t)m(yp)s(e)g(dynamics)f(satis\014es)h(a)h(P)m(oincar)m(\023)-43 b(e)42 b(inequalit)m(y)d(in)g(that)i(v)m(olume)0 626 y(with)34 b(that)i(b)s(oundary)d(condition.)55 b(Unfortunately)-8 b(,)36 b(w)m(e)g(are)g(not)f(able)g(to)h(pro)m(v)m(e)g(the)g(exp)s (onen)m(tial)e(deca)m(y)j(of)0 754 y(correlation)31 b(as)g(stated)h(in) e(Corollary)g(2.4)i(but)f(only)f(with)g(the)h Fn(L)2294 721 y Fr(1)2335 754 y Fo(\()p Fn(d\026)2472 710 y Fj(\021)2472 780 y Fr(\003)2527 754 y Fo(\))g(norm)g(whic)m(h)f(app)s(ears)g(on)h (the)g(RHS)0 881 y(of)g(\(2.7\))h(replaced)e(b)m(y)g(a)h(m)m(uc)m(h)f (stronger)h(norm;)e(therefore)i(w)m(e)g(do)f(not)h(ha)m(v)m(e)h(equiv) -5 b(alence)29 b(in)g(Theorem)i(2.2.)227 1009 y(The)26 b(argumen)m(t)g(leading)f(to)i(the)g(result)e(previously)e(outlined)i (is)g(w)m(ell{kno)m(wn)g(in)g(the)h(con)m(text)i(of)f(lattice)0 1136 y(\(b)s(ounded)34 b(or)i(un)m(b)s(ounded\))d(spin)h(systems,)j (see)f(e.g.)58 b([Ma,MO1,MO2,SZ1,SZ2,Y)-8 b(o,Ze].)61 b(Belo)m(w)36 b(w)m(e)g(stress)0 1263 y(the)31 b(main)e(di\013erences)h (in)f(the)h(con)m(tin)m(uous)g(case)i(w)m(e)e(are)h(dealing)e(with.)227 1391 y(Recalling)20 b(the)i(op)s(erator)f Fn(D)1196 1358 y Fr(+)1193 1414 y Fj(x)1257 1391 y Fo(,)i(de\014ned)d(in)g(\(2.3\),)25 b(w)m(e)d(in)m(tro)s(duce)e(the)i(follo)m(wing)d(semi-norm.)37 b(F)-8 b(or)22 b Fn(f)35 b Fm(2)24 b(F)3845 1405 y Fr(\003)0 1518 y Fo(w)m(e)31 b(set)1387 1645 y Fm(j)-15 b(j)g(j)p Fn(f)10 b Fm(j)-15 b(j)g(j)26 b Fo(:=)1679 1522 y Fg(Z)1730 1728 y Fr(\003)1784 1645 y Fn(dx)1913 1568 y Fg(\015)1913 1623 y(\015)1964 1645 y Fn(D)2042 1608 y Fr(+)2039 1668 y Fj(x)2102 1645 y Fn(f)2157 1568 y Fg(\015)2157 1623 y(\015)2207 1682 y Fj(L)2257 1664 y Fd(1)2323 1682 y Fr(\()p Fj(d\026)2433 1651 y Ff(\021)2433 1705 y Fh(\003)2480 1682 y Fr(\))3713 1645 y Fo(\(4)p Fn(:)p Fo(1\))0 1891 y(whic)m(h)20 b(is)g(the)i(con)m(tin)m(uous)f(analogous)g(of)h (Liggett's)g(triple)e(norm)g([Li,)j(Ch.I].)e(W)-8 b(e)23 b(sho)m(w)e(next)g(that)h(the)g(mapping)0 2018 y Fn(x)37 b Fm(!)g(k)p Fn(D)340 1985 y Fr(+)337 2041 y Fj(x)401 2018 y Fn(f)10 b Fm(k)501 2037 y Fj(L)551 2019 y Fd(1)617 2037 y Fr(\()p Fj(d\026)727 2007 y Ff(\021)727 2061 y Fh(\003)773 2037 y Fr(\))844 2018 y Fo(is)36 b(indeed)g(measurable.)62 b(W)-8 b(e)39 b(\014rst)d(notice)i(that)g(\()p Fn(x;)15 b(!)s Fo(\))39 b Fm(!)e Fn(D)3074 1985 y Fr(+)3071 2041 y Fj(x)3134 2018 y Fn(f)10 b Fo(\()p Fn(!)s Fo(\))37 b(is)g(measurable)0 2146 y(w.r.t.)i(the)24 b(pro)s(duct)g Fn(\033)s Fm(\000)p Fo(algebra)g Fm(B)s Fo(\()p Fk(R)1332 2113 y Fj(d)1381 2146 y Fo(\))8 b Fm(\012)g(F)h Fo(.)39 b(F)-8 b(or)25 b(this)f(purp)s(ose)e(it)i(is)g(enough)g(to)h(sho)m(w)f (that)h Fn(t)g Fo(:)h Fk(R)3466 2113 y Fj(d)3523 2146 y Fm(\002)8 b Fo(\012)25 b Fm(7!)g Fo(\012,)0 2273 y(de\014ned)32 b(b)m(y)g Fn(t)p Fo(\()p Fn(x;)15 b(!)s Fo(\))30 b(:=)g Fn(!)24 b Fm([)e Fn(x)32 b Fo(is)g(measurable.)47 b(Since)32 b Fm(F)42 b Fo(is)32 b(generated)i(b)m(y)f(the)g(functions)e(\()p Fn(N)3363 2287 y Fj(A)3423 2273 y Fo(\))3458 2291 y Fj(A)p Fi(2B)3610 2301 y Ff(b)3643 2291 y Fr(\()p Fa(R)3721 2273 y Ff(d)3755 2291 y Fr(\))3788 2273 y Fo(,)j(if)0 2400 y(w)m(e)i(sho)m(w)g(that)g(\()p Fn(t)640 2367 y Fi(\000)p Fr(1)761 2400 y Fm(\016)24 b Fn(N)913 2361 y Fi(\000)p Fr(1)903 2427 y Fj(A)1011 2400 y Fo(\))p Fm(f)p Fn(k)s Fm(g)35 b(2)f(B)s Fo(\()p Fk(R)1473 2367 y Fj(d)1522 2400 y Fo(\))24 b Fm(\012)f(F)45 b Fo(for)35 b(all)g(nonnegativ)m(e)h(in)m(tegers)g Fn(k)s Fo(,)h(the)f (measurabilit)m(y)d(of)j Fn(t)0 2527 y Fo(follo)m(ws.)k(But)256 2741 y(\()p Fn(t)324 2703 y Fi(\000)p Fr(1)442 2741 y Fm(\016)20 b Fn(N)590 2702 y Fi(\000)p Fr(1)580 2767 y Fj(A)688 2741 y Fo(\))p Fm(f)p Fn(k)s Fm(g)26 b Fo(=)f Fm(f)p Fo(\()p Fn(x;)15 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Fo(is)g(the)i(translation)d(b)m(y)i Fn(x)p Fo(,)h(and,)g(since)e(the)i(mapping)0 3393 y(\()p Fn(x;)15 b(!)s Fo(\))26 b Fm(!)f Fn(#)418 3407 y Fj(x)464 3393 y Fn(!)h Fo(is)d(measurable)g(\(see,)j(for)d(instance)h([MKK]\),)g (w)m(e)g(ha)m(v)m(e)h(that)g Fn(M)33 b Fo(is)23 b(measurable.)38 b(W)-8 b(e)24 b(ha)m(v)m(e)h(th)m(us)0 3520 y(sho)m(wn)37 b(that)i(\()p Fn(x;)15 