Content-Type: multipart/mixed; boundary="-------------0210290418193" This is a multi-part message in MIME format. ---------------0210290418193 Content-Type: text/plain; name="02-438.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-438.comments" AMS Subject Classification: 60J10, 60K35 ---------------0210290418193 Content-Type: text/plain; name="02-438.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-438.keywords" eigenvalue problem, diffusion process, metastability, relaxation time, potential theory ---------------0210290418193 Content-Type: application/postscript; name="eckhoff.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="eckhoff.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: myspectotal.dvi %%Pages: 43 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips myspectotal 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y(the)e(latter)i(prop)s(ert)m(y) d(henceforth)h(will)h(b)s(e)e(referred)h(to)h(as)f(a)g Fo(r)-5 b(e)g(gular)38 b Ft(set.)59 b(The)35 b Fs(\025)h Ft(-)h(capacit)m(y)h(of)f(the)50 949 y(capacitor)32 b(\()p Fs(A;)15 b(B)5 b Ft(\))31 b(is)g(giv)m(en)g(b)m(y)906 1209 y(cap)1042 1171 y Fr(\025)1042 1231 y(A)1101 1209 y Ft(\()p Fs(B)5 b Ft(\))26 b Fq(\021)f Fs(\017)1419 1085 y Fh(Z)1469 1291 y Fr(@)t(A)1586 1209 y Fs(e)1628 1171 y Fi(\000)p Fr(F)6 b(=\017)1810 1209 y Fs(@)1858 1223 y Fr(n)1907 1209 y Fs(h)1959 1171 y Fr(\025)1959 1231 y(A;B)2114 1209 y Fs(d\033)23 b Fq(\000)d Fs(\025)2395 1085 y Fh(Z)2446 1291 y Fr(A)2520 1209 y Fs(e)2562 1171 y Fi(\000)p Fr(F)6 b(=\017)2759 1209 y Fs(dy)672 b Ft(\(1)p Fs(:)p Ft(2\))50 1482 y(where)39 b Fs(n)g Ft(is)h(the)g(unit)g(normal)e (at)j Fs(@)5 b(A)27 b Fq([)f Fs(@)5 b(B)44 b Ft(p)s(oin)m(ting)c(to)m (w)m(ards)h Fs(A)27 b Fq([)f Fs(B)5 b Ft(.)68 b(Here)40 b Fs(h)3066 1449 y Fr(\025)3066 1507 y(A;B)3245 1482 y Ft(denotes)h(the)50 1627 y(electrostatic)33 b(equilibrium)27 b(p)s(oten)m(tial)k(of)e(the)g(capacitor,)j(i.e.)41 b(the)30 b(solution)f(of)h(the)f(Diric)m(hlet)i(problem)796 1877 y(\()p Fs(L)893 1891 y Fr(\017)948 1877 y Fq(\000)20 b Fs(\025)p Ft(\))p Fs(h)p Ft(\()p Fs(x)p Ft(\))27 b(=)e Fs(g)s Ft(\()p Fs(x)p Ft(\))92 b Fs(x)25 b Fq(2)g Ft(\000)p Fs(;)197 b(h)p Ft(\()p Fs(x)p Ft(\))27 b(=)d Fs(f)10 b Ft(\()p Fs(x)p Ft(\))91 b Fs(x)26 b Fq(2)e Fs(@)5 b Ft(\000)560 b(\(1)p Fs(:)p Ft(3\))50 2126 y(where)35 b(\000)f Fq(\021)f Fp(R)573 2093 y Fr(d)621 2126 y Fq(n)p Fs(A)p Fq(n)p Fs(B)5 b Ft(,)38 b Fs(g)g Fq(\021)33 b Ft(0)j(and)f Fs(f)43 b Fq(\021)34 b Ft(1)-23 b(I)1613 2140 y Fr(A)1672 2126 y Ft(.)56 b(W)-8 b(e)37 b(shall)f(use)f(the)h (con)m(v)m(en)m(tion)i(that)e Fs(h)3171 2093 y Fr(\025)3171 2151 y(A;B)3346 2126 y Ft(tak)m(es)h(the)50 2271 y(v)-5 b(alue)39 b(one)g(on)f Fs(A)h Ft(and)f(zero)i(on)e Fs(B)5 b Ft(.)65 b(T)-8 b(o)39 b(functions)g Fs(h)g Ft(satisfying)g(\(1.3\))h (for)f(some)f Fs(f)48 b Ft(and)38 b Fs(g)k Fq(\021)d Ft(0)g(w)m(e)50 2417 y(sometimes)29 b(refer)h(to)i(as)e Fs(\025)p Fo(-harmonic)i Ft(functions.)41 b(The)30 b(comm)m(unication)g (heigh)m(t)h(b)s(et)m(w)m(een)g(sets)g Fs(A)f Ft(and)50 2562 y Fs(B)35 b Ft(is)30 b(de\014ned)f(b)m(y)1174 2685 y(^)1154 2708 y Fs(F)13 b Ft(\()p Fs(A;)i(B)5 b Ft(\))26 b Fq(\021)192 b Ft(inf)1671 2768 y Fk(c)p Fl(:[0)p Fk(;)p Fl(1])p Fg(!)p Ff(R)1929 2755 y Fk(d)1610 2824 y(c)p Fl(\(0\))p Fg(2)p Fk(A;c)p Fl(\(1\))p Fg(2)p Fk(B)2058 2708 y Ft(max)15 b Fs(F)e Ft(\()p Fs(c)p Ft([0)p Fs(;)i Ft(1]\))920 b(\(1)p Fs(:)p Ft(4\))50 2978 y(where)27 b(the)i(in\014m)m(um)24 b(is)29 b(tak)m(en)g(o)m(v)m(er)g(all)g(con)m (tin)m(uous)g(curv)m(es.)40 b(F)-8 b(urthermore,)28 b(for)g(a)g (\014nite)g(set)h(of)f(p)s(oin)m(ts)50 3124 y Fs(I)f Fq([)20 b Fs(x)30 b Ft(suc)m(h)g(that)h Fs(B)751 3138 y Fr(I)5 b Fi([)p Fr(x)913 3124 y Ft(is)31 b(a)g(disjoin)m(t)f(union)g (of)g(op)s(en)g(balls,)h(where)1274 3373 y Fs(B)1343 3387 y Fr(J)1419 3373 y Fq(\021)1532 3287 y Fh([)1515 3482 y Fr(y)r Fi(2)p Fr(J)1665 3373 y Fs(B)5 b Ft(\()p Fs(y)s(;)15 b(\017=)p Ft(4\))p Fs(;)198 b(J)9 b Fq(\032)p Fp(R)2437 3336 y Fr(d)3523 3373 y Ft(\(1)p Fs(:)p Ft(5\))50 3679 y(w)m(e)31 b(in)m(tro)s(duce)1137 3825 y Fs(A)1205 3839 y Fr(x;I)1335 3825 y Fq(\021)25 b(f)p Fs(y)k Fq(2)c Fp(R)1695 3787 y Fr(d)1759 3825 y Fq(j)1820 3802 y Ft(^)1799 3825 y Fs(F)13 b Ft(\()p Fs(y)s(;)i(x)p Ft(\))26 b Fs(<)2223 3802 y Ft(^)2202 3825 y Fs(F)13 b Ft(\()p Fs(y)s(;)i(I)7 b Fq(n)p Fs(x)p Ft(\))p Fq(g)903 b Ft(\(1)p Fs(:)p Ft(6\))50 4032 y(In)29 b(analogy)j(to)f([E2])h(w)m(e)e(de\014ne)g(the)h(time)f (scales)1457 4334 y Fs(T)1510 4348 y Fr(x;I)1640 4334 y Fq(\021)1747 4175 y Fh(R)1790 4281 y Fr(A)1845 4290 y Fk(x;I)1958 4248 y Fs(e)2000 4215 y Fi(\000)p Fr(F)6 b(=\017)2196 4248 y Fs(dy)p 1747 4313 545 4 v 1771 4396 a Ft(cap)1907 4365 y Fv(0)1907 4423 y Fr(B)1962 4432 y Fk(x)2008 4396 y Ft(\()p Fs(B)2112 4413 y Fr(I)f Fi(n)p Fr(x)2232 4396 y Ft(\))3523 4334 y(\(1)p Fs(:)p Ft(7\))50 4613 y(W)-8 b(e)31 b(recall)h(from)d(Theorem)g([3.1)j(in)e(BEGK3])h (the)g(classical)h(Eyring)e(form)m(ula)g(for)g(the)g(capacit)m(y)-8 b(.)50 4832 y Fe(Theorem)41 b(1.1:)60 b Fo(Fix)37 b(r)-5 b(e)g(gular,)38 b(disjoint,)g(nonempty)g(sets)e Fs(A)g Fo(and)i Fs(B)5 b Fo(.)51 b(Assume)36 b(that)i(ther)-5 b(e)37 b(is)f(only)50 4977 y(one)j(solution)i(of)f Fs(F)13 b Ft(\()p Fs(z)845 4944 y Fi(\003)887 4977 y Ft(\))38 b(=)1088 4954 y(^)1068 4977 y Fs(F)13 b Ft(\()p Fs(A;)i(B)5 b Ft(\))38 b Fs(>)f Ft(max)15 b Fs(F)e Ft(\()p Fs(A)25 b Fq([)g Fs(B)5 b Ft(\))25 b(+)g Fs(R)q(\017)15 b Ft(log)r(\(1)p Fs(=\017)p Ft(\))41 b Fo(such)e(that)i Fs(z)3166 4944 y Fi(\003)3247 4977 y Fo(is)e(a)h(critic)-5 b(al)50 5122 y(p)g(oint)34 b(of)f Fs(F)13 b Fo(.)41 b(If)33 b(the)g(Hessian)g(at)g Fs(z)1271 5089 y Fi(\003)1345 5122 y Fo(of)g Fs(F)46 b Fo(is)32 b(nonde)-5 b(gener)g(ate,)35 b(then)e(for)g(some)h Fs(R)598 5395 y Ft(cap)734 5358 y Fv(0)734 5418 y Fr(A)794 5395 y Ft(\()p Fs(B)5 b Ft(\))25 b(=)g(\(1)c(+)f Fq(O)s Ft(\(1\))p Fs(\017)15 b Ft(log)t(\(1)p Fs(=\017)p Ft(\)\))1955 5334 y(\(2)p Fs(\031)s Ft(\))2125 5301 y Fr(d=)p Fv(2)p Fi(\000)p Fv(1)2335 5334 y Fq(j)p Fs(\025)2413 5301 y Fi(\003)2455 5334 y Fq(j)p 1855 5375 724 4 v 1855 5393 a Fh(p)p 1947 5393 633 4 v 1947 5470 a Fq(j)g Ft(det)h(Hess)g Fs(F)d Ft(\()p Fs(z)2477 5444 y Fi(\003)2519 5470 y Ft(\))p Fq(j)2591 5395 y Fs(\017)2628 5358 y Fr(d=)p Fv(2)2743 5395 y Fs(e)2785 5358 y Fi(\000)2858 5342 y Fv(^)2842 5358 y Fr(F)d Fv(\()p Fr(A;B)s Fv(\))p Fr(=\017)3523 5395 y Ft(\(1)p Fs(:)p Ft(8\))p eop %%Page: 6 6 6 5 bop 50 295 a Fm(6)3309 b Fj(Se)l(ction)27 b Fm(1)50 513 y Fo(wher)-5 b(e)31 b Fs(\025)357 480 y Fi(\003)429 513 y Fo(is)f(the)g(unique,)g(ne)-5 b(gative)30 b(eigenvalue)g(of)g (the)h(Hessian)f(at)g Fs(z)2520 480 y Fi(\003)2563 513 y Fo(.)40 b(The)31 b(mo)-5 b(dulus)31 b(of)g(the)f(L)-5 b(andau)50 658 y(symb)g(ol)34 b(is)e(dominate)-5 b(d)36 b(by)c(a)h(c)-5 b(onstant)34 b Fs(C)e Fq(\021)25 b Fs(C)7 b Ft(\()p Fs(d;)15 b(F)e Ft(\))p Fo(.)159 876 y Ft(Let)40 b Fq(M)f Ft(denote)h(the)f(set)h(of)f Fo(lo)-5 b(c)g(al)43 b(minima)d Ft(of)f Fs(F)13 b Ft(.)67 b(F)-8 b(or)40 b Fs(x)g Fq(2)g(M)f Ft(and)f Fs(I)7 b Fq(\032Mn)p Fs(x)40 b Ft(nonempt)m(y)e(with)50 1022 y(nondegenerate)23 b(Hessian)g(at)g Fs(x)g Ft(w)m(e)g(obtain)g(from)e(\(1.8\))j(that)f(the)g(time)f(scale)i (in)m(tro)s(duced)e(ab)s(o)m(v)m(e)h(satis\014es)648 1308 y Fs(T)701 1322 y Fr(x;I)831 1308 y Ft(=)i(\(1)c(+)f Fq(O)s Ft(\(1\))p Fs(\017)15 b Ft(log)s(\(1)p Fs(=\017)p Ft(\)\))1711 1144 y Fh(s)p 1804 1144 713 4 v 1844 1246 a Fq(j)g Ft(det)h(Hess)g Fs(F)d Ft(\()p Fs(z)2374 1220 y Fi(\003)2416 1246 y Ft(\))p