b(!)s Fo(\))38 b Fm(!)f Fn(D)954 3487 y Fr(+)951 3542 y Fj(x)1015 3520 y Fn(f)10 b Fo(\()p Fn(!)s Fo(\))37 b(is)g(measurable.)62 b(By)38 b(consequence)h Fn(x)e Fm(!)h(k)p Fn(D)2881 3487 y Fr(+)2878 3542 y Fj(x)2941 3520 y Fn(f)10 b Fm(k)3041 3539 y Fj(L)3091 3521 y Ff(p)3129 3539 y Fr(\()p Fj(d\026)3239 3508 y Ff(\021)3239 3562 y Fh(\003)3285 3539 y Fr(\))3356 3520 y Fo(is)37 b(measurable)0 3647 y(for)31 b(all)g Fn(p)c Fm(2)g Fo([1)p Fn(;)15 b Fm(1)p Fo(\).)45 b(Finally)-8 b(,)31 b Fm(k)p Fn(D)1187 3614 y Fr(+)1184 3669 y Fj(x)1248 3647 y Fn(f)10 b Fm(k)1348 3666 y Fj(L)1398 3648 y Fd(1)1464 3666 y Fr(\()p Fj(d\026)1574 3635 y Ff(\021)1574 3689 y Fh(\003)1620 3666 y Fr(\))1680 3647 y Fo(=)27 b(lim)1905 3661 y Fj(p)p Fi(!1)2107 3647 y Fm(k)p Fn(D)2230 3614 y Fr(+)2227 3669 y Fj(x)2290 3647 y Fn(f)10 b Fm(k)2390 3666 y Fj(L)2440 3648 y Ff(p)2478 3666 y Fr(\()p Fj(d\026)2588 3635 y Ff(\021)2588 3689 y Fh(\003)2634 3666 y Fr(\))2667 3647 y Fo(,)33 b(so)e Fn(x)d Fm(!)f(k)p Fn(D)3158 3614 y Fr(+)3155 3669 y Fj(x)3218 3647 y Fn(f)10 b Fm(k)3318 3666 y Fj(L)3368 3648 y Fd(1)3434 3666 y Fr(\()p Fj(d\026)3544 3635 y Ff(\021)3544 3689 y Fh(\003)3591 3666 y Fr(\))3655 3647 y Fo(is)31 b(also)0 3774 y(measurable.)0 3954 y(The)f(main)f(result)g(in)g(this)h(section)g (is:)0 4134 y Fp(Theorem)k(4.1.)53 b Fc(Let)31 b Fn(')25 b Fm(\025)g Fo(0)31 b Fc(b)s(e)f(of)g(\014nite)g(range)g Fn(r)s Fc(.)41 b(If)30 b(there)g(exists)g Fn(G)c(<)f Fm(1)30 b Fc(suc)m(h)g(that)1249 4359 y Fn(\026)1304 4315 y Fj(\021)1304 4385 y Fr(\003)1358 4359 y Fo(\()p Fn(f)5 b(;)15 b(f)10 b Fo(\))25 b Fm(\024)g Fn(G)15 b Fm(E)1836 4315 y Fj(\021)1828 4385 y Fr(\003)1883 4359 y Fo(\()p Fn(f)10 b Fo(\))182 b Fm(8)p Fn(f)34 b Fm(2)24 b(D)s Fo(\()p Fm(E)2569 4315 y Fj(\021)2561 4385 y Fr(\003)2616 4359 y Fo(\))0 4583 y Fc(then)31 b(there)h(are)h Fn(m)27 b Fo(=)g Fn(m)p Fo(\()p Fn(G;)15 b(r)m(;)g(z)t Fo(\))30 b Fn(>)e Fo(0)k Fc(and)f Fn(\013)d Fo(=)f Fn(\013)p Fo(\()p Fn(G;)15 b(r)m(;)g(z)t Fo(\))31 b Fn(<)c Fm(1)32 b Fc(suc)m(h)f(that)i (the)e(follo)m(wing)g(holds.)43 b(F)-8 b(or)33 b(an)m(y)0 4711 y Fo(\003)63 4725 y Fj(f)110 4711 y Fn(;)15 b Fo(\003)213 4725 y Fj(g)284 4711 y Fm(2)27 b(B)432 4725 y Fj(b)468 4711 y Fo(\()p Fk(R)563 4678 y Fj(d)611 4711 y Fo(\))32 b Fc(suc)m(h)g(that)g Fo(\003)1146 4725 y Fj(f)1221 4711 y Fm(\032)27 b Fo(\003)p Fc(,)32 b Fo(\003)1502 4725 y Fj(g)1572 4711 y Fm(\032)27 b Fo(\003)p Fc(,)33 b Fo(\003)1854 4725 y Fj(f)1922 4711 y Fm(\\)21 b Fo(\003)2067 4725 y Fj(g)2137 4711 y Fo(=)27 b Fm(;)p Fc(,)32 b(and)f Fm(j)2549 4688 y Fo(\026)2540 4711 y(\003)2603 4650 y Ff(r)2603 4736 y Fj(f)2651 4711 y Fm(j)21 b(^)g(j)2813 4688 y Fo(\026)2804 4711 y(\003)2867 4650 y Ff(r)2867 4733 y Fj(g)2910 4711 y Fm(j)27 b(\024)h Fo(exp)o(\()p Fn(m)15 b(d)p Fo(\(\003)3474 4725 y Fj(f)3523 4711 y Fn(;)g Fo(\003)3626 4725 y Fj(g)3669 4711 y Fo(\)\))p Fc(,)32 b(w)m(e)0 4838 y(ha)m(v)m(e)239 5062 y Fm(j)p Fn(\026)319 5019 y Fj(\021)319 5089 y Fr(\003)373 5062 y Fo(\()p Fn(f)5 b(;)15 b(g)s Fo(\))p Fm(j)26 b(\024)f Fn(\013)800 4961 y Fg(\020)854 5062 y Fm(k)p Fn(f)10 b Fm(k)999 5081 y Fj(L)1049 5063 y Fh(2)1085 5081 y Fr(\()p Fj(d\026)1195 5051 y Ff(\021)1195 5105 y Fh(\003)1241 5081 y Fr(\))1289 5062 y Fm(k)p Fn(g)s Fm(k)1425 5081 y Fj(L)1475 5063 y Fh(2)1512 5081 y Fr(\()p Fj(d\026)1622 5051 y Ff(\021)1622 5105 y Fh(\003)1669 5081 y Fr(\))1722 5062 y Fo(+)20 b Fn(z)f Fm(j)-15 b(j)g(j)p Fn(f)10 b Fm(j)-15 b(j)g(j)15 b(j)-15 b(j)g(j)p Fn(g)s Fm(j)g(j)g(j)2170 4961 y Fg(\021)2242 5062 y Fn(e)2284 5025 y Fi(\000)p Fj(m)12 b(d)p Fr(\(\003)2534 5035 y Ff(f)2575 5025 y Fj(;)p Fr(\003)2647 5034 y Ff(g)2684 5025 y Fr(\))2899 5062 y Fm(8)p Fn(f)34 b Fm(2)25 b(F)3180 5076 y Fr(\003)3230 5086 y Ff(f)3275 5062 y Fn(;)46 b(g)29 b Fm(2)c(F)3569 5076 y Fr(\003)3619 5085 y Ff(g)0 5340 y Fo(The)f(k)m(ey)g(ingredien)m (t)f(in)g(pro)m(ving)g(the)h(ab)s(o)m(v)m(e)h(Theorem)f(is)f(the)h (follo)m(wing)f Fe(\014nite)k(sp)-5 b(e)g(e)g(d)28 b(of)f(pr)-5 b(op)g(agation)28 b Fo(lemma.)0 5558 y Fh(30)p Ff(=mag)r(g)r(io=)p Fh(2000)i Fr([13])1281 b Fo(4.2)p eop %%Page: 14 14 14 13 bop 0 91 a Fp(Lemma)28 