Fq(j)p 1815 1287 692 4 v 1815 1370 a(j)p Fs(\025)1893 1344 y Fi(\003)1935 1370 y Fq(j)i Ft(det)h(Hess)f Fs(F)e Ft(\()p Fs(x)p Ft(\))2517 1308 y Fs(e)2559 1270 y Fv(\()2603 1254 y(^)2587 1270 y Fr(F)e Fv(\()p Fr(x;I)5 b Fv(\))p Fi(\000)p Fr(F)10 b Fv(\()p Fr(x)p Fv(\)\))p Fr(=\017)3523 1308 y Ft(\(1)p Fs(:)p Ft(9\))50 1574 y(Let)36 b(us)g(remark)e(that)j(the)f(conditions) h(under)d(whic)m(h)i(\(1.9\))h(holds)f(are)g(not)h(at)f(all)h(b)s (orderline)e(of)h(our)50 1719 y(approac)m(h.)50 b(In)33 b(fact,)i(the)f(only)g(condition)g(w)m(e)g(need)f(is)g(that)i(either)f Fs(T)2537 1736 y Fr(x;I)5 b Fi(n)p Fr(x)2750 1719 y Fs(>>)30 b(T)2975 1736 y Fr(y)r(;I)5 b Fi(n)p Fr(y)3187 1719 y Ft(or)34 b Fs(T)3355 1736 y Fr(x;I)5 b Fi(n)p Fr(x)3568 1719 y Fs(<<)50 1865 y(T)103 1881 y Fr(y)r(;I)g Fi(n)p Fr(y)282 1865 y Ft(,)30 b(where)f Fs(x;)15 b(y)28 b Fq(2)d(M)p Ft(,)30 b Fs(x)25 b Fq(6)p Ft(=)g Fs(y)s Ft(,)30 b Fs(I)7 b Fq(\032M)p Ft(.)40 b(In)29 b(particular,)h(as)g(long)g(as)g(this)f (condition)h(is)g(v)-5 b(alid)29 b(w)m(e)h(can)50 2010 y(deal)k(with)f(all)i(kinds)e(of)h(degeneracies)h(of)f Fs(F)46 b Ft(in)34 b(the)g(relev)-5 b(an)m(t)35 b(regions.)51 b(This)33 b(just)g(leads)h(to)g(di\013eren)m(t)50 2156 y(asymptotic)e(b)s(eha)m(viors)h(of)f Fs(T)1087 2170 y Fr(x;I)1192 2156 y Ft(.)47 b(W)-8 b(e)33 b(refer)f(the)h(reader)f(to) h([E1,2])h(where)e(in)g(the)g(con)m(text)j(of)d(Mark)m(o)m(v)50 2301 y(c)m(hains)39 b(precise,)j(minimal)c(conditions)h(on)g(the)g (time)g(scales)i(are)e(giv)m(en.)68 b(As)39 b(w)m(e)h(only)f(fo)s(cus)g (on)g(the)50 2447 y(new)28 b(tec)m(hnical)j(complications)f(in)f(the)g (con)m(tin)m(uous)h(state)g(space)g(setting)g(w)m(e)g(do)f(not)g(aim)f (at)i(the)f(most)50 2592 y(general)i(conditions)g(under)e(whic)m(h)h (the)h(analysis)f(w)m(orks.)159 2810 y(Let)41 b(us)g(no)m(w)f(describ)s (e)h(the)g(main)f(results)g(of)h(this)g(pap)s(er.)72 b(W)-8 b(e)42 b(ha)m(v)m(e)g(to)f(in)m(tro)s(duce)g(some)g(more)50 2956 y(notation.)h(Let)31 b(us)e(de\014ne)h(for)g Fs(y)e Fq(2)d Fp(R)1349 2923 y Fr(d)911 3185 y Fs(\016)s Ft(\()p Fs(y)s Ft(\))h Fq(\021)f Fs(\016)1234 3199 y Fr(F)r(;\017)1339 3185 y Ft(\()p Fs(y)s Ft(\))h Fq(\021)f Ft(sup)1716 3111 y Fh(\010)1769 3185 y Fs(\016)k(>)c Ft(0)15 b Fq(j)g Ft(8)p Fs(\016)91 b Ft(sup)2139 3265 y Fr(B)s Fv(\()p Fr(y)r(;)p Fv(8)p Fr(\017\016)r Fv(\))2432 3185 y Fq(jr)p Fs(F)13 b Fq(j)26 b Fs(<)e Ft(1)2795 3111 y Fh(\011)3477 3185 y Ft(\(1)p Fs(:)p Ft(10\))50 3460 y(F)-8 b(or)31 b(a)g(subset)e(\006)p Fq(\032)p Fp(R)764 3427 y Fr(d)842 3460 y Ft(and)h(a)h(function)f Fs(r)e Ft(:)d Fp(R)1630 3427 y Fr(d)1703 3460 y Fq(!)h Ft(\(0)p Fs(;)15 b Fq(1)p Ft(\))31 b(w)m(e)g(in)m(tro)s(duce)f(its)h Fs(r)i Ft(-)e(neigh)m(b)s (orho)s(o)s(d)e(b)m(y)1218 3689 y(\006)1284 3651 y Fr(r)1349 3689 y Fq(\021)c(f)p Fs(x)h Fq(2)f Fp(R)1714 3651 y Fr(d)1777 3689 y Fq(j)31 b Ft(dist\()p Fs(x;)15 b Ft(\006\))26 b Fs(<)f(r)s Ft(\()p Fs(x)p Ft(\))p Fq(g)936 b Ft(\(1)p Fs(:)p Ft(11\))50 3918 y(In)29 b(the)i(sequel)g(w)m(e)g(imp)s(ose)e (the)h(follo)m(wing)i(conditions)f(on)f Fs(F)13 b Ft(.)50 4136 y Fe(Assumption)41 b(1.2:)60 b Fs(F)44 b Ft(:)32 b Fp(R)1145 4103 y Fr(d)1225 4136 y Fq(!)f Fp(R)44 b Fo(is)36 b(c)-5 b(ontinuously)37 b(di\013er)-5 b(entiable.)53 b(Ther)-5 b(e)36 b(ar)-5 b(e)37 b(c)-5 b(onstants)38 b Fs(c)31 b(>)g Ft(0)50 4281 y Fo(and)i Fs(C)291 4295 y Fv(0)365 4281 y Fo(satisfying)g Ft(inf)880 4298 y Fi(f)p Fr(F)10 b(>C)1082 4307 y Fl(0)1117 4298 y Fi(g)1174 4281 y Fq(jr)p Fs(F)j Fq(j)26 b(\025)f Fs(c)p Fo(.)42 b(Ther)-5 b(e)33 b(ar)-5 b(e)33 b Fs(\017)2054 4295 y Fv(0)2120 4281 y Fs(>)25 b Ft(0)33 b Fo(and)h Fs(C)2536 4295 y Fv(1)2609 4281 y Fo(such)f(that)1340 4474 y Ft(sup)1477 4496 y Fr(c)1509 4478 y Fl(4)p Fk(\017\016)1623 4474 y Fq(jr)p Fs(F)13 b Fq(j)p 1331 4515 499 4 v 1331 4598 a Ft(inf)1442 4615 y Fr(c)p Fv(\(0)p Fr(;l)p Fv(\))1632 4598 y Fq(jr)p Fs(F)g Fq(j)1865 4536 y Fs(<)2133 4474 y(C)2198 4488 y Fv(1)2239 4474 y Fs(l)p 1972 4515 457 4 v 1972 4598 a Fq(j)p Fs(c)p Ft(\()p Fs(l)r Ft(\))22 b Fq(\000)e Fs(c)p Ft(\(0\))p Fq(j)3477 4536 y Ft(\(1)p Fs(:)p Ft(12\))50 4800 y Fo(for)40 b(al)5 b(l)39 b Ft(0)f Fs(<)e(\017)h(<)g(\017)747 4814 y Fv(0)827 4800 y Fo(and)j(every)f (\015ow)h(line)g Fs(c)d Ft(:)g([0)p Fs(;)15 b(l)r Ft(])39 b Fq(!)e Fp(R)2168 4767 y Fr(d)2256 4800 y Fo(of)i Fs(F)52 b Fo(of)40 b(euclide)-5 b(an)39 b(length)h Fs(l)h Fo(satisfying)50 4946 y Fs(F)13 b Ft(\()p Fs(c)p Ft(\()p Fs(l)r Ft(\)\))p Fs(;)i(F)e Ft(\()p Fs(c)p Ft(\(0\)\))32 b Fs(>)27 b(C)859 4960 y Fv(0)933 4946 y Fo(and)35 b Fq(j)p Fs(c)p Ft(\()p Fs(l)r Ft(\))22 b Fq(\000)f Fs(c)p Ft(\(0\))p Fq(j)29 b(\024)e Ft(4)p Fs(\017\016)s Ft(\()p Fs(x)p Ft(\))p Fo(,)36 b(wher)-5 b(e)35 b Fs(\016)c Ft(:)c Fp(R)2446 4913 y Fr(d)2522 4946 y Fq(!)g Ft(\(0)p Fs(;)15 b Fq(1)p Ft(\))35 b Fo(is)f(de\014ne)-5 b(d)35 b(in)e(\(1.10\),)50 5091 y Fs(c)89 5058 y Fv(4)p Fr(\017\016)228 5091 y Fo(is)g(de\014ne)-5 b(d)33 b(in)g(\(1.11\))h(and)g(wher)-5 b(e)33 b Fs(x)25 b Ft(=)g(\(1)p Fs(=)p Ft(2\))p Fs(c)p Ft(\(0\))f(+)c(\(1)p Fs(=)p Ft(2\))p Fs(c)p Ft(\()p Fs(l)r Ft(\))p Fo(.)159 5309 y Ft(The)35 b(condition)g(inf)865 5326 y Fi(f)p Fr(F)10 b(>C)1067 5335 y Fl(0)1103 5326 y Fi(g)1160 5309 y Fq(jr)p Fs(F)j Fq(j)33 b(\025)g Fs(c)j Ft(can)f(b)s(e)g(relaxed)g(to) h(the)g(existence)g(of)g(a)f(subset)3375 5286 y(~)3331 5309 y Fq(M)g Ft(of)h(the)50 5455 y(set)e(of)f(lo)s(cal)i(minima)c Fq(M)j Ft(of)f Fs(F)47 b Ft(with)33 b(the)h(prop)s(ert)m(y)e(that)i (min)2300 5482 y Fr(x)p Fi(2)2424 5466 y Fv(~)2390 5482 y Fi(M)2495 5455 y Fs(T)2548 5482 y Fr(x;)p Fi(Mn)2768 5466 y Fv(~)2733 5482 y Fi(M)2854 5455 y Fs(<<)c Ft(sup)3163 5482 y Fr(y)r Fi(2Mn)3408 5466 y Fv(~)3373 5482 y Fi(M)3479 5455 y Fs(T)3532 5482 y Fr(x;)3629 5466 y Fv(~)3595 5482 y Fi(M)3685 5455 y Ft(.)p eop %%Page: 7 7 7 6 bop 50 295 a Fj(Sp)l(e)l(ctr)l(a)3378 b Fm(7)50 513 y Ft(In)34 b(particular,)i Fs(F)47 b Ft(ma)m(y)35 b(ha)m(v)m(e)g (in\014nite)g(man)m(y)e(lo)s(cal)j(minima)d(where)h(the)g(minima)f(in)h Fq(Mn)3323 490 y Ft(~)3278 513 y Fq(M)i Ft(are)e(not)50 658 y(as)c("deep")h(as)g(those)g(in)965 635 y(~)920 658 y Fq(M)p Ft(.)159 876 y(The)22 b(\014rst)f(result,)k(stated)e(in)f(4.2) h(and)f(referred)g(to)h(as)g(the)f(sharp)f(uncertain)m(t)m(y)j (principle,)g(is)e(strikingly)50 1022 y(reminiscen)m(t)k(of)g(the)g (uncertain)m(t)m(y)h(priniciple)f(in)g(quan)m(tum)e(mec)m(hanics.)40 b(W)-8 b(e)27 b(recall)g(that)g(the)f(tunneling)50 1167 y(time)36 b(of)g(a)g(quan)m(tum)f(mec)m(hanical)h(particle)h(mo)m(ving) f(in)g(a)g(double)g(w)m(ell)h(p)s(oten)m(tial)g(appro)m(ximately)g(is) 50 1313 y(giv)m(en)28 b(b)m(y)f(the)g(in)m(v)m(erse)h(of)f(the)g(sp)s (ectral)g(gap.)40 b(Let)28 b Fs(\025)p Ft(\(\012\))f(b)s(e)g(the)g (principal)g(eigen)m(v)-5 b(alue)28 b(of)g(the)f(Diric)m(hlet)50 1458 y(op)s(erator)j Fs(L)478 1425 y Fv(\012)478 1481 y Fr(\017)565 1458 y Ft(with)g(zero)h(b)s(oundary)e(conditions)i(on)f Fp(R)1994 1425 y Fr(d)2043 1458 y Fq(n)p Ft(\012,)h(where)e(\012)h(is)h (an)f(op)s(en,)g(regular)h(set.)50 1676 y Fe(Theorem)36 b(1.3:)51 b Fo(Assume)32 b(that)i Fs(F)45 b Fo(satis\014es)34 b(1.2.)42 b(Then)33 b(ther)-5 b(e)33 b(exists)g Fs(N)i Fq(\021)25 b Fs(N)10 b Ft(\()p Fs(d)p Ft(\))34 b Fo(such)e(that)i(for)f (al)5 b(l)50 1822 y Fs(\032)25 b(>)g(N)10 b(\017)15 b Ft(log)r(\(1)p Fs(=\017)p Ft(\))34 b Fo(satisfying)g Fs(T)1159 1836 y Fr(x;I)1289 1822 y Ft(=)25 b Fs(T)1438 1836 y Fr(I)1505 1822 y Fq(\021)g Ft(max)1770 1838 y Fr(y)r Fi(2Mn)p Fr(I)2038 1822 y Fs(T)2091 1836 y Fr(y)r(;I)2218 1822 y Fq(\025)g Fs(e)2356 1789 