b(4.2.)53 b Fc(Let)27 b Fn(')f Fm(\025)f Fo(0)i Fc(b)s(e)f(of)h(\014nite)e(range)i Fn(r)s Fc(.)39 b(Then)26 b(there)h(are)g Fn(\016)i Fo(=)c Fn(\016)s Fo(\()p Fn(r)m(;)15 b(z)t Fo(\))28 b Fn(>)d Fo(0)i Fc(and)f Fn(M)35 b Fo(=)25 b Fn(M)10 b Fo(\()p Fn(r)m(;)15 b(z)t Fo(\))27 b Fn(<)e Fm(1)0 218 y Fc(suc)m(h)32 b(that)i(the)f(follo)m(wing)e(holds.)47 b(F)-8 b(or)33 b(an)m(y)g Fo(\003)1635 232 y Fj(f)1683 218 y Fn(;)15 b Fo(\003)1786 232 y Fj(g)1858 218 y Fm(2)29 b(B)2008 232 y Fj(b)2044 218 y Fo(\()p Fk(R)2139 185 y Fj(d)2187 218 y Fo(\))k Fc(suc)m(h)g(that)g Fo(\003)2725 232 y Fj(f)2802 218 y Fm(\032)c Fo(\003)p Fc(,)34 b Fo(\003)3087 232 y Fj(g)3158 218 y Fm(\032)29 b Fo(\003)p Fc(,)34 b Fo(\003)3443 232 y Fj(f)3512 218 y Fm(\\)22 b Fo(\003)3658 232 y Fj(g)3729 218 y Fo(=)29 b Fm(;)p Fc(,)0 346 y(and)h Fm(j)211 323 y Fo(\026)202 346 y(\003)265 286 y Ff(r)265 372 y Fj(f)312 346 y Fm(j)21 b(^)f(j)473 323 y Fo(\026)464 346 y(\003)527 286 y Ff(r)527 368 y Fj(g)569 346 y Fm(j)26 b(\024)f Fo(exp\()p Fn(\016)19 b(d)p Fo(\(\003)1094 360 y Fj(f)1142 346 y Fn(;)c Fo(\003)1245 360 y Fj(g)1288 346 y Fo(\)\))p Fc(,)31 b(w)m(e)g(ha)m(v)m(e)745 537 y Fg(\014)745 592 y(\014)745 646 y(\014)775 642 y Fn(\026)830 598 y Fj(\021)830 668 y Fr(\003)899 541 y Fg(\020)953 642 y Fn(P)1024 598 y Fr(\003)p Fj(;\021)1011 664 y(t)1140 642 y Fo(\()p Fn(f)10 b(g)s Fo(\))21 b Fm(\000)f Fn(P)1494 598 y Fr(\003)p Fj(;\021)1481 664 y(t)1609 642 y Fn(f)25 b(P)1750 598 y Fr(\003)p Fj(;\021)1737 664 y(t)1865 642 y Fn(g)1911 541 y Fg(\021)1966 537 y(\014)1966 592 y(\014)1966 646 y(\014)2022 642 y Fm(\024)g Fn(M)g(z)20 b(e)2335 604 y Fj(M)7 b(t)p Fi(\000)p Fj(\016)15 b(d)p Fr(\(\003)2659 614 y Ff(f)2700 604 y Fj(;)p Fr(\003)2772 613 y Ff(g)2809 604 y Fr(\))2858 642 y Fm(j)-15 b(j)g(j)p Fn(f)10 b Fm(j)-15 b(j)g(j)15 b(j)-15 b(j)g(j)p Fn(g)s Fm(j)g(j)g(j)559 b Fo(\(4)p Fn(:)p Fo(2\))0 928 y Fc(for)30 b(an)m(y)h Fn(f)j Fm(2)25 b(F)541 942 y Fr(\003)591 952 y Ff(f)657 928 y Fm(\\)20 b Fn(L)800 895 y Fr(2)841 928 y Fo(\()p Fn(d\026)978 884 y Fj(\021)978 955 y Fr(\003)1032 928 y Fo(\))31 b Fc(and)f Fn(g)f Fm(2)24 b(F)1497 942 y Fr(\003)1547 951 y Ff(g)1610 928 y Fm(\\)c Fn(L)1753 895 y Fr(2)1794 928 y Fo(\()p Fn(d\026)1931 884 y Fj(\021)1931 955 y Fr(\003)1986 928 y Fo(\))p Fc(.)0 1118 y Fe(R)-5 b(emark.)34 b Fo(F)-8 b(or)33 b(compact)h(or)e(\(suitable\))g(un)m(b)s(ounded)e (spin)g(systems,)j(one)g(can)g(pro)m(v)m(e)g(a)g(b)s(ound)d(analogous)j (to)0 1246 y(\(4.2\))42 b(with)c(its)h(LHS)h(replaced)f(b)m(y)g Fm(k)q Fn(P)1402 1260 y Fj(t)1434 1246 y Fo(\()p Fn(f)10 b(g)s Fo(\))21 b Fm(\000)f Fn(P)1775 1260 y Fj(t)1807 1246 y Fn(f)25 b(P)1935 1260 y Fj(t)1967 1246 y Fn(g)s Fm(k)2059 1273 y Fj(L)2109 1255 y Fd(1)2176 1273 y Fr(\()p Fj(d\026)2286 1242 y Ff(\021)2286 1297 y Fh(\003)2332 1273 y Fr(\))2365 1246 y Fo(;)45 b(see)40 b(e.g.)70 b([Li,)42 b(I)d(Prop.)69 b(4.18])41 b(and)e([Y)-8 b(o])0 1373 y(resp)s(ectiv)m (ely)g(.)48 b(In)32 b(the)g(con)m(tin)m(uous)h(case)h(w)m(e)f(do)g(not) f(get)i(suc)m(h)f(a)g(stronger)g(b)s(ound;)f(ho)m(w)m(ev)m(er)i(Lemma)f (4.2)h(as)0 1501 y(stated)d(is)f(precisely)f(what)h(w)m(e)h(need)f(to)h (pro)m(v)m(e)g(Theorem)f(4.1.)0 1692 y Fe(Pr)-5 b(o)g(of)34 b(of)e(The)-5 b(or)g(em)35 b(4.1.)30 b Fo(W)-8 b(e)32 b(can)e(assume)g Fn(\026)1598 1648 y Fj(\021)1598 1718 y Fr(\003)1652 1692 y Fn(f)35 b Fo(=)25 b Fn(\026)1883 1648 y Fj(\021)1883 1718 y Fr(\003)1937 1692 y Fn(g)k Fo(=)24 b(0.)42 b(Since)29 b Fn(\026)2508 1648 y Fj(\021)2508 1718 y Fr(\003)2592 1692 y Fo(is)g(the)h(in)m(v)-5 b(arian)m(t)29 b(measure)i(for)e Fn(P)3784 1648 y Fr(\003)p Fj(;\021)3771 1714 y(t)0 1819 y Fo(w)m(e)i(ha)m(v)m(e)349 2059 y Fm(j)p Fn(\026)429 2016 y Fj(\021)429 2086 y Fr(\003)498 2059 y Fo(\()q Fn(f)5 b(;)15 b(g)s Fo(\))p Fm(j)26 b Fo(=)852 1982 y Fg(\014)852 2037 y(\014)882 2059 y Fn(\026)937 2016 y Fj(\021)937 2086 y Fr(\003)991 2059 y Fo(\()p Fn(P)1097 2016 y Fr(\003)p Fj(;\021)1084 2082 y(t)1213 2059 y Fo(\()p Fn(f)10 b(g)s Fo(\)\))1419 1982 y Fg(\014)1419 2037 y(\014)1476 2059 y Fo(=)1572 1982 y Fg(\014)1572 2037 y(\014)1602 2059 y Fn(\026)1657 2016 y Fj(\021)1657 2086 y Fr(\003)1711 2059 y Fo(\()p Fn(P)1817 2016 y Fr(\003)p Fj(;\021)1804 2082 y(t)1933 2059 y Fn(f)25 b(P)2074 2016 y Fr(\003)p Fj(;\021)2061 2082 y(t)2189 2059 y Fn(g)s Fo(\))c(+)f Fn(\026)2437 2016 y Fj(\021)2437 2086 y Fr(\003)2491 2059 y Fo(\()p Fn(P)2597 2016 y Fr(\003)p Fj(;\021)2584 2082 y(t)2713 2059 y Fo(\()p Fn(f)10 b(g)s Fo(\))21 b Fm(\000)f Fn(P)3067 2016 y Fr(\003)p Fj(;\021)3054 2082 y(t)3183 2059 y Fn(f)k(P)3323 2016 y Fr(\003)p Fj(;\021)3310 2082 y(t)3439 2059 y Fn(g)s Fo(\))3520 1982 y Fg(\014)3520 2037 y(\014)756 2214 y Fm(\024)852 2137 y Fg(\015)852 2191 y(\015)902 2214 y Fn(P)973 2170 y Fr(\003)p Fj(;\021)960 2236 y(t)1089 2214 y Fn(f)1144 2137 y Fg(\015)1144 2191 y(\015)1194 2250 y Fj(L)1244 2232 y Fh(2)1279 2250 y Fr(\()p Fj(d\026)1389 2220 y Ff(\021)1389 2274 y Fh(\003)1436 2250 y Fr(\))1469 2137 y Fg(\015)1469 2191 y(\015)1519 2214 y Fn(P)1590 2170 y Fr(\003)p Fj(;\021)1577 2236 y(t)1706 2214 y Fn(g)1752 2137 y Fg(\015)1752 2191 y(\015)1803 2250 y Fj(L)1853 2232 y Fh(2)1888 2250 y Fr(\()p Fj(d\026)1998 2220 y Ff(\021)1998 2274 y Fh(\003)2045 2250 y Fr(\))2098 2214 y Fo(+)2189 2137 y Fg(\014)2189 2191 y(\014)2219 2214 y Fn(\026)2274 2170 y Fj(\021)2274 2241 y Fr(\003)2328 2214 y Fo(\()p Fn(P)2434 2170 y Fr(\003)p Fj(;\021)2421 2236 y(t)2550 2214 y Fo(\()p Fn(f)10 b(g)s Fo(\))21 b Fm(\000)f Fn(P)2904 2170 y Fr(\003)p Fj(;\021)2891 2236 y(t)3019 2214 y Fn(f)25 b(P)3160 2170 y Fr(\003)p Fj(;\021)3147 2236 y(t)3275 2214 y Fn(g)s Fo(\))3356 2137 y Fg(\014)3356 2191 y(\014)0 2501 y Fo(where)32 b(w)m(e)g(used)g(Sc)m(h)m(w)m(arz)h (inequalit)m(y)-8 b(.)45 b(F)-8 b(or)33 b Fn(\016)j Fo(and)c Fn(M)42 b Fo(as)32 b(in)f(Lemma)h(4.2,)i(c)m(ho)s(ose)g Fn(t)28 b Fo(=)g Fn(\016)19 b Fo(\(2)p Fn(M)10 b Fo(\))3412 2468 y Fi(\000)p Fr(1)3526 2501 y Fn(d)p Fo(\(\003)3671 2515 y Fj(f)3719 2501 y Fn(;)15 b Fo(\003)3822 2515 y Fj(g)3865 2501 y Fo(\))0 2628 y(and)30 b(apply)f Fm(k)p Fn(P)543 2585 y Fr(\003)p Fj(;\021)530 2651 y(t)659 2628 y Fn(f)10 b Fm(k)759 2647 y Fj(L)809 2629 y Fh(2)844 2647 y Fr(\()p Fj(d\026)954 2617 y Ff(\021)954 2671 y Fh(\003)1000 2647 y Fr(\))1059 2628 y Fm(\024)25 b Fn(e)1197 2595 y Fi(\000)p Fj(t=G)1380 2628 y Fm(k)p Fn(f)10 b Fm(k)1525 2647 y Fj(L)1575 2629 y Fh(2)1610 2647 y Fr(\()p Fj(d\026)1720 2617 y Ff(\021)1720 2671 y Fh(\003)1766 2647 y Fr(\))1830 2628 y Fo(together)32 b(with)d(the)h(b)s(ound)f (\(4.2\))j(to)f(get)g(the)g(result.)90 b Fb(\003)0 2819 y Fe(Pr)-5 b(o)g(of)29 b(of)f(L)-5 b(emma)29 b(4.2.)d Fo(Since)e(the)h(v)m(olume)g(\003)g(and)f(the)i(b)s(oundary)d (condition)g Fn(\021)29 b Fo(are)c(k)m(ept)h(\014xed)e(w)m(e)i(drop)e (them)0 2947 y(from)30 b(the)g(notation.)41 b(W)-8 b(e)32 b(claim)d(for)i(eac)m(h)g Fn(f)5 b(;)15 b(g)29 b Fm(2)c(D)s Fo(\()p Fm(E)8 b Fo(\))1096 3263 y Fn(\026)15 b Fo(\()p Fn(P)1259 3277 y Fj(t)1292 3263 y Fo(\()p Fn(f)10 b(g)s Fo(\))20 b Fm(\000)g Fn(P)1632 3277 y Fj(t)1665 3263 y Fn(f)k(P)1792 3277 y Fj(t)1824 3263 y Fn(g)s Fo(\))j(=)e(2)2088 3139 y Fg(Z)2179 3161 y Fj(t)2139 3346 y Fr(0)2211 3263 y Fn(ds)30 b Fm(E)24 b Fo(\()p Fn(P)2496 3277 y Fj(s)2535 3263 y Fn(f)5 b(;)15 b(P)2683 3277 y Fj(s)2722 3263 y Fn(g)s Fo(\))910 b(\(4)p Fn(:)p Fo(3\))0 3562 y(whic)m(h)36 b(is)g(a)i(general)g(iden)m(tit)m(y)e(for)h(self{adjoin)m(t)g(Mark)m(o) m(v)i(semigroups.)61 b(In)36 b(order)h(to)h(v)m(erify)f(it,)i(let)e(us) g(\014rst)0 3689 y(consider)42 b Fn(f)5 b(;)15 b(g)49 b Fm(2)c Fn(L)717 3656 y Fi(1)794 3689 y Fo(\()p Fn(d\026)p Fo(\))29 b Fm(\\)g(D)s Fo(\()p Fm(E)8 b Fo(\))43 b(and)f(appro)m (ximate)h(the)g(generator)h Fn(L)e Fo(b)m(y)h(the)g(b)s(ounded)d(\(in)i Fn(L)3651 3656 y Fr(2)3692 3689 y Fo(\()p Fn(d\026)p Fo(\)\))0 3817 y(op)s(erator)e Fn(L)438 3831 y Fj(k)522 3817 y Fo(de\014ned)e(b)m(y)h Fn(L)1041 3831 y Fj(k)1086 3817 y Fn(f)50 b Fo(:=)40 b Fm(\000)1403 3744 y Fg(R)1464 3766 y Fj(k)1446 3849 y Fr(0)1524 3817 y Fn(\025)15 b(dE)1706 3831 y Fj(\025)1754 3817 y Fo(\()p Fn(f)10 b Fo(\))39 b(where)g Fm(f)p Fn(E)2302 3831 y Fj(\025)2350 3817 y Fn(;)31 b(\025)41 b Fm(2)f Fo([0)p Fn(;)15 b Fm(1)p Fo(\))p Fm(g)41 b Fo(is)d(the)i(family)e(of)i(sp)s(ectral)0 3944 y(pro)5 b(jectors)32 