y Fi(\000)p Fr(\032=\017)2523 1822 y Fs(T)2576 1836 y Fr(I)5 b Fi([)p Fr(x)2741 1822 y Fo(it)32 b(fol)5 b(lows)559 2092 y Fs(\025)p Ft(\()p Fp(R)707 2055 y Fr(d)756 2092 y Fq(n)p Fs(B)870 2106 y Fr(I)912 2092 y Ft(\))25 b(=)g(\(1)c(+)f Fq(O)s Ft(\(1\))p Fs(\017)1487 2055 y Fi(\000)p Fr(N)1614 2092 y Fs(e)1656 2055 y Fi(\000)p Fr(\032=\017)1823 2092 y Ft(\))1894 2031 y(1)p 1869 2071 95 4 v 1869 2155 a Fs(T)1922 2169 y Fr(I)2000 2092 y Ft(=)25 b(\(1)c(+)f Fq(O)s Ft(\(1\))p Fs(\017)2515 2055 y Fi(\000)p Fr(N)2642 2092 y Fs(e)2684 2055 y Fi(\000)p Fr(\032=\017)2851 2092 y Ft(\))3021 2031 y(1)p 2897 2071 293 4 v 2897 2155 a Fp(E)2952 2128 y Fr(x)3003 2155 y Ft([)p Fs(\034)3068 2169 y Fr(B)3123 2178 y Fk(I)3164 2155 y Ft(])3477 2092 y(\(1)p Fs(:)p Ft(13\))50 2364 y Fo(wher)-5 b(e)35 b Fp(E)363 2331 y Fr(x)414 2364 y Ft([)p Fs(\034)479 2378 y Fr(B)534 2387 y Fk(I)575 2364 y Ft(])g Fo(is)f(the)h(exp)-5 b(e)g(cte)g(d)35 b(time)g(to)f(r)-5 b(e)g(ach)36 b Fs(B)1867 2378 y Fr(I)1943 2364 y Fo(starting)f(fr)-5 b(om)36 b Fs(x)e Fo(of)g(the)h(di\013usion)f (gener)-5 b(ate)g(d)36 b(by)50 2509 y Fs(L)112 2523 y Fr(\017)147 2509 y Fo(.)41 b(Her)-5 b(e)33 b(the)g(mo)-5 b(dulus)34 b(of)f(the)g(L)-5 b(andau)34 b(symb)-5 b(ol)34 b(is)e(dominate)-5 b(d)36 b(by)c(a)h(c)-5 b(onstant)34 b Fs(C)e Fq(\021)25 b Fs(C)7 b Ft(\()p Fs(d;)15 b(N)5 b(;)15 b(F)e Ft(\))p Fo(.)159 2727 y Ft(W)-8 b(e)35 b(are)g(also)h (able)f(to)g(analyze)g(all)h(other)e(exp)s(onen)m(tially)i(small)e (eigen)m(v)-5 b(alues)36 b(and)e(relate)i(them)d(to)50 2873 y(the)d(metastable)h(structure)f(giv)m(en)i(b)m(y)e Fs(F)13 b Ft(.)41 b(Let)31 b(us)e(de\014ne)h Fs(\032)25 b Fq(\021)g Fs(\032)p Ft(\()p Fs(F)s(;)15 b(\017)p Ft(\))31 b(b)m(y)986 3145 y Fs(e)1028 3108 y Fr(\032=\017)1164 3145 y Fq(\021)25 b Ft(max)1429 3017 y Fh(\032)1508 3077 y Fs(T)1561 3094 y Fr(x;I)5 b Fi(n)p Fr(x)p 1508 3124 237 4 v 1510 3208 a Fs(T)1563 3224 y Fr(y)r(;I)g Fi(n)p Fr(y)1770 3013 y Fh(\014)1770 3068 y(\014)1770 3122 y(\014)1770 3177 y(\014)1816 3145 y Fs(x;)15 b(y)28 b Fq(2)d(M)p Fs(;)15 b(x)26 b Fq(6)p Ft(=)f Fs(y)s(;)15 b(I)7 b Fq(\032M)2705 3017 y Fh(\033)3477 3145 y Ft(\(1)p Fs(:)p Ft(14\))50 3426 y(W)-8 b(e)31 b(then)f(ha)m(v)m(e)i(\(see)f(4.6)h(for)e(a)h(more)e (detailed)j(v)m(ersion\))50 3644 y Fe(Theorem)39 b(1.4:)58 b Fo(Assume)34 b(that)i Fs(F)48 b Fo(satis\014es)36 b(1.2.)50 b(Ther)-5 b(e)36 b(exists)f Fs(N)k Fq(\021)30 b Fs(N)10 b Ft(\()p Fs(d)p Ft(\))35 b Fo(and)h Fs(C)g Fq(\021)30 b Fs(C)7 b Ft(\()p Fs(d;)15 b(F)s(;)g(N)10 b Ft(\))50 3790 y Fo(such)33 b(that)g(for)h(al)5 b(l)33 b Fs(\032)25 b(>)g(N)10 b(\017)15 b Ft(log)r(\(1)p Fs(=\017)p Ft(\))34 b Fo(the)f(fol)5 b(lowing)34 b(holds.)10 4008 y(\(i\))47 b(F)-7 b(or)34 b(every)e Fs(x)25 b Fq(2)g(M)33 b Fo(ther)-5 b(e)33 b(exists)g(a)g(simple)h(eigenvalue)e Fs(\025)2204 4022 y Fr(x)2282 4008 y Fo(of)h Fs(L)2451 4022 y Fr(\017)2518 4008 y Fo(such)g(that)1162 4262 y Fs(\025)1215 4276 y Fr(x)1286 4262 y Ft(=)25 b(\(1)c(+)f Fq(O)s Ft(\(1\))p Fs(\017)1801 4225 y Fi(\000)p Fr(N)1928 4262 y Fs(e)1970 4225 y Fi(\000)p Fr(\032=\017)2137 4262 y Ft(\))p Fs(\025)p Ft(\()p Fp(R)2320 4225 y Fr(d)2369 4262 y Fq(n)p Fs(B)2483 4279 y Fi(M)p Fv(\()p Fr(x)p Fv(\))2671 4262 y Ft(\))771 b(\(1)p Fs(:)p Ft(15\))159 4517 y Fo(wher)-5 b(e)34 b Fq(M)p Ft(\()p Fs(x)p Ft(\))26 b Fq(\021)f(f)p Fs(y)j Fq(2)d(M)15 b(j)g Fs(F)e Ft(\()p Fs(y)s Ft(\))27 b Fs(<)e(F)13 b Ft(\()p Fs(x)p Ft(\))p Fq(g)p Fo(.)-18 4735 y(\(ii\))47 b(L)-5 b(et)28 b Fq(M)420 4749 y Fr(x)491 4735 y Fq(\021)d(f)p Fs(y)k Fq(2)c(M)15 b(j)g Fs(\025)1009 4749 y Fr(y)1078 4735 y Fs(<)25 b(\025)1227 4749 y Fr(x)1273 4735 y Fq(g)p Fo(.)41 b(F)-7 b(or)28 b(every)g(eigenfunction)f Fs(\036)2398 4749 y Fr(x)2472 4735 y Fo(c)-5 b(orr)g(esp)g(onding)31 b(to)d Fs(\025)3198 4749 y Fr(x)3272 4735 y Fo(ther)-5 b(e)28 b(exists)159 4880 y(a)33 b(set)424 4857 y Ft(~)380 4880 y Fq(M)489 4894 y Fr(x)567 4880 y Fo(such)g(that)h Fs(B)5 b Ft(\()p Fs(y)s(;)1157 4815 y Fq(p)p 1233 4815 37 4 v 65 x Fs(\017)o Ft(\))21 b Fq(\\)1450 4857 y Ft(~)1406 4880 y Fq(M)1515 4894 y Fr(x)1593 4880 y Fo(is)33 b(a)g(singleton)g (such)g(that)h(for)f(al)5 b(l)33 b Fs(z)d Fq(2)25 b(f)p Fs(F)38 b(<)25 b(C)3281 4894 y Fv(0)3322 4880 y Fq(g)457 5135 y Fs(\036)511 5149 y Fr(x)557 5135 y Ft(\()p Fs(z)t Ft(\))h(=)f(\(1)c(+)f Fq(O)s Ft(\(1\))p Fs(\017)1214 5097 y Fi(\000)p Fr(N)1341 5135 y Fs(e)1383 5097 y Fi(\000)p Fr(\032=\017)1550 5135 y Ft(\))p Fs(h)1637 5097 y Fv(0)1637 5157 y Fr(B)1692 5166 y Fk(x)1733 5157 y Fr(;B)1837 5166 y Fl(~)1810 5177 y Fg(M)1881 5186 y Fk(x)1930 5135 y Ft(\()p Fs(y)s Ft(\))h(+)f Fq(O)s Ft(\(1\))p Fs(\017)2387 5097 y Fi(\000)p Fr(N)2514 5135 y Fs(e)2556 5097 y Fi(\000)p Fr(\032=\017)2723 5135 y Fs(h)2775 5097 y Fv(0)2775 5157 y Fr(B)2858 5166 y Fl(~)2830 5177 y Fg(M)2901 5186 y Fk(x)2946 5157 y Fr(;)p Fi([)p Fr(x)3063 5135 y Ft(\()p Fs(z)t Ft(\))298 b(\(1)p Fs(:)p Ft(16\))159 5462 y Fo(Her)-5 b(e)32 b(the)h(L)-5 b(andau)35 b(symb)-5 b(ols)34 b(ar)-5 b(e)33 b(b)-5 b(ounde)g(d)34 b(by)f Fs(C)39 b Fo(in)32 b(absolute)i(value.)p eop %%Page: 8 8 8 7 bop 50 295 a Fm(8)3309 b Fj(Se)l(ction)27 b Fm(1)-46 513 y Fo(\(iii\))1327 658 y Fs(\033)s Ft(\()p Fs(L)1479 672 y Fr(\017)1514 658 y Ft(\))21 b Fq(\\)f Ft([0)p Fs(;)15 b(\017)1798 621 y Fr(N)1868 658 y Ft(\))25 b(=)g Fq(f)p Fs(\025)2122 672 y Fr(x)2184 658 y Fq(j)15 b Fs(x)25 b Fq(2)g(Mg)936 b Ft(\(1)p Fs(:)p Ft(17\))159 1085 y(\(1.15\))27 b(in)e(com)m(bination)g(with)g(\(1.13\))i(relates)f(exp)s(onen)m (tially)g(small)e(eigen)m(v)-5 b(alues)27 b(to)f(the)f(metastable)50 1231 y(structure)35 b(of)h(the)f(di\013usion)g(generated)i(b)m(y)e Fs(L)1700 1245 y Fr(\017)1735 1231 y Ft(.)56 b(F)-8 b(urthermore,)36 b(under)e(the)h(conditions)h(required)f(for)50 1376 y(\(1.9\))d(w)m(e)f (ha)m(v)m(e)g(determined)e(the)i(leading)g(asymptotic)g(in)f(\(1.13\).) 159 1595 y(Let)41 b(us)f(\014nally)g(describ)s(e)g(the)h(organization)i (of)d(the)h(pap)s(er.)70 b(Using)41 b(Harnac)m(k)h(and)e(H\177)-45 b(older)41 b(t)m(yp)s(e)50 1740 y(estimates,)d(in)d(c)m(hapter)h(t)m(w) m(o)h(w)m(e)f(deriv)m(e)h(analogous)g(estimates)f(for)f(a)h(priori)f (non-p)s(ositiv)m(e)i(harmonic)50 1885 y(functions.)i(As)25 b(a)i(result)f(w)m(e)g(gain)g(in)g(Lemma)e(2.3)j(p)s(oin)m(t)m(wise)g (con)m(trol)g(on)f(the)g(oscillation)i(of)e(eigenfunc-)50 2031 y(tions)f(corresp)s(onding)f(to)i(small)e(eigen)m(v)-5 b(alues)27 b(in)e(terms)e(of)i(suprema)e(o)m(v)m(er)k(suitable)e(small) g(balls)g(close)h(to)50 2176 y(the)j(lo)s(cal)g(minima)e(of)i Fs(F)13 b Ft(.)40 b(In)28 b(c)m(hapter)h(three)g(w)m(e)g(pro)m(v)m(e)g (b)s(ounds)e(of)h(those)i(suprema)c(b)m(y)j(exploiting)h(the)50 2322 y(strong)f(drift)f(of)h(the)g(di\013usion)f(to)m(w)m(ards)i(lo)s (cal)g(minima)d(of)i Fs(F)13 b Ft(.)40 b(The)28 b(a)h(priori)g(input)f (then)g(giv)m(es)i(precise)50 2467 y(con)m(trol)41 b(of)f (eigenfunctions)g(in)g(compact)g(sets.)69 b(As)40 b(so)s(on)g(as)g(w)m (e)g(ha)m(v)m(e)h(established)f(this)g(structural)50 2613 y(information)e(w)m(e)h(are)f(in)h(a)f(p)s(osition)h(to)g (generalize)i(the)d(analysis)i(dev)m(elop)s(ed)e(in)h(the)f(discrete)h (state)50 2758 y(space)c(setting)g(to)g(the)g(di\013usion)e(setting.)54 b(In)34 b(particular,)i(in)e(c)m(hapter)h(four)e(w)m(e)i(relate)h(the)e (lo)m(w)h(lying)50 2904 y(sp)s(ectrum)29 b(to)j(the)f(capacit)m(y)i (matrix)e(as)g(in)m(tro)s(duced)g(in)f([J])i(and)e(deriv)m(e)i(the)f (asymptotic)g(information)50 3049 y(in)f(terms)f(of)i(the)f(time)g (scales)i(in)m(tro)s(duced)e(in)g(\(1.7\).)p eop %%Page: 9 9 9 8 bop 50 295 a