b(asso)s(ciated)h(to)g Fm(\000)p Fn(L)p Fo(.)45 b(W)-8 b(e)33 b(also)g(let)f Fn(P)1727 3911 y Fj(k)1714 3967 y(t)1799 3944 y Fo(:=)c(exp\()p Fn(L)2159 3958 y Fj(k)2204 3944 y Fn(t)p Fo(\).)46 b(Since)31 b Fn(L)2644 3958 y Fj(k)2721 3944 y Fo(is)g(b)s(ounded,)g(a)h(straigh)m (tforw)m(ard)0 4072 y(computation)e(sho)m(ws,)h(see)g([Li,)f(I)g(Prop.) 40 b(4.4],)32 b(sho)m(ws)440 4388 y Fn(P)511 4350 y Fj(k)498 4410 y(t)556 4388 y Fo(\()p Fn(f)10 b(g)s Fo(\))20 b Fm(\000)g Fn(P)909 4350 y Fj(k)896 4410 y(t)954 4388 y Fn(f)k(P)1094 4350 y Fj(k)1081 4410 y(t)1139 4388 y Fn(g)29 b Fo(=)1307 4264 y Fg(Z)1398 4287 y Fj(t)1358 4471 y Fr(0)1430 4388 y Fn(ds)h(P)1621 4350 y Fj(k)1608 4410 y(t)p Fi(\000)p Fj(s)1747 4314 y Fg(\002)1785 4388 y Fn(L)1847 4402 y Fj(k)1906 4314 y Fg(\000)1948 4388 y Fn(P)2019 4350 y Fj(k)2006 4410 y(s)2064 4388 y Fn(f)10 b(P)2190 4350 y Fj(k)2177 4410 y(s)2234 4388 y Fn(g)2280 4314 y Fg(\001)2342 4388 y Fm(\000)20 b Fn(P)2504 4350 y Fj(k)2491 4410 y(s)2549 4388 y Fn(f)10 b(L)2666 4402 y Fj(k)2710 4388 y Fn(P)2781 4350 y Fj(k)2768 4410 y(s)2826 4388 y Fn(g)23 b Fm(\000)d Fn(P)3054 4350 y Fj(k)3041 4410 y(s)3099 4388 y Fn(g)s(L)3207 4402 y Fj(k)3252 4388 y Fn(P)3323 4350 y Fj(k)3310 4410 y(s)3368 4388 y Fn(f)3423 4314 y Fg(\003)0 4692 y Fo(T)-8 b(aking)36 b(exp)s(ectation)h(w.r.t.)58 b Fn(\026)36 b Fo(and)g(using)f(self{adjoin)m(tness)g(of)h Fn(L)2415 4706 y Fj(k)2496 4692 y Fo(w)m(e)h(get)g(\(4.3\))h(for)f(the) f(appro)m(ximating)0 4820 y(semigroup.)48 b(F)-8 b(or)34 b Fn(f)5 b(;)15 b(g)33 b Fm(2)c Fn(L)963 4787 y Fi(1)1040 4820 y Fo(\()p Fn(d\026)p Fo(\))23 b Fm(\\)f(D)s Fo(\()p Fm(E)8 b Fo(\))33 b(w)m(e)h(no)m(w)f(tak)m(e)i(the)e(limit)e Fn(k)j Fm(!)29 b(1)k Fo(whic)m(h)f(giv)m(es)i(\(4.3\);)i(\014nally)31 b(w)m(e)0 4947 y(extend)f(it)g(to)h(an)m(y)g Fn(f)5 b(;)15 b(g)29 b Fm(2)c(D)s Fo(\()p Fm(E)8 b Fo(\))30 b(b)m(y)h(densit)m(y)-8 b(.)227 5078 y(W)g(e)39 b(no)m(w)f(\014x)f Fn(f)5 b(;)15 b(g)40 b Fm(2)d(D)1065 5092 y Fr(0)1106 5078 y Fo(\()p Fn(L)p Fo(\))h(\()p Fm(D)1381 5092 y Fr(0)1422 5078 y Fo(\()p Fn(L)p Fo(\))g(w)m(as)g(de\014ned)e(in)g(\(2.5\)\))k(and)d(pro) m(v)m(e)h(the)g(b)s(ound)e(\(4.2\))j(for)e(suc)m(h)0 5206 y(functions.)47 b(The)33 b(Lemma)g(follo)m(ws)f(then)h(b)m(y)g (densit)m(y)f(\(see)i(the)f(pro)s(of)g(of)g(Prop)s(osition)e(2.1)j(for) f(an)g(analogous)0 5333 y(argumen)m(t\).)49 b(Giv)m(en)32 b Fn(f)42 b Fo(as)33 b(ab)s(o)m(v)m(e)h(w)m(e)g(de\014ne)e Fn(F)1671 5347 y Fj(t)1732 5333 y Fm(2)d Fn(L)1884 5300 y Fi(1)1961 5333 y Fo(\(\003)p Fn(;)15 b(dx)p Fo(\))34 b(as)f Fn(F)2439 5347 y Fj(t)2472 5333 y Fo(\()p Fn(x)p Fo(\))d(:=)f Fm(k)p Fn(D)2872 5300 y Fr(+)2869 5356 y Fj(x)2933 5333 y Fn(P)2991 5347 y Fj(t)3023 5333 y Fn(f)10 b Fm(k)3123 5360 y Fj(L)3173 5342 y Fd(1)3239 5360 y Fr(\()p Fj(d\026)p Fr(\))3382 5333 y Fo(.)48 b(W)-8 b(e)34 b(also)f(let)0 5558 y Fh(30)p Ff(=mag)r(g)r(io=)p Fh(2000)d Fr([14])1281 b Fo(4.3)p eop %%Page: 15 15 15 14 bop 0 91 a Fn(G)71 105 y Fj(t)139 91 y Fo(b)s(e)34 b(de\014ned)g(in)g(the)h(same)h(w)m(a)m(y)g(with)e Fn(f)45 b Fo(replaced)34 b(b)m(y)h Fn(g)s Fo(.)56 b(Then,)36 b(recalling)e(\(2\))i(in)e(Prop)s(osition)f(2.1,)38 b(the)0 218 y(iden)m(tit)m(y)30 b(\(4.3\))i(implies)101 523 y Fm(j)p Fn(\026)15 b Fo(\()q Fn(P)290 537 y Fj(t)322 523 y Fo(\()p Fn(f)10 b(g)s Fo(\))21 b Fm(\000)f Fn(P)663 537 y Fj(t)695 523 y Fn(f)25 b(P)823 537 y Fj(t)855 523 y Fn(g)s Fo(\))q Fm(j)g Fo(=)g(2)15 b Fn(z)1189 418 y Fg(\014)1189 473 y(\014)1189 527 y(\014)1221 399 y(Z)1312 421 y Fj(t)1271 605 y Fr(0)1344 523 y Fn(ds)1449 399 y Fg(Z)1499 605 y Fr(\003)1553 523 y Fn(dx)g(\026)1737 449 y Fg(\000)1779 523 y Fn(D)1857 485 y Fr(+)1854 545 y Fj(x)1918 523 y Fn(P)1976 537 y Fj(s)2015 523 y Fn(f)24 b(D)2162 485 y Fr(+)2159 545 y Fj(x)2222 523 y Fn(P)2280 537 y Fj(s)2320 523 y Fn(g)2366 449 y Fg(\001)2408 418 y(\014)2408 473 y(\014)2408 527 y(\014)2463 523 y Fm(\024)h Fo(2)15 b Fn(z)2681 399 y Fg(Z)2772 421 y Fj(t)2732 605 y Fr(0)2804 523 y Fn(ds)2909 399 y Fg(Z)2960 605 y Fr(\003)3014 523 y Fn(dx)31 b(F)3202 537 y Fj(s)3241 523 y Fo(\()p Fn(x)p Fo(\))15 b Fn(G)3449 537 y Fj(s)3489 523 y Fo(\()p Fn(x)p Fo(\))102 b(\(4)p Fn(:)p Fo(4\))0 815 y(W)-8 b(e)32 b(claim)d(there)i(are)f Fn(\016)f Fo(=)c Fn(\016)s Fo(\()p Fn(z)t(;)15 b(r)s Fo(\))27 b Fn(>)e Fo(0)31 b(and)f Fn(M)35 b Fo(=)25 b Fn(M)10 b Fo(\()p Fn(z)t(;)15 b(r)s Fo(\))26 b Fn(<)f Fm(1)31 b Fo(suc)m(h)f(that)1267 1103 y Fn(F)1325 1117 y Fj(t)1357 1103 y Fo(\()p Fn(x)p Fo(\))c Fm(\024)f Fn(M)10 b(e)1741 1065 y Fj(M)d(t)1865 979 y Fg(Z)1915 1186 y Fr(\003)1970 1103 y Fn(dy)18 b(e)2122 1065 y Fi(\000)p Fj(\016)r(d)p Fr(\()p Fj(x;y)r Fr(\))2416 1103 y Fn(F)2474 1117 y Fr(0)2515 1103 y Fo(\()p Fn(y)s Fo(\))1080 b(\(4)p Fn(:)p Fo(5\))0 1395 y(P)m(ostp)s(oning)32 b(its)h(pro)s(of,)g(let)h (us)e(\014rst)h(conclude)f(the)i(Lemma.)49 b(Since)32 b Fn(F)2522 1409 y Fr(0)2564 1395 y Fo(\()p Fn(y)s Fo(\))e(=)g(0)k(if)e Fn(y)h Fm(62)c Fo(\003)3209 1409 y Fj(f)3257 1395 y Fo(,)34 b(from)e(\(4.4\))k(and)0 1522 y(\(4.5\))c(w)m(e)f(get)194 1810 y Fm(j)p Fn(\026)15 b Fo(\()p Fn(P)382 1824 y Fj(t)415 1810 y Fo(\()p Fn(f)10 b(g)s Fo(\))21 b Fm(\000)f Fn(P)756 1824 y Fj(t)788 1810 y Fn(f)k(P)915 1824 y Fj(t)948 1810 y Fn(g)s Fo(\))q Fm(j)h(\024)g Fo(2)15 b Fn(z)20 b(M)1396 1773 y Fr(2)1452 1810 y Fn(e)1494 1773 y Fr(2)p Fj(M)7 b(t)1654 1687 y Fg(Z)1705 1893 y Fr(\003)1759 1810 y Fn(dy)18 b(F)1927 1824 y Fr(0)1968 1810 y Fo(\()p Fn(y)s Fo(\))2101 1687 y Fg(Z)2153 1893 y Fr(\003)2207 1810 y Fn(dy)2302 1773 y Fi(0)2343 1810 y Fn(G)2414 1824 y Fr(0)2455 1810 y Fo(\()p Fn(y)2538 1773 y Fi(0)2564 1810 y Fo(\))49 b(sup)2636 1874 y Ff(y)r Fd(2)p Fh(\003)2755 1890 y Ff(f)2625 1948 y(y)2660 1935 y Fd(0)2685 1948 y(2)p Fh(\003)2769 1957 y Ff(g)2833 1687 y Fg(Z)2883 1893 y Fr(\003)2938 1810 y Fn(dx)15 b(e)3094 1773 y Fi(\000)p Fj(\016)r Fr([)p Fj(d)p Fr(\()p Fj(x;y)r Fr(\)+)p Fj(d)p Fr(\()p Fj(x;y)3627 1745 y Fd(0)3653 1773 y Fr(\)])1080 2124 y Fm(\024)25 b Fn(z)20 b(C)h(M)1422 2087 y Fr(2)1478 2124 y Fn(e)1520 2087 y Fr(2)p Fj(M)7 b(t)1680 2124 y Fm(j)p Fo(\003)1768 2138 y Fj(f)1816 2124 y Fm(j)20 b(^)g(j)p Fo(\003)2030 2138 y Fj(g)2073 2124 y Fm(jj)-15 b(j)g(j)p Fn(f)10 b Fm(j)-15 b(j)g(j)15 b(j)-15 b(j)g(j)p Fn(g)s Fm(j)g(j)g(j)15 b Fn(e)2451 2087 y Fi(\000)p Fj(\016)i(d)p Fr(\(\003)2674 2097 y Ff(f)2715 2087 y Fj(;)p Fr(\003)2787 2096 y Ff(g)2824 2087 y Fr(\))p Fj(=)p Fr(2)0 2366 y Fo(for)30 b(some)h(constan)m(t)h Fn(C)f Fo(=)25 b Fn(C)7 b Fo(\()p Fn(\016)s Fo(\).)42 b(Rede\014ning)29 b Fn(\016)34 b Fo(and)c Fn(M)10 b Fo(,)30 b(the)h(b)s(ound)d(\(4.2\))k(follo)m(ws.)0 2521 y(It)e(remains)g(to)h(pro)m(v)m(e)g(\(4.5\).)42 b(W)-8 b(e)32 b(ha)m(v)m(e)626 2748 y Fn(d)p 610 2788 81 4 v 610 2872 a(dt)716 2809 y(D)794 2772 y Fr(+)791 2832 y Fj(x)854 2809 y Fn(P)912 2823 y Fj(t)944 2809 y Fn(f)j Fo(=)25 b Fn(LD)1260 2772 y Fr(+)1257 2832 y Fj(x)1320 2809 y Fn(P)1378 2823 y Fj(t)1410 2809 y Fn(f)30 b Fo(+)1576 2736 y Fg(\002)1613 2809 y Fn(D)1691 2772 y Fr(+)1688 2832 y Fj(x)1752 2809 y Fn(;)15 b(L)1854 2736 y Fg(\003)1907 2809 y Fn(P)1965 2823 y Fj(t)1997 2809 y Fn(f)1024 3059 y Fo(=)25 b Fn(LD)1260 3021 y Fr(+)1257 3081 y Fj(x)1320 3059 y Fn(P)1378 3073 y Fj(t)1410 3059 y Fn(f)30 b Fm(\000)20 b Fn(D)1654 3021 y Fr(+)1651 3081 y Fj(x)1714 3059 y Fn(P)1772 3073 y Fj(t)1804 3059 y Fn(f)30 b Fo(+)19 b Fn(z)2031 2935 y Fg(Z)2081 3142 y Fr(\003)2136 3059 y Fn(dy)2261 2958 y Fg(\020)2315 3059 y Fn(D)2393 3021 y Fr(+)2390 3081 y Fj(x)2454 3059 y Fn(e)2496 3021 y Fi(\000)p Fj(\014)s(D)2658 2994 y Fh(+)2656 3040 y Ff(y)2710 3021 y Fj(H)2770 3030 y Fh(\003)2820 2958 y Fg(\021)2905 3059 y Fn(T)2958 3073 y Fj(x)3019 3059 y Fn(D)3097 3021 y Fr(+)3094 3081 y Fj(y)3157 3059 y Fn(P)3215 3073 y Fj(t)3247 3059 y Fn(f)3713 2936 y Fo(\(4)p Fn(:)p Fo(6\))0 3356 y(where)30 b([)q Fn(D)367 3323 y Fr(+)364 3378 y Fj(x)427 3356 y Fn(;)15 b(L)p Fo(])31 b(denotes)g(the)g(comm)m(utator)h(and)e(\()q Fn(T)1850 3370 y Fj(x)1896 3356 y Fn(f)10 b Fo(\))k(\()p Fn(!)s Fo(\))27 b(:=)f Fn(f)e Fo(\()q Fn(!)f Fm([)c(f)p Fn(x)p Fm(g)p Fo(\))r(.)41 b(The)31 b(second)f(iden)m(tit)m(y)h(in)e (\(4.6\))0 3483 y(follo)m(ws)g(b)m(y)i(a)g(direct)e(computation)i(from) f(\(2.4\).)42 b(By)31 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943 y Fr(\()p