Fj(Sp)l(e)l(ctr)l(a)3378 b Fm(9)50 513 y Fw(2.)55 b(Regularit)m(y)40 b(of)h Fs(\025)p Fw(-harmonic)f (functions)h(c)m(hanging)g(sign)159 731 y Ft(This)21 b(c)m(hapter)i(is)f(dev)m(oted)h(to)g(the)f(follo)m(wing)h(simple)e (idea.)39 b(A)22 b(solution)h Fs(\036)f Ft(of)g(the)g(eigen)m(v)-5 b(alue)24 b(problem)913 981 y(\()p Fs(L)1010 995 y Fr(\017)1065 981 y Fq(\000)c Fs(\025)p Ft(\))p Fs(\036)p Ft(\()p Fs(x)p Ft(\))27 b(=)e(0)91 b Fs(x)25 b Fq(2)g Ft(\012)p Fs(;)197 b(\036)p Ft(\()p Fs(x)p Ft(\))26 b(=)f(0)91 b Fs(x)26 b Fq(2)f Fs(@)5 b Ft(\012)676 b(\(2)p Fs(:)p Ft(1\))50 1231 y(in)29 b(a)h(regular,)h(op)s(en)e(set)h(\012)f(with)h(small)f (energy)h Fs(\025)f Ft(cannot)i(create)g(large)g(oscillations)h(ev)m (erywhere)e(in)f(a)50 1376 y(region)i(where)f Fs(F)43 b Ft(is)30 b(small.)41 b(W)-8 b(e)31 b(start)g(with)50 1594 y Fu(A)51 b(priori)e(b)s(ounds)i(on)h(principal)c(eigen)m(v)-6 b(alues:)67 b Ft(On)43 b(sev)m(eral)j(o)s(ccasions)f(w)m(e)g(shall)f (meet)h(the)50 1740 y(follo)m(wing)32 b(ob)m(vious)f(represen)m(tation) h(form)m(ula.)41 b(The)30 b(P)m(oisson-Diric)m(hlet)k(problem)29 b(\(1.3\))k(for)d(a)h(regular,)50 1885 y(op)s(en)f(set)g(\000)h(in)f(a) 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Ft(with)f(Diric)m(hlet)j(conditions)f(on)f Fs(@)5 b Ft(\006.)63 b(Recall)40 b(the)e(de\014nition)50 3030 y(of)d(the)g Fs(\025)g Ft(-)g(equilibrium)e(p)s(oten)m(tial)k Fs(h)1400 2997 y Fr(\025)1400 3055 y(A;B)1539 3030 y Ft(,)f Fs(A)p Ft(,)h Fs(B)i Ft(closed)d(and)e(regular,)j Fs(\025)32 b Fq(\025)h Ft(0,)k(in)m(tro)s(duced)d(in)h(\(1.3\).)50 3175 y(F)-8 b(urthermore,)22 b(let)h Fs(w)785 3143 y Fr(\025)783 3201 y(A;B)943 3175 y Ft(b)s(e)e(the)h(solution)g(of)g(the) f(P)m(oisson's)i(problem)d(\(1.3\))j(with)e Fs(f)35 b Fq(\021)25 b Ft(0)d(and)f Fs(g)28 b Fq(\021)d Fs(h)3545 3143 y Fr(\025)3545 3201 y(A;B)3685 3175 y Ft(.)50 3321 y(W)-8 b(e)31 b(also)h(shall)e(use)g(the)h(con)m(v)m(en)m(tion)i Fs(h)1422 3288 y Fr(\025)1422 3346 y(A)1506 3321 y Fq(\021)25 b Fs(h)1654 3288 y Fr(\025)1654 3346 y(A;A)1820 3321 y Ft(and)30 b Fs(w)2064 3288 y Fr(\025)2062 3346 y(A)2147 3321 y Fq(\021)25 b Fs(w)2310 3288 y Fr(\025)2308 3346 y(A;A)2443 3321 y Ft(.)41 b(De\014ne)31 b(for)f Fs(K)7 b Fq(\032)o Ft(\()p Fs(A)21 b Fq([)f Fs(B)5 b Ft(\))3398 3288 y Fr(c)1448 3622 y Fs(s)1491 3584 y Fr(\025)1491 3644 y(K)1560 3622 y Ft(\()p Fs(A;)15 b(B)5 b Ft(\))26 b Fq(\021)f Ft(sup)1970 3698 y Fr(K)2097 3549 y Fs(w)2164 3516 y Fr(\025)2162 3574 y(A;B)p 2097 3601 204 4 v 2103 3687 a Fs(h)2155 3656 y Fr(\025)2155 3714 y(A;B)3523 3622 y Ft(\(2)p Fs(:)p Ft(4\))50 3903 y(W)-8 b(e)31 b(abbreviate)174 4153 y Fs(s)217 4115 y Fr(\025)265 4153 y Ft(\()p Fs(A;)15 b(B)5 b Ft(\))26 b Fq(\021)f Fs(s)682 4115 y Fr(\025)682 4178 y Fv(\()p Fr(A)p Fi([)p Fr(B)s Fv(\))900 4160 y Fk(c)938 4153 y Ft(\()p Fs(A;)15 b(B)5 b Ft(\))p Fs(;)198 b(s)1456 4115 y Fr(\025)1456 4175 y(K)1526 4153 y Ft(\()p Fs(A)p Ft(\))26 b Fq(\021)f Fs(s)1829 4115 y Fr(\025)1829 4175 y(K)1899 4153 y Ft(\()p Fs(A;)15 b(B)31 b Fq(\021)25 b Fs(A)p Ft(\))p Fs(;)197 b(s)2606 4115 y Fr(\025)2606 4175 y(A)2661 4157 y Fk(c)2724 4153 y Fq(\021)25 b Fs(s)2863 4115 y Fr(\025)2863 4175 y(A)2918 4157 y Fk(c)2956 4153 y Ft(\()p Fs(A;)15 b(B)31 b Fq(\021)25 b Fs(A)p Ft(\))125 b(\(2)p Fs(:)p Ft(5\))50 4403 y(and)30 b(denote)h(the)f(principal)g (eigen)m(v)-5 b(alue)32 b(of)f(the)g(Diric)m(hlet)h(op)s(erator)e Fs(L)2553 4370 y Fv(\006)2553 4425 y Fr(\017)2640 4403 y Ft(b)m(y)1540 4653 y Fs(\025)p Ft(\(\006\))25 b Fq(\021)g Ft(inf)d Fs(\033)s Ft(\()p Fs(L)2128 4615 y Fv(\006)2128 4675 y Fr(\017)2185 4653 y Ft(\))1303 b(\(2)p Fs(:)p Ft(6\))50 4903 y(W)-8 b(e)25 b(refer)e(to)i(the)f(follo)m(wing)h(lo)m (w)m(er)g(b)s(ound)d(on)i(the)g(principal)f(eigen)m(v)-5 b(alue)26 b(as)e(the)g(uncertain)m(t)m(y)h(principle.)50 5121 y Fe(Lemma)33 b(2.1:)49 b Fo(L)-5 b(et)31 b Ft(\006)f Fo(b)-5 b(e)31 b(a)g(b)-5 b(ounde)g(d,)32 b(r)-5 b(e)g(gular,)32 b(op)-5 b(en)32 b(set.)41 b(Then)31 b(for)g(al)5 b(l)31 b(r)-5 b(e)g(gular,)32 b(close)-5 b(d)32 b(sets)f Fs(A)p Fo(,)f Fs(B)5 b Fo(,)50 5266 y(such)33 b(that)g Fs(A)21 b Fq([)f Fs(B)29 b Ft(=)c(\006)872 5233 y Fr(c)942 5266 y Fo(it)32 b(fol)5 b(lows)1545 5413 y Fs(\025)p Ft(\(\006\))26 b Fq(\025)2012 5352 y Ft(1)p 1867 5392 337 4 v 1867 5475 a Fs(s)1910 5449 y Fv(0)1950 5475 y Ft(\()p Fs(A;)15 b(B)5 b Ft(\))3523 5413 y(\(2)p Fs(:)p Ft(7\))p eop %%Page: 10 10 10 9 bop 50 295 a Fm(10)3270 b Fj(Se)l(ction)27 b Fm(2)50 731 y Fd(Pro)s(of:)55 b Ft(The)33 b(follo)m(wing)i(v)-5 b(ariational)35 b(c)m(haracterization)i(of)d(Donsk)m(er)g(and)f(V)-8 b(aradhan)34 b(is)g(w)m(ell-kno)m(wn,)50 876 y(see)d([D)m(V])h(or)e (Theorem)f([7.1)j(in)e(P].)904 1126 y Fs(\025)p Ft(\(\006\))25 b(=)225 b(inf)1347 1190 y Fk(f)5 b Fg(2C)1464 1177 y Fl(1)1499 1190 y(\()p 1524 1148 44 4 v(\006)o(\))1225 1274 y Fk(f)g Fg(j)p Fk(@)s Fl(\006=0)p Fk(;)1462 1212 y Fh(R)1505 1318 y Fl(\006)1564 1274 y Fk(f)1600 1261 y Fl(2)1636 1274 y(=1)1856 1126 y Ft(sup)1801 1208 y Fk(u)p Fg(2C)1920 1195 y Fl(2)1955 1208 y(\()p 1980 1166 V(\006)o(\))1750 1264 y Fk(u)p Fl(\()p Fk(x)p Fl(\))p Fk(>)p Fl(0)p Fk(;x)p Fg(2)p Fl(\006)2124 1002 y Fh(Z)2174 1209 y Fv(\006)2257 1064 y Fs(L)2319 1078 y Fr(\017)2353 1064 y Fs(u)p Ft(\()p Fs(x)p Ft(\))p 2257 1105 272 4 v 2305 1188 a Fs(u)p Ft(\()p Fs(x)p Ft(\))2539 1126 y Fs(f)10 b Ft(\()p Fs(x)p Ft(\))2716 1088 y Fv(2)2757 1126 y Fs(dx)667 b Ft(\(2)p Fs(:)p Ft(8\))50 1498 y(T)-8 b(o)31 b(obtain)f(\(2.7\),)j(w)m(e)e(simply)e(insert)h Fs(u)25 b Fq(\021)g Fs(w)1627 1465 y Fv(0)1625 1523 y Fr(A;B)1794 1498 y Ft(and)30 b(use)g Fs(L)2190 1512 y Fr(\017)2225 1498 y Fs(u)25 b Ft(=)g Fs(h)2450 1465 y Fv(0)2450 1523 y Fr(A;B)2620 1498 y Ft(on)30 b(\006.)40 b Fq(})159 1716 y Ft(F)-8 b(rom)38 b(the)h(w)m(ell-kno)m(wn)h(Min-Max)f (Principle)g(w)m(e)g(obtain)h(the)f(follo)m(wing)h(as)f(w)m(e)g(shall)g (see)g(in)g(4.2)50 1861 y(sharp)29 b(upp)s(er)f(b)s(ound.)50 2080 y Fe(Lemma)37 b(2.2:)54 b Fo(L)-5 b(et)34 b Ft(\006)g Fo(b)-5 b(e)33 b(a)i(r)-5 b(e)g(gular,)35 b(op)-5 b(en)35 b(set)f(such)g(that)h Ft(dist\()p Fq(M)21 b(\\)g Ft(\006)p Fs(;)15 b(@)5 b Ft(\006\))28 b Fs(>)f(\032)33 b Fo(for)i(some)f Fs(\032)27 b(>)g Ft(0)p Fo(.)50 2225 y(Then)33 b(for)g(some)g Fs(C)f Fq(\021)25 b Fs(C)7 b Ft(\()p Fs(d;)15 b(F)e Fq(j)p Ft(\006)p Fs(;)i(\032)p Ft(\))34 b Fo(and)f(al)5 b(l)33 b Fs(x)26 b Fq(2)f(M)20 b(\\)g Ft(\006)1118 2499 y Fs(\025)p Ft(\(\006\))26 b Fq(\024)f Ft(\(1)c(+)f Fs(C)7 b(e)1735 2462 y Fi(\000)p Fr(\014)s(=\017)1908 2499 y Fs(=\017)p Ft(\))2072 2419 y(cap)2209 2386 y Fv(0)2209 2448 y Fr(B)s Fv(\()p Fr(x;\017)p Fv(\))2421 2419 y Ft(\(\006)2522 2386 y Fr(c)2559 2419 y Ft(\))p 2036 2478 594 4 v 2036 2490 a Fh(R)2079 2596 y Fr(A)2134 2565 y Fk(\014)2134 2621 y(x;)p Fl(\006)2235 2608 y Fk(c)2293 2563 y Fs(e)2335 2537 y Fi(\000)p Fr(F)f(=\017)2531 2563 y Fs(dx)3523 2499 y Ft(\(2)p Fs(:)p Ft(9\))50 2815 y Fo(wher)-5 b(e)41 b(we)e(have)i(de\014ne)-5 b(d)40 b Fs(A)1053 2771 y Fr(\014)1053 2842 y(x;)p Fv(\006)1168 2823 y Fk(c)1245 2815 y Fq(\021)e(f)p Fs(y)j Fq(2)d Ft(\006)15 b Fq(j)1726 2792 y Ft(^)1705 2815 y Fs(F)e Ft(\()p Fs(y)s(;)i(x)p Ft(\)\))40 b Fs(<)2190 2792 y Ft(^)2169 2815 y Fs(F)14 b Ft(\()p Fs(y)s(;)h Ft(\006)2430 2782 y Fr(c)2467 2815 y Ft(\))26 b Fq(\000)f Fs(\014)5 b Fq(g)40 b Fo(for)g(al)5 b(l)40 b Fs(\014)k(>)38 b Ft(0)i Fo(such)f(that)50 2960 y Fs(B)5 b Ft(\()p Fs(x;)15 b(\032=)p Ft(2\))p Fq(\032)q Fs(A)p Fo(.)42 b(Her)-5 b(e)871 2937 y Ft(^)850 2960 y Fs(F)46 b Fo(denotes)34 b(the)f(c)-5 b(ommunic)g(ation)34 b(height)g(intr)-5 