Fj(!)r Fr(\))1816 980 y Fn(D)1894 943 y Fi(\000)1891 1003 y Fj(x)1955 980 y Fn(f)10 b Fo(\()p Fn(!)s Fo(\))20 b(+)g Fn(z)2312 856 y Fg(Z)2363 1063 y Fr(\003)2432 980 y Fn(D)2510 943 y Fr(+)2507 1003 y Fj(x)2570 980 y Fn(f)10 b Fo(\()p Fn(!)s Fo(\))15 b Fn(dx)182 b(!)29 b Fm(2)24 b Fo(\012)15 b Fn(:)385 b Fo(\(5)p Fn(:)p Fo(4\))0 1375 y(whic)m(h)23 b(corresp)s(onds)g(to)i(a)g(pro)s(cess)f (where)g(particle)g(app)s(ear)g(with)f(rate)i Fn(z)t Fo(,)h(and)e(disapp)s(ear)f(with)g(rate)i Fn(e)3589 1342 y Fi(\000)p Fj(\014)s(D)3751 1315 y Fd(\000)3749 1360 y Ff(x)3806 1342 y Fj(H)3875 1375 y Fo(.)0 1504 y(The)30 b(asso)s(ciated)h(Dirc)m(hlet)f(form)g(is)1401 1754 y(^)1386 1777 y Fm(E)1442 1734 y Fj(\021)1434 1804 y Fr(\003)1488 1777 y Fo(\()p Fn(f)10 b Fo(\))26 b(=)e Fn(z)1796 1653 y Fg(Z)1846 1860 y Fr(\003)1916 1777 y Fn(dx)15 b(\026)2085 1734 y Fj(\021)2085 1804 y Fr(\003)2139 1777 y Fo(\(\()p Fn(D)2287 1740 y Fr(+)2284 1800 y Fj(x)2348 1777 y Fn(f)10 b Fo(\))2438 1740 y Fr(2)2479 1777 y Fo(\))0 2128 y(and,)31 b(since)442 2105 y(^)426 2128 y Fm(E)482 2084 y Fj(\021)474 2155 y Fr(\003)555 2128 y Fm(\025)26 b(E)708 2084 y Fj(\021)700 2155 y Fr(\003)754 2128 y Fo(,)31 b(the)g(generator)1383 2105 y(^)1372 2128 y Fn(L)1434 2084 y Fj(\021)1434 2155 y Fr(\003)1519 2128 y Fo(has)g(a)h(sp)s(ectral)e(gap)h(greater)h(than)f (or)g(equal)f(to)i(the)f(sp)s(ectral)g(gap)0 2256 y(of)g Fn(L)166 2213 y Fj(\021)166 2283 y Fr(\003)220 2256 y Fo(.)40 b(Inequalit)m(y)30 b(\(5.2\))i(b)s(ecomes)1147 2530 y Fn(\032)p Fo([)p Fn(n;)15 b Fm(1)p Fo(\))g(log)i Fn(\032)p Fo([)p Fn(n;)e Fm(1)p Fo(\))1881 2492 y Fi(\000)p Fr(1)2004 2530 y Fm(\024)25 b Fn(c)2139 2544 y Fj(s)2194 2530 y Fn(z)19 b Fm(j)p Fo(\003)p Fm(j)c Fn(\032)p Fo(\()p Fn(n)21 b Fm(\000)f Fo(1\))15 b Fn(:)961 b Fo(\(5)p Fn(:)p Fo(5\))0 2804 y(While)31 b(\(5.5\))j(is)d(equiv)-5 b(alen)m(t)32 b(to)h(\(5.2\))h(when)d Fn(')e Fo(=)f(0,)33 b(if)e(for)h Fn(')h Fo(in)e(some)i(appropriate)e(class)h(of)g(p)s(oten)m(tials)g (one)0 2932 y(knew)i(that)h(the)g(particle)f(distribution)d Fn(\032)j Fo(b)s(eha)m(v)m(es)h(lik)m(e)f Fn(e)2071 2899 y Fi(\000)p Fj(cn)2205 2872 y Fh(2)2280 2932 y Fo(for)g(large)h Fn(n)p Fo(,)g(then)g(\(5.5\))h(w)m(ould)d(hold)h(and)g(a)0 3061 y(LSI)29 b(cannot)i(b)s(e)e(ruled)f(out.)41 b(F)-8 b(or)31 b(sup)s(erstable)d(p)s(oten)m(tials)h(\(see)i([Ru]\))f(it)f(is) g(kno)m(wn)h(that)g Fn(\032)p Fo(\()p Fn(n)p Fo(\))c Fm(\024)f Fn(e)3496 3028 y Fi(\000)p Fj(c)3585 3037 y Fh(1)3621 3028 y Fj(n)3666 3000 y Fh(2)3701 3028 y Fr(+)p Fj(c)3789 3037 y Fh(2)3825 3028 y Fj(n)3875 3061 y Fo(,)0 3189 y(so)31 b(it)f(could)f(b)s(e)h(in)m(teresting,)g(in)f(this)g (case,)j(to)f(in)m(v)m(estigate)g(the)g(p)s(ossibilit)m(y)c(of)j(ha)m (ving)g(a)h(LSI)e(for)3488 3166 y(^)3478 3189 y Fn(L)3540 3145 y Fj(\021)3540 3216 y Fr(\003)3594 3189 y Fo(.)0 5558 y Fh(30)p Ff(=mag)r(g)r(io=)p Fh(2000)h Fr([17])1281 b Fo(5.2)p eop %%Page: 18 18 18 17 bop 0 91 a Fp(References)-101 247 y Fo([CMM])46 b(F.)29 b(Cesi,)e(C.)g(Maes,)i(and)e(F.)i(Martinelli,)c Fe(R)-5 b(elaxation)33 b(of)d(disor)-5 b(der)g(e)g(d)34 b(magnets)d(in)f(the)g(Gri\016ths)i(r)-5 b(e)g(gime)p Fo(,)227 375 y(Comm)m(un.)30 b(Math.)h(Ph)m(ys.)f Fp(188)p Fo(,)i(135{173)h(\(1997\))16 503 y([Da])47 b(E.)31 b(B.)g(Da)m(vies,)g Fe(He)-5 b(at)33 b(kernels)g(and)g(sp)-5 b(e)g(ctr)g(al)35 b(the)-5 b(ory)p Fo(,)33 b(Cam)m(bridge)c(Univ.)h(Press,)g(1989)-56 631 y([FF)m(G])47 b(R.)23 b(F)-8 b(ern\023)-45 b(andez,)24 b(P)-8 b(.A.)24 b(F)-8 b(errari)22 b(and)g(N.L.)i(Garcia,)f Fe(Perfe)-5 b(ct)26 b(simulation)h(for)f(inter)-5 b(acting)27 b(p)-5 b(oint)27 b(pr)-5 b(o)g(c)g(esses,)227 758 y(loss)34 b(networks)g(and)f(Ising)g(mo)-5 b(dels)p Fo(,)32 b(Preprin)m(t)d (\(1999\))-11 886 y([Le1])46 b(M.)22 b(Ledoux,)f Fe(Conc)-5 b(entr)g(ation)27 b(of)e(me)-5 b(asur)g(e)26 b(and)f(lo)-5 b(garithmic)26 b(Sob)-5 b(olev)26 b(ine)-5 b(qualities)p Fo(,)22 b(S)m(\023)-43 b(eminaire)20 b(de)h(Prob-)227 1014 y(abilit)m(\023)-43 b(es.)30 b(XXXIII.)h(Lecture)g(Notes)g(in)e (Mathematics)j(1709,)g(Springer,)c(Berlin,)h(1999,)k(pp.)c(120{216)-11 1142 y([Le2])46 b(M.)37 b(Ledoux,)f Fe(L)-5 