b(o)g(duc)g(e)g(d)35 b(in)d(\(1.4\).)50 3179 y Fd(Pro)s(of:)56 b Ft(Insert)34 b Fs(u)e Fq(\021)g Fs(h)892 3146 y Fv(0)892 3208 y Fr(B)s Fv(\()p Fr(x;\017)p Fv(\))p Fr(;)p Fv(\006)1173 3190 y Fk(c)1247 3179 y Ft(with)i Fs(x)e Fq(2)f(M)23 b(\\)g Ft(\006)34 b(and)f(b)m(y)i(con)m(v)m(en)m(tion)h Fs(h)2771 3146 y Fv(0)2771 3208 y Fr(B)s Fv(\()p Fr(x;\017)p Fv(\))p Fr(;)p Fv(\006)3052 3190 y Fk(c)3124 3179 y Fq(\021)31 b Ft(1)k(on)f Fs(B)5 b Ft(\()p Fs(x;)15 b(\017)p Ft(\))50 3324 y(in)m(to)1151 3482 y Fs(\025)p Ft(\(\006\))25 b(=)190 b(inf)1461 3557 y Fr(u)p Fi(2)p Fr(W)1631 3526 y Fl(1)p Fk(;)p Fl(2)1621 3579 y(0)1719 3557 y Fv(\(\006\))p Fi(n)p Fv(0)1927 3341 y Fh(R)1970 3446 y Fv(\006)2041 3414 y Fs(e)2083 3381 y Fi(\000)p Fr(F)6 b(=\017)2265 3414 y Fq(jr)p Fs(u)p Fq(j)2443 3381 y Fv(2)2499 3414 y Fs(dx)p 1927 3461 672 4 v 1965 3472 a Fh(R)2008 3578 y Fv(\006)2079 3546 y Fs(e)2121 3519 y Fi(\000)p Fr(F)g(=\017)2302 3546 y Fq(j)p Fs(u)p Fq(j)2404 3519 y Fv(2)2461 3546 y Fs(dx)3477 3482 y Ft(\(2)p Fs(:)p Ft(10\))50 3714 y(to)31 b(obtain)1175 3878 y Fs(\025)p Ft(\(\006\))25 b Fq(\024)1774 3797 y Ft(cap)1910 3764 y Fv(0)1910 3827 y Fr(B)s Fv(\()p Fr(x;\017)p Fv(\))2123 3797 y Ft(\(\006)2224 3764 y Fr(c)2261 3797 y Ft(\))p 1496 3857 1078 4 v 1496 3869 a Fh(R)1539 3974 y Fr(A)1594 3944 y Fk(\014)1594 4000 y(x;)p Fl(\006)1695 3987 y Fk(c)1752 3942 y Fs(e)1794 3916 y Fi(\000)p Fr(F)6 b(=\017)1976 3942 y Ft(\()p Fs(h)2063 3911 y Fv(0)2063 3973 y Fr(B)s Fv(\()p Fr(x;\017)p Fv(\))p Fr(;)p Fv(\006)2344 3955 y Fk(c)2383 3942 y Ft(\))2418 3916 y Fv(2)2475 3942 y Fs(dx)3477 3878 y Ft(\(2)p Fs(:)p Ft(11\))50 4153 y(In)m(v)m(oking)48 b(Corollary)i([4.8)f(in)f(BEGK3],)53 b(w)m(e)48 b(deriv)m(e)g 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Fk(c)2765 4444 y Ft(.)46 b(This)31 b(establishes)i(\(2.9\).)50 4589 y Fq(})50 4807 y Fu(Uniform)g(regularit)m(y)f(estimates)f(for)j Fs(\025)p Fu(-harmonic)f(functions)g(c)m(hanging)h(sign:)39 b Ft(W)-8 b(e)30 b(are)g(no)m(w)50 4953 y(in)g(a)h(p)s(osition)f(to)h (turn)e(the)i(idea)g(men)m(tioned)f(in)g(the)h(b)s(eginning)e(of)i (this)f(c)m(hapter)h(in)m(to)50 5171 y Fe(Lemma)48 b(2.3:)74 b Fo(L)-5 b(et)42 b Fs(h)f Fq(2)g Fs(W)1180 5138 y Fv(2)p Fr(;n)1287 5171 y Ft(\()p Fp(R)1382 5138 y Fr(d)1430 5171 y Ft(\))h Fo(b)-5 b(e)41 b(a)g(str)-5 b(ong)43 b(solution)g(of)e (the)h(e)-5 b(quation)42 b Ft(\()p Fs(L)3092 5185 y Fr(\017)3154 5171 y Fq(\000)26 b Fs(\025)p Ft(\))p Fs(h)41 b Ft(=)g(0)h Fo(in)50 5316 y Fs(B)5 b Ft(\()p Fs(x;)15 b Ft(2)296 5251 y Fq(p)p 372 5251 37 4 v 65 x Fs(\017)p Ft(\))p Fo(,)38 b(wher)-5 b(e)38 b Fs(x)32 b Fq(2)h(M)j Fo(and)i Ft(0)33 b Fq(\024)g Fs(\025)f Fq(\024)h Fs(\017)p Fo(.)53 b(Then)37 b(ther)-5 b(e)38 b(exists)43 b Ft(~)-51 b Fs(x)33 b Fq(2)f Fs(B)5 b Ft(\()p Fs(x;)2868 5251 y Fq(p)p 2944 5251 V 65 x Fs(\017)p Ft(\))37 b Fo(such)g(that)h Fs(h)f Fo(do)-5 b(es)50 5462 y(not)33 b(change)g(sign)g(in)f Fs(B)5 b Ft(\()h(~)-51 b Fs(x;)15 b(\017)p Ft(\))p Fo(.)p eop %%Page: 11 11 11 10 bop 50 295 a Fj(Sp)l(e)l(ctr)l(a)3340 b Fm(11)159 513 y Fo(L)-5 b(et)46 b Ft(\000)p Fo(,)j Ft(\006)p Fq(\032)o Ft(\000)p Fo(,)g(b)-5 b(e)46 b(a)g(r)-5 b(e)g(gular,)50 b(op)-5 b(en)47 b(sets.)82 b(L)-5 b(et)46 b Fs(g)53 b Fq(2)c Fs(L)2217 480 y Fi(1)2217 538 y Fv(lo)r(c)2312 513 y Ft(\()p Fp(R)2407 480 y Fr(d)2456 513 y Ft(\))d Fo(and)g(let)g Fs(h)h Fo(b)-5 b(e)45 b(a)h(non-ne)-5 b(gative,)50 658 y(str)g(ong)38 b(solution)f(of)g(the)g(e)-5 b(quation)37 b Ft(\()p Fs(L)1403 672 y Fr(\017)1462 658 y Fq(\000)22 b Fs(\025)p Ft(\))p Fs(h)33 b Ft(=)f Fs(g)41 b Fo(in)36 b Ft(\000)p Fo(.)53 b(Assume)37 b(that)g(ther)-5 b(e)38 b(is)e Ft(0)d Fs(<)f(r)j(<)d Ft(1)p Fs(=)p Ft(2)38 b Fo(and)50 804 y Fs(B)5 b Ft(\()p Fs(x;)15 b Ft(2)296 738 y Fq(p)p 372 738 37 4 v 66 x 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Fv(\))2399 1213 y Fs(h)21 b Ft(+)f Fs(C)7 b(\017)2672 1175 y Fr(d=)p Fv(2)2886 1213 y Ft(sup)2802 1298 y Fr(B)s Fv(\()p Fr(x;)p Fv(2)2987 1252 y Fi(p)p 3047 1252 31 4 v 46 x Fr(\017)q Fv(\))3121 1213 y Fq(j)p Fs(g)s Fq(j)260 b Ft(\(2)p Fs(:)p Ft(12\))159 1694 y(Ha)m(ving)47 b(established)e(p)s (ositivit)m(y)i(of)f(eigenfunctions)g(in)f(vicinities)i(of)f(the)g(lo)s (cal)g(minima)e(of)h Fs(F)13 b Ft(,)50 1840 y(w)m(e)40 b(ma)m(y)f(use)g(strong)h(p)s(oin)m(t)m(wise)g(regularit)m(y)h(suc)m(h) f(as)f(the)h(lo)s(cal)h(\(Boundary\))f(Maxim)m(um)e(Principle)50 1985 y(Theorem)e([9.20)i(in)f(GT])g(\(Theorem)f([9.26)j(in)e(GT]\),)h (the)f(Harnac)m(k's)h(inequalit)m(y)g(Theorem)e([9.22)j(in)50 2131 y(GT])29 b(and)f(the)i(\(Boundary\))f(H\177)-45 b(older's)29 b(estimates)h(Corollary)h([9.24)f(in)f(GT])g(\(Corollary)i ([9.28)g(in)e(GT]\).)159 2349 y(Recall)g(the)e(de\014nition)g(\(1.10\)) j(of)d(the)h(radius)e Fs(\016)s Ft(\()p Fs(x)p Ft(\),)j Fs(x)d Fq(2)f Fp(R)2210 2316 y Fr(d)2259 2349 y Ft(.)39 b(Clearly)-8 b(,)29 b Fs(\017\016)i Ft(only)d(dep)s(ends)d(on)i Fq(jr)p Fs(F)13 b Fq(j)p Fs(=\017)50 2494 y Ft(and)30 b(that)h(8)p Fs(\017\016)s Ft(\()p Fs(y)s Ft(\))15 b(sup)821 2516 y Fr(B)s Fv(\()p Fr(y)r(;)p Fv(4)p Fr(\017\016)r Fv(\()p Fr(y)r Fv(\)\))1214 2494 y Fq(jr)p Fs(F)e Fq(j)p Fs(=\017)26 b Ft(=)f(1.)42 b(Com)m(bination)30 b(of)h(Harnac)m(k's)g (and)f(H\177)-45 b(older's)31 b(principles)50 2640 y(giv)m(es)50 2858 y Fe(Theorem)k(2.4:)51 b Fo(Fix)33 b Ft(0)25 b Fs(<)g(\032)g Fq(\024)g Fs(\017)p Fo(.)42 b(L)-5 b(et)32 b Ft(0)26 b Fq(\024)f Fs(h)h Fq(2)f Fs(W)1953 2825 y Fv(2)p Fr(;n)2059 2858 y Ft(\()p Fp(R)2154 2825 y Fr(d)2203 2858 y Ft(\))32 b Fo(b)-5 b(e)32 b(a)h(str)-5 b(ong,)33 b(non-ne)-5 b(gative)33 b(solution)g(of)50 3003 y(the)k(e)-5 b(quation)37 b Ft(\()p Fs(L)668 3017 y Fr(\017)726 3003 y Fq(\000)23 b Fs(\025)p Ft(\))p Fs(h)33 b Ft(=)f(0)37 b Fo(in)f Fs(B)5 b Ft(\()p Fs(x;)15 b Ft(2)1539 2947 y Fq(p)p 1615 2947 48 4 v 56 x Fs(\032)p Ft(\))p Fo(,)38 b(wher)-5 b(e)38 b Fs(x)32 b Fq(2)g(M)k Fo(and)i Ft(0)32 b Fq(\024)g Fs(\025)h Fq(\024)e Ft(1)p Fo(.)54 b(Then)37 b(ther)-5 b(e)37 b(exists)50 3149 y Fs(C)32 b Ft(=)25 b Fs(C)7 b Ft(\()p Fs(F)13 b Fq(j)p Fs(B)5 b Ft(\()p Fs(x;)15 b Ft(2)692 3092 y Fq(p)p 768 3092 V 57 x Fs(\032)p Ft(\)\))33 b Fo(and)h Fs(\013)26 b Ft(=)f Fs(\013)p Ft(\()p Fs(F)13 b Fq(j)p Fs(B)5 b Ft(\()p Fs(x;)15 b Ft(2)1710 3092 y Fq(p)p 1787 3092 V 1787 3149 a Fs(\032)p Ft(\)\))26 b Fs(>)f Ft(0)33 b Fo(such)g(that)h(for)f(al)5 b(l)33 b Ft(0)26 b Fs(<)f(r)j(<)3103 3092 y Fq(p)p 3179 3092 V 57 x Fs(\032)1265 3412 y Ft(osc)1386 3429 y Fr(B)s Fv(\()p Fr(x;r)r Fv(\))1620 3412 y Fs(h)d Fq(\024)g Fs(C)7 b Ft(\()p Fs(r)s(=)1989 3351 y Fq(p)p 2065 3351 V 61 x Fs(\032)p Ft(\))2147 3375 y Fr(\013)2265 3412 y Ft(inf)2214 3475 y Fr(B)s Fv(\()p Fr(x;r)r Fv(\))2443 3412 y Fs(h)982 b Ft(\(2)p Fs(:)p Ft(13\))50 3704 y Fo(Assume)28 b(that)h Ft(0)d Fq(\024)f Fs(h)h Fq(2)f Fs(W)996 3671 y Fv(2)p Fr(;n)1102 3704 y Ft(\()p Fp(R)1197 3671 y Fr(d)1246 3704 y Ft(\))j Fo(is)g(a)h(str)-5 b(ong,)30 b(non-ne)-5 b(gative)29 b(solution)h(of)e(the)h(e)-5 b(quation)29 b Ft(\()p Fs(L)3346 3718 y Fr(\017)3392 3704 y Fq(\000)11 b Fs(\025)p Ft(\))p Fs(h)25 b Ft(=)50 3850 y Fs(f)44 b Fo(in)34 b Ft(\006)p Fo(,)h(wher)-5 b(e)35 b Ft(\006)f Fo(is)h(an)g(op)-5 b(en,)36 b(r)-5 b(e)g(gular)36 b(set,)f Ft(0)29 b Fq(\024)g Fs(\025)g Fq(\024)f Ft(1)35 b Fo(and)h Fs(f)44 b Fo(is)34 b(in)h Fs(L)2669 3817 y Fr(d)2711 3850 y Ft(\(\006\))p Fo(.)48 b(Ther)-5 b(e)35 b(ar)-5 b(e)36 b(c)-5 b(onstants)50 3995 y Fs(C)50 b Ft(=)44 b Fs(C)7 b Ft(\()p Fs(d)p Ft(\))43 b Fo(and)h Fs(\013)g Ft(=)f Fs(\013)p Ft(\()p Fs(d)p Ft(\))j Fs(>)d Ft(0)g Fo(such)g(that)h(for)g(al)5 b(l)43 b Fs(x)h Fq(2)f Ft(\006)g Fo(and)h(al)5 b(l)43 b Ft(0)h Fs(<)g(\032)f(<)h(\017\016)s Ft(\()p Fs(x)p Ft(\))g Fo(satisfying)50 4141 y Fs(B)5 b Ft(\()p Fs(x;)15 b Ft(4)p Fs(\032)p Ft(\))p Fq(\032)q Ft(\006)32 b Fo(and)h(al)5 b(l)33 b Ft(0)26 b Fs(<)f(r)j(<)d(\032)900 4404 y Ft(osc)1022 4421 y Fr(B)s Fv(\()p Fr(x;r)r Fv(\))1255 4404 y Fs(h)h Fq(\024)f Fs(C)7 b Ft(\()p Fs(r)s(=\032)p Ft(\))1707 4367 y Fr(\013)1759 4331 y Fh(\000)1852 4404 y Ft(inf)1800 4467 y Fr(B)s Fv(\()p Fr(x;r)r Fv(\))2029 4404 y Fs(h)20 b Ft(+)g Fq(jj)p Fs(f)10 b Fq(jj)2347 4421 y Fr(L)2397 4403 y Fk(n)2442 4421 y Fv(\()p Fr(B)s Fv(\()p Fr(x;r)r Fv(\))p Fi(\\)p Fv(\006\))2818 4331 y Fh(\001)3477 4404 y Ft(\(2)p Fs(:)p Ft(14\))50 4688 y Fo(F)-7 b(or)36 b Fs(x)29 b Fq(2)h Fs(@)5 b Ft(\006)34 b Fo(let)i Fs(V)731 4702 y Fr(x)811 4688 y Fo(b)-5 b(e)35 b(the)h(exterior)g(c)-5 b(one)35 b(at)h Fs(x)p Fo(.)48 b(We)35 b(stil)5 b(l)35 b(have)h(for)f(some)h(c)-5 b(onstant)37 b Fs(C)f Fq(\021)29 b Fs(C)7 b Ft(\()p Fs(d;)15 b(V)3628 4702 y Fr(x)3674 4688 y Ft(\))50 4833 y Fo(and)33 b Fs(\013)26 b Fq(\021)f Fs(\013)p Ft(\()p Fs(d;)15 b(V)639 4847 y Fr(x)686 4833 y Ft(\))26 b Fs(>)f Ft(0)33 b Fo(and)h(al)5 b(l)33 b Ft(0)25 b Fs(<)g(r)j(<)d(\032)g(<)g(\017\016)s Ft(\()p Fs(x)p Ft(\))640 5060 y(osc)761 5077 y Fr(B)s Fv(\()p Fr(x;r)r Fv(\))p Fi(\\)p Fv(\006)1095 5060 y Fs(h)h Fq(\024)p Fs(C)7 b Ft(\()p Fs(r)s(=\032)p Ft(\))1522 5023 y Fr(\013)1574 4986 y Fh(\000)1615 5060 y Ft(osc)1737 5077 y Fr(B)s Fv(\()p Fr(x;\032)p Fv(\))p Fi(\\)p Fv(\006)2073 5060 y Fs(h)21 b Ft(+)e Fq(jj)p Fs(f)30 b Fq(\000)20 b Fs(\025h)p Fq(jj)2607 5077 y Fr(L)2657 5058 y Fk(n)2703 5077 y Fv(\()p Fr(B)s Fv(\()p Fr(x;r)r Fv(\))p Fi(\\)p Fv(\006\))3078 4986 y Fh(\001)1173 5233 y Ft(+)p Fs(C)h Ft(osc)1452 5249 y Fr(B)s Fv(\()p Fr(x;)1601 5209 y Fi(p)p 1661 5209 74 4 v 40 x Fr(r)r(\032)o Fv(\))p Fi(\\)p Fr(@)t Fv(\006)1926 5233 y Fs(h)3477 5149 y Ft(\(2)p Fs(:)p Ft(15\))50 5460 y Fo(wher)-5 b(e)34 b Ft(osc)428 5476 y Fr(B)s Fv(\()p Fr(x;r)r Fv(\))p Fi(\\)p Fr(@)t Fv(\006)805 5460 y Fs(h)26 b Fq(\021)f Ft(lim)15 b(sup)1257 5482 y Fr(x)p Fi(!)p Fr(B)s Fv(\()p Fr(x;r)r Fv(\))p Fi(\\)p Fr(@)t Fv(\006)1747 5460 y Fs(h)21 b Fq(\000)f Ft(lim)15 b(inf)2163 5476 y Fr(x)p Fi(!)p Fr(B)s Fv(\()p Fr(x;r)r Fv(\))p Fi(\\)p Fr(@)t Fv(\006)2653 5460 y Fs(h)p Fo(.)p eop %%Page: 12 12 12 11 bop 50 295 a Fm(12)3270 b Fj(Se)l(ction)27 b Fm(2)159 513 y Ft(W)-8 b(e)26 b(also)g(need)f(the)g(b)s(oundary)e(Harnac)m(k's)j (inequalit)m(y)-8 b(,)29 b(whic)m(h)24 b(is)h(a)h(consequence)g(of)f (Theorem)f([8.0.1)50 658 y(in)30 b(P].)50 876 y Fe(Theorem)49 b(2.5:)76 b Fo(Assume)41 b(that)j Fq(r)p Fs(F)54 b Fo(is)42 b(lo)-5 b(c)g(al)5 b(ly)44 b(Lipschitz)e(c)-5 b(ontinuous.)71 b(L)-5 b(et)42 b Ft(\006)g Fo(b)-5 b(e)41 b(an)i(op)-5 b(en)43 b(set)50 1022 y(with)f(uniformly)g(Lipschitz)g(c)-5 b(ontinuous)42 b(b)-5 b(oundary.)68 b(Ther)-5 b(e)42 b(exist)f Fs(C)47 b Fq(\021)40 b Fs(C)7 b Ft(\()p Fs(d)p Ft(\))p Fo(,)43 b Fs(\032)e Fq(\021)f Fs(\032)p Ft(\()p Fs(d)p Ft(\))h Fs(>)f Ft(0)i Fo(and)50 1167 y(a)e(function)g Fs(R)g Ft(:)f Fs(@)5 b Ft(\006)39 b Fq(!)f Ft(\(0)p Fs(;)15 b Fq(1)p Ft(\))p Fo(,)44 b Fs(R)39 b Fq(\024)f Fs(\016)44 b Fo(with)d(the)g(fol)5 b(lowing)41 b(pr)-5 b(op)g(erties.)66 b(Fix)41 b Fs(z)i Fq(2)38 b Fs(@)5 b Ft(\006)40 b 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Ft(\()p Fs(x)p Ft(\))1134 1865 y Fq(\024)25 b Fs(C)1312 1803 y(u)p Ft(\()p Fs(y)s Ft(\))p 1312 1844 171 4 v 1314 1927 a Fs(v)s Ft(\()p Fs(y)s Ft(\))1494 1865 y Fs(;)201 b(x;)15 b(y)28 b Fq(2)d Ft(\006)20 b Fq(\\)g Fs(B)2207 1879 y Fr(z)2269 1865 y Fq(\\)g Fs(B)5 b Ft(\()p Fs(z)t(;)15 b(r)s(\017R)q Ft(\()p Fs(z)t Ft(\)\))630 b(\(2)p Fs(:)p Ft(16\))50 2328 y Fd(Pro)s(of:)46 b Ft(Denote)30 b(b)m(y)f(1)p Fs(=\015)5 b Ft(\()p Fs(z)t Ft(\))31 b(the)d(b)s(est)h(Lipsc)m(hitz)g (constan)m(t)h(of)f Fs(\037)f Ft(at)i Fs(z)j Ft(in)28 b Fs(B)5 b Ft(\()p Fs(z)t(;)15 b Ft(8)p Fs(\017)p Ft(\))30 b(and)e(let)i(1)p Fs(=\014)5 b Ft(\()p Fs(z)t Ft(\))30 b(b)s(e)50 2473 y(the)22 b(b)s(est)f(Lipsc)m(hitz)i(constan)m(t)g(of)f Fq(r)p Fs(F)34 b Ft(in)22 b Fs(B)5 b Ft(\()p Fs(z)t(;)15 b Ft(8)p Fs(\017)p Ft(\).)39 b(De\014ne)22 b Fs(R)q Ft(\()p Fs(z)t Ft(\))k Fq(\021)f Ft(min)n(\()p Fs(\014)5 b Ft(\()p Fs(z)t Ft(\))p Fs(;)15 b(\015)5 b Ft(\()p Fs(z)t Ft(\))p Fs(;)15 b(\016)s Ft(\()p Fs(z)t Ft(\)\),)30 b(where)21 b Fs(\016)s Ft(\()p Fs(z)t Ft(\))50 2619 y(is)h(giv)m(en)h(in)f (\(1.10\).)40 b(Let)22 b(us)g(in)m(tro)s(duce)g(the)g(function)28 b(~)-51 b Fs(u)p Ft(\()6 b(~)-51 b Fs(x)p Ft(\))26 b Fq(\021)f Fs(u)p Ft(\()p Fs(x)p Ft(\),)30 b(~)-51 b Fs(x)25 b Fq(\021)g Ft(\()p Fs(x)t Fq(\000)t Fs(z)t Ft(\))p Fs(=)p Ft(\()p Fs(\017R)q Ft(\()p Fs(z)t Ft(\)\),)h(and)21 b(lik)m(ewise)53 2764 y(~)-48 b Fs(v)s Ft(.)39 b(F)-8 b(urthermore,)25 b(let)845 2741 y(~)834 2764 y Fs(L)g Fq(\021)g(\000)p Ft(\001)8 b(+)1248 2740 y(~)1251 2764 y Fs(b)h Fq(\001)f(r)p Ft(,)26 b(where)1713 2740 y(~)1716 2764 y Fs(b)p Ft(\()6 b(~)-51 b Fs(x)p Ft(\))26 b Fq(\021)1996 2740 y Ft(~)1999 2764 y Fs(b)2038 2780 y Fr(\017R)p Fv(\()p Fr(z)s Fv(\))2222 2764 y Ft(\()6 b(~)-51 b Fs(x)p Ft(\))26 b Fq(\021)f(r)p Fs(F)13 b Ft(\()p Fs(x)p Ft(\).)39 b(Fix)25 b Fs(r)j(>)d Ft(0)g(and)e(let)j Fs(u)e Ft(and)50 2909 y Fs(v)i Ft(b)s(e)d Fs(L)299 2923 y Fr(\017)357 2909 y Ft(-)h(harmonic)e(in)h Fs(B)5 b Ft(\()p Fs(z)t(;)15 b Ft(8)p 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Fs(v)35 b Ft(=)d(0)j(on)f Fs(B)5 b Ft(\(0)p Fs(;)15 b Ft(8)p Fs(r)s Ft(\))24 b Fq(\\)2060 3177 y Ft(~)2039 3200 y Fs(B)2108 3214 y Fr(z)2172 3200 y Fq(\\)f Fs(@)2319 3177 y Ft(~)2309 3200 y(\006.)53 b(Note)35 b(that)g(b)m(y)f(de\014nition)h(of)f Fs(R)q Ft(\()p Fs(z)t Ft(\))50 3346 y(under)39 b(this)i(transformation,)i(the) e(b)s(est)f(Lipsc)m(hitz)i(constan)m(t)g(of)52 b(~)-56 b Fs(\037)p Ft(\()6 b(~)-51 b Fs(x)p Ft(\))43 b Fq(\021)g Fs(\037)p Ft(\()p Fs(x)p Ft(\))e(at)h Fs(z)j Ft(and)40 b(the)h(b)s(est)50 3491 y(Lipsc)m(hitz)28 b(constan)m(t)h(of)897 3467 y(~)900 3491 y Fs(b)f Ft(in)f Fs(B)5 b Ft(\(0)p Fs(;)15 b Ft(8\))29 b(are)f(b)s(ounded)e(b)m(y)h(one.)40 b(Moreo)m(v)m(er,)31 b(the)d(suprem)m(um)c(norm)i(of)3564 3467 y(~)3567 3491 y Fs(b)i Ft(in)50 3637 y Fs(B)5 b Ft(\(0)p Fs(;)15 b Ft(8\))28 b(is)g(that)f(of)h Fq(r)p Fs(F)40 b Ft(in)26 b Fs(B)5 b Ft(\()p Fs(z)t(;)15 b Ft(8)p Fs(\017R)q Ft(\()p Fs(z)t Ft(\)\))29 b(and)e(hence)g(is)g(b)s(ounded)e (b)m(y)i(one)h(b)m(y)f(de\014nition)g(of)g Fs(\016)s Ft(\()p Fs(z)t Ft(\).)41 b(The)50 3782 y(b)s(oundary)25 b(Harnac)m(k's)k(principle)e(Theorem)f([8.0.1)j(in)e(P])g(applied)h(to) g Fs(D)g Fq(\021)d Fs(B)5 b Ft(\(0)p Fs(;)15 b Ft(8\))f Fq(\\)3093 3759 y Ft(~)3070 3782 y Fs(B)3139 3796 y Fr(z)3196 3782 y Fq(\\)3281 3759 y Ft(~)3271 3782 y(\006)27 b(giv)m(es)i(the)50 3928 y(existence)j(of)f Fs(C)h Fq(\021)26 b Fs(C)7 b Ft(\()p Fs(d)p Ft(\))31 b(and)f Fs(\032)c Fq(\021)f Fs(\032)p Ft(\()p Fs(d)p Ft(\))i Fs(>)e Ft(0)31 b(suc)m(h)g(that)37 b(~)-51 b Fs(u)p Ft(\()p Fs(x)p Ft(\))p Fs(=)s Ft(~)j Fs(v)t Ft(\()p Fs(x)p Ft(\))27 b Fq(\024)e Fs(C)12 b Ft(~)-50 b Fs(u)p Ft(\()p Fs(y)s Ft(\))p Fs(=)s Ft(~)i Fs(v)t Ft(\()p Fs(y)s Ft(\))31 b(for)g(all)g(0)c Fs(<)e(r)j Fq(\024)e Fs(\032)50 4073 y Ft(and)35 b(~)-50 b Fs(x;)21 b Ft(~)-51 b Fs(y)28 b Fq(2)d Fs(B)5 b Ft(\(0)p Fs(;)15 b(r)s Ft(\))21 b Fq(\\)874 4050 y Ft(~)853 4073 y Fs(B)922 4087 y Fr(z)984 4073 y Fq(\\)1074 4050 y Ft(~)1064 4073 y(\006.)41 b Fq(})159 4291 y Ft(Sev)m(eral)31 b(times)f(in)g(the)h (sequel)g(w)m(e)f(shall)h(use)f(the)h(follo)m(wing)g(ob)m(vious)g (consequence)g(of)g(\(2.2\).)713 4536 y(sup)749 4612 y Fr(K)865 4536 y Fq(j)p Fs(h)p Fq(j)26 b(\024)f Fs(s)1132 4499 y Fv(0)1132 4559 y Fr(K)1202 4536 y Ft(\()p Fs(@)5 b Ft(\006\))15 b(sup)1449 4612 y Fv(\006)1544 4536 y Ft(\()p Fs(\025)p Fq(j)p Fs(h)p Fq(j)21 b Ft(+)f Fq(j)p Fs(g)s Fq(j)p Ft(\))i(+)e(sup)2125 4612 y Fr(K)2242 4536 y Fs(H)2325 4499 y Fv(0)2318 4559 y(\006)2374 4536 y Fq(j)p Fs(h)p Fq(j)p Fs(;)198 b(K)7 b Fq(\032)o Ft(\006)p Fq(\032)p Ft(\000)430 b(\(2)p Fs(:)p Ft(17\))50 4807 y(The)33 b(follo)m(wing)h(consequence)h(of)e(Theorem)f(2.4)i(serv)m(es) g(for)f(uniform)f(a)h(priori)g(con)m(trol)i(in)e Fp(R)3346 4774 y Fr(d)3394 4807 y Ft(.)50 b(It)33 b(is)g(a)50 4953 y(crucial)e(ingredien)m(t)f(for)g(the)g(pro)s(of)f(of)h(gro)m(wth)g (estimates)h(of)f Fs(\025)g Ft(-)g(harmonic)f(functions)g(in)h(Prop)s (osition)50 5098 y(3.6.)50 5316 y Fe(Lemma)47 b(2.6:)74 b Fo(Assume)40 b(1.2.)67 b(L)-5 b(et)42 b Fs(c)e Ft(:)h([0)p Fs(;)15 b(l)r Ft(])41 b Fq(!)f Fp(R)1988 5283 y Fr(d)2077 5316 y Fo(b)-5 b(e)41 b(a)g(\015ow)h(line)f(of)g(euclide)-5 b(an)41 b(length)h Fs(l)h Fo(such)50 5462 y(that)d Fs(F)13 b Ft(\()p Fs(c)p Ft(\()p Fs(l)r Ft(\)\))40 b Fs(>)d(F)13 b Ft(\()p Fs(c)p Ft(\(0\)\))40 b Fs(>)d(C)1177 5476 y Fv(0)1218 5462 y Fo(,)k(wher)-5 b(e)41 b Fs(C)1616 5476 y Fv(0)1696 5462 y Fo(is)e(sp)-5 b(e)g(ci\014e)g(d)41 b(in)e(1.2.)63 b(L)-5 b(et)40 b Ft(0)e Fq(\024)f Fs(h)h Fq(2)g Fs(W)3132 5429 y Fv(2)p Fr(;n)3238 5462 y Ft(\()p Fs(B)3347 5429 y Fr(c)3384 5462 y Ft(\))i Fo(satisfy)p eop %%Page: 13 13 13 12 bop 50 295 a Fj(Sp)l(e)l(ctr)l(a)3340 b Fm(13)50 513 y Ft(\()p Fs(L)147 527 y Fr(\017)202 513 y Fq(\000)20 b Fs(\025)p Ft(\))p Fs(h)26 b Fq(\025)f Ft(0)p Fo(,)33 b Ft(0)26 b Fq(\024)e Fs(\025)i Fq(\024)f Ft(1)p Fo(,)33 b(in)f Fs(c)1258 480 y Fv(4)p Fr(r)r(\017\016)1401 513 y Fo(,)g Ft(0)26 b Fs(<)f(r)i Fq(\024)e Ft(1)p 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535 y Fr(B)s Fv(\()p Fr(x;)p Fv(4)p Fr(r)r(\017\016)r Fv(\()p Fr(x)p Fv(\)\))p Fi(\\)p Fv(\012)2535 545 y Fk(k)2597 513 y Fs(h)2649 480 y Fr(\025)2649 538 y Fv(\012)2701 516 y Fk(c)2701 562 y(k)2774 513 y Ft(if)28 b(it)g(attains)h(its)g(maximal)50 658 y(v)-5 b(alue)31 b(at)g(some)f Fs(x)c Fq(2)f Ft(\012)854 672 y Fr(k)898 658 y Fq(n)p Ft(\012)1009 672 y Fr(k)r Fi(\000)p Fv(2)1147 658 y Ft(.)42 b(It)30 b(hence)h(su\016ces)g(to)g(consider)f Fs(k)2387 672 y Fv(0)2454 658 y Fs(>)25 b Ft(1.)42 b(The)30 b(de\014nition)h(of)f Fs(r)k Ft(implies)50 804 y(sup)187 826 y Fv(\000)232 836 y Fk(k)291 804 y Fs(h)343 818 y Fr(k)r Fi(\000)p Fv(1)519 804 y Fs(<)j Ft(1)p Fs(=)p Ft(2)i(for)e(all)i(1)e Fq(\024)g Fs(k)k Fq(\024)c Fs(K)44 b Ft(b)m(y)38 b(\(2.15\))i(for)d Fs(h)2190 818 y Fr(k)2273 804 y Fq(\021)g Fs(h)2433 771 y Fv(0)2433 833 y(\000)2478 843 y Fk(k)q Fg(\000)p Fl(1)2598 833 y Fr(;)p Fi(f)p Fr(F)10 b Fi(\024)p Fr(\013)p Fi(g)2896 804 y Fq(\024)37 b Ft(1.)63 b(Application)39 b(of)50 949 y(\(2.17\))32 b(to)g Fs(w)g Ft(in)e(\000)c Fq(\021)e Ft(\012)871 963 y Fr(k)910 972 y Fl(0)945 963 y Fi(\000)p Fv(1)1073 949 y Ft(for)30 b Fs(K)i Fq(\021)25 b Ft(\012)1483 963 y Fr(k)1522 972 y Fl(0)1592 949 y Ft(then)30 b(giv)m(es)1143 1185 y(sup)1148 1261 y Fv(\012)1200 1271 y Fk(k)1234 1285 y Fl(0)1295 1185 y Fs(w)1362 1147 y Fr(\025)1435 1185 y Fq(\024)25 b Fs(s)1574 1147 y Fr(\025)1574 1207 y Fv(\012)1626 1185 y Fk(c)1626 1232 y(k)1660 1246 y Fl(0)1695 1232 y Fg(\000)p Fl(1)1805 1185 y Ft(+)20 b(sup)1905 1261 y Fv(\000)1950 1271 y Fk(k)1984 1285 y Fl(0)2048 1185 y Fs(h)2100 1199 y Fr(k)2139 1208 y Fl(0)2174 1199 y Fi(\000)p Fv(1)2318 1185 y Ft(sup)2287 1261 y Fv(\000)2332 1271 y Fk(k)2366 1285 y Fl(0)2402 1271 y Fg(\000)p Fl(1)2502 1185 y Fs(w)2569 1147 y Fr(\025)3477 1185 y Ft(\(2)p Fs(:)p Ft(24\))50 1470 y(so)35 b(that)g(sup)504 1492 y Fv(\006)576 1470 y Fs(w)643 1437 y Fr(\025)723 1470 y Fq(\024)e Ft(2)p Fs(s)915 1484 y Fv(0)955 1470 y Ft(,)k(where)d Fs(s)1327 1484 y Fv(0)1400 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b(\(2.34\))k(w)m(e)e(obtain)g(the)g(estimate)g(since)50 3298 y Fs(c)89 3312 y Fr(n)164 3298 y Fq(\024)25 b Fs(M)10 b Ft(.)41 b Fq(})p eop %%Page: 17 17 17 16 bop 50 295 a Fj(Sp)l(e)l(ctr)l(a)3340 b Fm(17)50 513 y Fw(3.)55 b(Gro)m(wth)41 b(estimates)f(at)g(in\014nit)m(y)159 731 y Ft(The)e(strong)h(drift)f(of)h Fq(r)p Fs(F)51 b Ft(to)m(w)m(ards)40 b(the)f(lo)s(cal)h(minima)c(of)j Fs(F)52 b Ft(sho)m(ws)38 b(that)i(the)f(in\015uence)f(of)h(the)50 876 y(v)-5 b(alues)31 b(of)f(a)h(solution)g Fs(\036)f Ft(of)h(\(2.1\))h(at)f(in\014nit)m(y)f(on)g(its)h(v)-5 b(alues)31 b(in)f(compact)h(sets)f(can)h(b)s(e)f(neglected.)50 1095 y Fu(A)38 b(priori)f(b)s(ounds)j(on)e(conditioned,)h(exp)s(ected)g (exit)e(times)g(from)i(b)s(ounded)g(sets:)47 b Ft(In)33 b(this)50 1240 y(section)c(w)m(e)g(pro)m(v)m(e)h(an)e(estimate)h(on)f (the)h(suprema)d Fs(s)1897 1207 y Fv(0)1897 1265 y Fr(K)1967 1240 y Ft(\()p Fs(A;)15 b(B)5 b Ft(\))29 b(for)f(regular,)i(closed)f (sets)g Fs(A)f Ft(and)g Fs(B)33 b Ft(with)50 1385 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b(the)g(Hopf)7 b('s)35 b(Maxim)m(um)f(Principle)h(w)m(e)g(again)h(ma)m(y)f(tak)m(e)i(the)e (suprem)m(um)c(o)m(v)m(er)37 b Fp(R)3136 4920 y Fr(d)3185 4953 y Fq(n)p Fs(A)p Fq(n)p Fs(B)j Ft(in)35 b(this)50 5098 y(inequalit)m(y)-8 b(.)42 b(The)30 b(assertion)h(th)m(us)f(follo)m (ws)i(from)d(\(3.33\))j(and)e(\(3.3\).)43 b Fq(})159 5316 y Ft(W)-8 b(e)28 b(no)m(w)f(turn)f(to)h(the)g(in)m(v)m(estigation) j(of)d(Laplace)i(transforms)c(in)h(the)h(non-compact)g(case.)41 b(Exploit-)50 5462 y(ing)30 b(the)h(strong)f(drift)g(to)m(w)m(ards)h (the)g(lo)s(cal)h(minima)c(of)i Fs(F)13 b Ft(,)31 b(w)m(e)g(can)g(pro)m (v)m(e)p eop %%Page: 24 24 24 23 bop 50 295 a Fm(24)3270 b Fj(Se)l(ction)27 b Fm(3)50 513 y Fe(Prop)s(osition)51 b(3.6:)59 b Fo(Assume)41 b(that)i Fs(F)54 b Fo(satis\014es)42 b(hyp)-5 b(othesis)44 b(1.2)d(and)i(let)e Ft(\012)g Fo(b)-5 b(e)41 b(a)g(r)-5 b(e)g(gular,)45 b(op)-5 b(en,)50 658 y(c)g(onne)g(cte)g(d)36 b(set)e Ft(\012)f Fo(indep)-5 b(endent)36 b(of)f Fs(\017)27 b(>)h Ft(0)35 b Fo(and)g(c)-5 b(ontaining)35 b Fq(f)p Fs(F)42 b(<)27 b(C)2494 672 y Fv(0)2535 658 y Fq(g)p Fo(,)35 b(wher)-5 b(e)35 b Fs(C)2966 672 y Fv(0)3041 658 y Fo(is)f(de\014ne)-5 b(d)35 b(in)f(1.2.)50 804 y(Fix)g Fs(I)7 b Fq(\032)o(M)34 b Fo(and)g(let)g Ft(\012)845 818 y Fv(0)919 804 y Fo(b)-5 b(e)33 b(a)h(union)f(of)h Fq(j)p Fs(I)7 b Fq(j)34 b Fo(b)-5 b(al)5 b(ls)34 b Fs(B)1882 818 y Fr(y)1953 804 y Fq(\021)26 b Fs(B)5 b Ft(\()h(~)-51 b Fs(y)s(;)15 b(\017=)p Ft(4\))p Fq(\032)q Fs(B)5 b Ft(\()p Fs(y)s(;)2678 738 y Fq(p)p 2754 738 37 4 v 66 x Fs(\017)p Ft(\))p Fo(,)34 b Fs(y)29 b Fq(2)e Fs(I)7 b Fo(.)44 b(Ther)-5 b(e)34 b(is)f Fs(N)k Fq(\021)50 949 y Fs(N)10 b Ft(\()p Fs(d)p Ft(\))35 b Fo(such)f(that)i(for)f(al)5 b(l)35 b Fs(C)1025 963 y Fv(2)1094 949 y Fs(>)27 b(C)1257 963 y Fv(0)1332 949 y Fo(and)36 b Fs(R)29 b(>)f Ft(0)34 b Fo(we)h(\014nd)f Fs(C)h Fq(\021)28 b Fs(C)7 b Ft(\()p Fs(d;)15 b(F)e Fq(jf)p Fs(F)43 b(<)28 b(C)2905 963 y Fv(2)2946 949 y Fq(g)p Fs(;)15 b Fq(jf)p Fs(F)42 b(<)28 b(C)3365 963 y Fv(2)3406 949 y Fq(gj)p Fs(;)15 b(R)q(;)g(\032)p Ft(\))50 1095 y Fo(with)27 b(the)g(fol)5 b(lowing)28 b(pr)-5 b(op)g(erty.)43 b(F)-7 b(or)27 b(al)5 b(l)28 b Ft(0)d Fq(\024)g Fs(\025)h Fq(\024)f Fs(\017)1823 1062 y Fr(N)1891 1095 y Fs(=T)1989 1109 y Fr(I)2032 1095 y Fo(,)i(al)5 b(l)27 b(non-ne)-5 b(gative)28 b Fs(f)34 b Fq(2)25 b Fs(L)2965 1062 y Fi(1)3042 1095 y Ft(\()p Fp(R)3137 1062 y Fr(d)3185 1095 y Ft(\))p Fq(n)p Ft(0)j Fo(satisfying)50 1240 y Ft(sup)187 1262 y Fr(@)t(B)285 1271 y Fk(y)343 1240 y Fs(f)35 b Fq(\024)25 b Fs(R)16 b Ft(inf)714 1254 y Fr(@)t(B)812 1263 y Fk(y)871 1240 y Fs(f)42 b Fo(and)34 b(al)5 b(l)33 b Fs(x)25 b Fq(2)g Ft(\000)g Fq(\021)g Ft(\012)p Fq(nf)p Fs(F)39 b Fq(\025)25 b Fs(C)2020 1254 y Fv(0)2060 1240 y Fq(gn)p 2150 1167 66 4 v Ft(\012)2217 1254 y Fv(0)2290 1240 y Fo(it)33 b(fol)5 b(lows)636 1457 y Fs(H)719 1420 y Fr(\025)712 1496 y Fv(\012)p Fi(n)p 801 1442 52 4 v Fv(\012)853 1505 y Fl(0)893 1457 y Ft(1)-23 b(I)948 1471 y Fr(@)t Fv(\012)1043 1480 y Fl(0)1082 1457 y Fs(f)10 b Ft(\()p Fs(x)p Ft(\))26 b Fq(\024)f Ft(\(1)c(+)f Fs(C)7 b(\017)1682 1420 y Fi(\000)p Fr(N)1807 1457 y Ft(\()p Fs(\025T)1948 1471 y Fr(I)2010 1457 y Ft(+)20 b Fs(e)2143 1420 y Fi(\000)p Fv(\()p Fr(C)2280 1429 y Fl(2)2316 1420 y Fi(\000)p Fr(C)2425 1429 y Fl(0)2461 1420 y Fv(\))p Fr(=\017)2561 1457 y Ft(\)\))p Fs(H)2714 1420 y Fv(0)2707 1480 y(\000)2757 1457 y Ft(1)-23 b(I)2812 1471 y Fr(@)t Fv(\012)2907 1480 y Fl(0)2947 1457 y Fs(f)10 b Ft(\()p Fs(x)p Ft(\))353 b(\(3)p Fs(:)p Ft(35\))50 1674 y Fo(and)33 b(for)h(al)5 b(l)33 b Fs(y)28 b Fq(2)d Fs(I)137 1892 y(w)204 1854 y Fr(\025)202 1914 y(B)257 1923 y Fk(y)296 1914 y Fr(;B)373 1923 y Fk(I)409 1914 y Fi([)p Fv(\012)510 1896 y Fk(c)549 1892 y Ft(\()p Fs(x)p Ft(\))h Fq(\024)f Ft(\(1)c(+)f Fs(C)7 b(\017)1094 1854 y Fi(\000)p Fr(N)1219 1892 y Fs(\025T)1325 1906 y Fr(I)1367 1892 y Ft(\))p Fs(w)1469 1854 y Fv(0)1467 1914 y Fr(B)1522 1923 y Fk(y)1561 1914 y Fr(;B)1638 1923 y Fk(I)1675 1914 y Fi([)p Fv(\012)1776 1892 y Fk(c)1776 1936 y Fl(2)1815 1892 y Ft(\()p Fs(x)p Ft(\))21 b(+)f Fs(C)i Ft(sup)2164 1968 y Fv(\000)2209 1977 y Fl(1)2288 1892 y Fs(w)2355 1854 y Fv(0)2353 1914 y Fr(B)2408 1923 y Fk(y)2447 1914 y Fr(;B)2524 1923 y Fk(I)2560 1914 y Fi([)p Fv(\012)2661 1892 y Fk(c)2661 1936 y Fl(2)2701 1892 y Fs(\017)2738 1854 y Fi(\000)p Fr(N)2864 1892 y Fs(e)2906 1854 y Fi(\000)p Fv(\()p Fr(C)3043 1863 y Fl(2)3079 1854 y Fi(\000)p Fr(F)10 b Fv(\()p Fr(x)p Fv(\)\))p Fr(=\017)3477 1892 y Ft(\(3)p Fs(:)p Ft(36\))159 2327 y(Before)28 b(w)m(e)f(turn)e(to)j(the)f(pro)s (of)f(of)h(this)f(Prop)s(osition,)i(w)m(e)f(note)g(the)g(follo)m(wing)h (a)g(priori)e(lo)m(w)m(er)i(b)s(ound)50 2473 y(on)c(principal)g(eigen)m (v)-5 b(alues)26 b(in)e(un)m(b)s(ounded)e(domains,)i(whic)m(h)g(is)g (an)g(immediate)g(consequence)h(of)f(\(3.35\).)50 2691 y Fe(Corollary)40 b(3.7:)51 b Fo(In)33 b(the)g(situation)g(of)g(the)g (pr)-5 b(evious)34 b(Pr)-5 b(op)g(osition)35 b(we)e(have)1196 2929 y Fs(\025)p Ft(\(\012)p Fq(n)p 1395 2856 74 4 v Fs(B)1478 2940 y Fv(~)1469 2957 y Fr(I)1511 2929 y Ft(\))26 b Fq(\025)1678 2868 y Fs(\017)1715 2835 y Fi(\000)p Fr(N)p 1678 2908 163 4 v 1712 2991 a Fs(T)1765 3005 y Fr(I)1852 2929 y Fs(;)201 b(B)2155 2940 y Fv(~)2147 2957 y Fr(I)2214 2929 y Fq(\021)2322 2843 y Fh([)2310 3038 y Fr(y)r Fi(2)p Fr(I)2451 2929 y Fs(B)2520 2943 y Fr(y)3477 2929 y Ft(\(3)p Fs(:)p Ft(37\))159 3364 y(Let)44 b(us)f(no)m(w)h(sp)s(ecify)f(ab)s(out) h(the)g(c)m(hoice)h(of)f(\(\012)1927 3378 y Fr(k)1972 3364 y Ft(\))2007 3378 y Fv(0)p Fi(\024)p Fr(k)r Fi(\024)p Fr(L)p Fv(+1)2387 3364 y Ft(for)g(whic)m(h)f(w)m(e)h(w)m(an)m(t)h(to)f (apply)g(the)50 3510 y(random)26 b(w)m(alk)j(represen)m(tation)g (written)f(in)f(\(3.24\).)42 b(Dep)s(ending)28 b(on)f(\012)h(and)f Fs(I)7 b Fq(\032M)28 b Ft(w)m(e)g(de\014ne)g(\012)3457 3524 y Fv(0)3525 3510 y Ft(as)g(in)50 3655 y(Prop)s(osition)i(3.6,)i (i.e.)1231 3801 y(\012)1297 3815 y Fv(0)1363 3801 y Fq(\021)1471 3714 y Fh([)1459 3909 y Fr(y)r Fi(2)p Fr(I)1599 3801 y Fs(B)1668 3815 y Fr(y)1712 3801 y Fs(;)197 b(B)2003 3815 y Fr(y)2072 3801 y Fq(\021)25 b Fs(B)5 b Ft(\()h(~)-51 b Fs(y)s(;)15 b(\017=)p Ft(4\))950 b(\(3)p Fs(:)p Ft(38\))50 4072 y(F)-8 b(or)36 b Fs(\015)k(>)34 b Ft(0)i(let)h(\()p Fs(z)705 4086 y Fr(k)751 4072 y Ft(\))786 4086 y Fr(k)r Fi(\025)p Fv(1)960 4072 y Ft(b)s(e)e(a)h(sequence)g(in)p 1664 3999 66 4 v 36 w(\012)1730 4015 y Fr(c)1730 4095 y Fv(0)1806 4072 y Ft(suc)m(h)g(that)g(inf)2330 4086 y Fr(k)r Fi(\025)p Fv(2)2483 4072 y Fs(F)13 b Ft(\()p Fs(z)2631 4086 y Fr(k)r Fv(+1)2768 4072 y Ft(\))24 b Fq(\000)g Fs(F)13 b Ft(\()p Fs(z)3070 4086 y Fr(k)3115 4072 y Ft(\))35 b Fq(\025)f Fs(\015)40 b(>)34 b Ft(0)i(and)50 4218 y(suc)m(h)30 b(that)h Fs(F)13 b Ft(\()p Fs(z)600 4232 y Fv(1)641 4218 y Ft(\))26 b(=)f Fs(C)863 4232 y Fv(0)934 4218 y Ft(and)30 b Fs(F)13 b Ft(\()p Fs(z)1259 4232 y Fv(2)1300 4218 y Ft(\))26 b(=)f Fs(C)1522 4232 y Fv(2)1588 4218 y Fq(\025)g Fs(C)1749 4232 y Fv(0)1810 4218 y Fq(\000)20 b Fs(\015)5 b Ft(.)40 b(W)-8 b(e)32 b(de\014ne)1310 4435 y(\012)1376 4449 y Fr(k)1446 4435 y Fq(\021)25 b(f)p Fs(z)30 b Fq(2)25 b Ft(\012)15 b Fq(j)g Fs(F)e Ft(\()p Fs(z)t Ft(\))26 b Fs(<)f(F)13 b Ft(\()p Fs(z)2323 4449 y Fr(k)2369 4435 y Ft(\))p Fq(g)1028 b Ft(\(3)p Fs(:)p Ft(39\))50 4652 y(The)42 b(follo)m(wing)j(consequence)f (of)f(Prop)s(osition)g([4.7)i(in)e(BEGK3])h(can)f(b)s(e)g(used)f(to)i (sho)m(w)f(that)h(the)50 4798 y(random)28 b(w)m(alk)k(expansion)e (\(3.24\))i(con)m(v)m(erges)h(exp)s(onen)m(tially)e(fast.)50 5016 y Fe(Lemma)k(3.8:)50 b Fo(Ther)-5 b(e)33 b(is)e Fs(C)h Fq(\021)25 b Fs(C)7 b Ft(\()p Fs(d)p Ft(\))32 b Fo(such)g(that)h(for)g(al)5 b(l)32 b Fs(\013)26 b(<)f(\014)f Ft(+)18 b Fs(\017)d Ft(log)i Fs(\017)32 b Fo(and)h(al)5 b(l)32 b(r)-5 b(e)g(gular,)33 b(op)-5 b(en)33 b(sets)50 5161 y Ft(\006)p Fq(\032)o Fs(K)7 b Fq(\032)p Ft(\000)32 b Fo(satisfying)h Fs(K)7 b Fq(\032f)p Fs(F)38 b Fq(\024)25 b Fs(\013)p Fq(g)34 b Fo(and)f Fq(f)p Fs(F)39 b(<)25 b(\014)5 b Fq(g\032)q Ft(\000)965 5379 y Fq(jj)p Ft(1)-23 b(I)1071 5409 y Fr(K)t Fi(n)p 1173 5355 52 4 v Fv(\006)1230 5379 y Fs(H)1313 5341 y Fv(0)1306 5417 y(\000)p Fi(n)p 1388 5363 V Fv(\006)1444 5379 y Ft(1)g(I)1499 5393 y Fr(@)t Fv(\000)1592 5379 y Fq(jj)25 b Ft(=)34 b(sup)1763 5473 y Fr(K)t Fi(n)p 1865 5419 V Fv(\006)1933 5379 y Fs(h)1985 5341 y Fv(0)1985 5401 y(\000)2030 5383 y Fk(c)2065 5401 y Fr(;)p Fv(\006)2168 5379 y Fq(\024)25 b Fs(C)7 b(e)2378 5341 y Fi(\000)p Fv(\()p Fr(\014)s Fi(\000)p Fr(\013)p Fv(\))p Fr(=\017)2712 5379 y Fs(=\017)683 b Ft(\(3)p Fs(:)p Ft(40\))p eop %%Page: 25 25 25 24 bop 50 295 a Fj(Sp)l(e)l(ctr)l(a)3340 b Fm(25)50 731 y Fd(Pro)s(of:)47 b Ft(It)29 b(follo)m(ws)h(from)e(Prop)s(ositions) i([4.3)g(and)f(4.7)h(in)f(BEGK3])i(the)e(existence)i(of)e Fs(C)7 b Ft(\()p Fs(d)p Ft(\))30 b(suc)m(h)f(that)50 876 y(for)h(all)h Fs(x)25 b Fq(2)g Fs(K)344 1074 y(h)396 1037 y Fv(0)396 1097 y(\000)441 1078 y Fk(c)476 1097 y Fr(;)p Fv(\006)554 1074 y Ft(\()p Fs(x)p Ft(\))g Fq(\024)g Fs(C)963 1001 y Ft(cap)1100 1023 y Fr(B)s Fv(\()p Fr(x;\016)r Fv(\()p Fr(x)p Fv(\))p Fr(\017)p Fv(\))1445 1001 y Ft(\(\000)1537 968 y Fr(c)1575 1001 y Ft(\))p 880 1053 814 4 v 880 1137 a(cap)1016 1159 y Fr(B)s Fv(\()p Fr(x;\016)r Fv(\()p Fr(x)p Fv(\))p Fr(\017)p Fv(\))1362 1137 y Ft(\(\000)1454 1110 y Fr(c)1511 1137 y Fq([)20 b Ft(\006\))1729 1074 y Fq(\024)2239 1013 y Fs(C)p 1836 1053 877 4 v 1836 1137 a(\016)s Ft(\()p Fs(x)p Ft(\))15 b(sup)2155 1159 y Fr(B)s Fv(\()p Fr(x;\017\016)r Fv(\()p Fr(x)p Fv(\)\))2516 1137 y Fq(jr)p Fs(F)e Fq(j)2724 1074 y Fs(e)2766 1037 y Fi(\000)p Fv(\()p Fr(\014)s Fi(\000)p Fr(\013)p Fv(\))p Fr(=\017)3101 1074 y Fs(=\017)294 b Ft(\(3)p Fs(:)p Ft(41\))50 1295 y(\(3.40\))32 b(holds)e(since)h Fs(\016)s Ft(\()p Fs(x)p Ft(\))15 b(sup)1092 1318 y Fr(B)s Fv(\()p Fr(x;\017\016)r Fv(\()p Fr(x)p Fv(\)\))1453 1295 y Fq(jr)p Fs(F)e Fq(j)25 b Ft(=)g(1)p Fs(=)p Ft(8.)42 b Fq(})50 1514 y Fd(Pro)s(of)36 b(of)f(Prop)s(osition)f(3.6:)87 b Ft(Cho)s(ose)1615 1491 y(~)1589 1514 y