b(o)g(garithmic)41 b(Sob)-5 b(olev)39 b(ine)-5 b(qualities)39 b(for)f(unb)-5 b(ounde)g(d)40 b(spin)f(systems)g(r)-5 b(evisite)g(d)p Fo(,)37 b(Preprin)m(t)227 1269 y(\(1999\))49 1397 y([Li])45 b(T.)30 b(M.)h(Liggett,)h Fe(Inter)-5 b(acting)34 b(p)-5 b(article)34 b(systems.)d Fo(Berlin:)e(Springer)f(1985)14 1525 y([L)-8 b(Y])46 b(S.)33 b(L.)f(Lu)g(and)g(H.)h(T.)g(Y)-8 b(au,)33 b Fe(Sp)-5 b(e)g(ctr)g(al)38 b(Gap)d(and)h(L)-5 b(o)g(garithmic)37 b(Sob)-5 b(olev)36 b(Ine)-5 b(quality)35 b(for)h(Kawasaki)g(and)227 1653 y(Glaub)-5 b(er)34 b(Dynamics)p Fo(,)d(Comm)m(un.)f(Math.)h(Ph)m(ys.)f Fp(156)p Fo(,)i(399{433)h (\(1993\))3 1781 y([Ma])46 b(F.)35 b(Martinelli,)c Fe(L)-5 b(e)g(ctur)g(es)37 b(on)f(Glaub)-5 b(er)37 b(dynamics)g(for)g(discr)-5 b(ete)36 b(spin)g(mo)-5 b(dels)p Fo(,)36 b(Lectures)e(on)g(probabil-) 227 1908 y(it)m(y)h(theory)g(and)f(statistics)h(\(Sain)m(t-Flour,)f (1997\).)j(Lecture)e(Notes)h(in)d(Mathematics)j(1717,)h(Springer,)227 2035 y(Berlin,)29 b(1999,)k(pp.)c(93{191)-93 2164 y([MKK])45 b(K.)26 b(Matthes,)i(J.)d(Kerstan,)i(and)e(J.)h(Mec)m(k)m(e,)i Fe(In\014nitely)i(divisible)e(p)-5 b(oint)30 b(pr)-5 b(o)g(c)g(esses)p Fo(,)28 b(John)e(Wiley)f(&)g(Sons,)227 2291 y(Chic)m(hester-New)31 b(Y)-8 b(ork-Brisbane,)30 b(1978)-68 2419 y([MO1])46 b(F.)32 b(Martinelli)c(and)j(E.)g(Olivieri,) c Fe(Appr)-5 b(o)g(ach)36 b(to)e(e)-5 b(quilibrium)33 b(of)g(Glaub)-5 b(er)34 b(dynamics)h(in)e(the)g(one)h(phase)227 2546 y(r)-5 b(e)g(gion)34 b(I:)e(The)h(attr)-5 b(active)34 b(c)-5 b(ase)p Fo(,)31 b(Comm)m(un.)f(Math.)h(Ph)m(ys.)g Fp(161)p Fo(,)g(447)h(\(1994\))-68 2675 y([MO2])46 b(F.)32 b(Martinelli)c(and)j(E.)g(Olivieri,)c Fe(Appr)-5 b(o)g(ach)36 b(to)e(e)-5 b(quilibrium)33 b(of)g(Glaub)-5 b(er)34 b(dynamics)h(in)e (the)g(one)h(phase)227 2802 y(r)-5 b(e)g(gion)34 b(II:)e(The)h(gener)-5 b(al)34 b(c)-5 b(ase)p Fo(,)31 b(Comm)m(un.)f(Math.)h(Ph)m(ys.)f Fp(161)p Fo(,)h(487)h(\(1994\))-7 2930 y([Pre])46 b(C.)29 b(Preston,)g Fe(Sp)-5 b(atial)34 b(birth-and-de)-5 b(ath)34 b(pr)-5 b(o)g(c)g(esses)p Fo(,)31 b(Pro)s(ceedings)d(of)h(the)h(40th)f (Session)f(of)h(the)h(In)m(terna-)227 3057 y(tional)g(Statistical)g (Institute)f(\(W)-8 b(arsa)m(w,)33 b(1975\).)f(Bull.)d(Inst.)h(In)m (ternat.)h(Statist.)g Fp(46)p Fo(,)g(371{391,)j(\(1975\))14 3185 y([Ru])45 b(D.)37 b(Ruelle,)f Fe(Sup)-5 b(erstable)39 b(inter)-5 b(actions)40 b(in)e(classic)-5 b(al)40 b(statistic)-5 b(al)40 b(me)-5 b(chanics)p Fo(,)38 b(Comm)m(un.)e(Math.)h(Ph)m(ys)227 3313 y Fp(18)p Fo(,)31 b(127{159)j(\(1970\))30 3441 y([Sp])45 b(H.)26 b(Sp)s(ohn,)d Fe(Equilibrium)k(\015uctuations)i(for)f(inter)-5 b(acting)28 b(Br)-5 b(ownian)30 b(p)-5 b(articles)p Fo(,)27 b(Comm)m(un.)d(Math.)i(Ph)m(ys.)227 3568 y Fp(103)p Fo(,)32 b(1{33)g(\(1986\))-20 3696 y([SZ1])45 b(D.)33 b(W.)f(Stro)s(o)s(c)m(k)g (and)f(B.)h(Zegarlinski,)e Fe(The)k(e)-5 b(quivalenc)g(e)33 b(of)i(the)f(lo)-5 b(garithmic)36 b(Sob)-5 b(olev)35 b(ine)-5 b(quality)34 b(and)227 3824 y(the)f(Dobrushin{Shlosman)j (mixing)d(c)-5 b(ondition)p Fo(,)32 b(Comm)m(un.)e(Math.)h(Ph)m(ys.)f Fp(144)p Fo(,)i(303{323)h(\(1992\))-20 3952 y([SZ2])45 b(D.)34 b(W.)g(Stro)s(o)s(c)m(k)g(and)e(B.)i(Zegarlinski,)d Fe(The)36 b(lo)-5 b(garithmic)38 b(Sob)-5 b(olev)36 b(ine)-5 b(quality)36 b(for)g(discr)-5 b(ete)36 b(spin)g(on)g(a)227 4079 y(lattic)-5 b(e)p Fo(,)32 b(Comm)m(un.)d(Math.)j(Ph)m(ys.)e Fp(149)p Fo(,)h(175)h(\(1992\))25 4207 y([Y)-8 b(o])47 b(N.)21 b(Y)-8 b(oshida,)21 b Fe(The)j(e)-5 b(quivalenc)g(e)24 b(of)g(the)g(lo)-5 b(g-Sob)g(olev)25 b(ine)-5 b(quality)25 b(and)g(a)f(mixing)g(c)-5 b(ondition)25 b(for)f(unb)-5 b(ounde)g(d)227 4335 y(spin)33 b(systems)h(on)f(the)g(lattic)-5 b(e)p Fo(,)32 b(Preprin)m(t)c(\(1998\))35 4463 y([Ze])46 b(B.)28 b(Zegarlinski,)c Fe(The)29 b(str)-5 b(ong)31 b(de)-5 b(c)g(ay)30 b(to)g(e)-5 b(quilibrium)29 b(for)h(the)g(sto)-5 b(chastic)31 b(dynamics)f(of)g(unb)-5 b(ounde)g(d)30 b(spin)227 4590 y(systems)k(on)f(a)g(lattic)-5 b(e)p Fo(,)31 b(Comm)m(un.)f(Math.)h(Ph)m(ys.)g Fp(175)p Fo(,)g(401{432)i (\(1996\))0 5558 y Fh(30)p Ff(=mag)r(g)r(io=)p Fh(2000)d Fr([18])1281 b Fo(5.3)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF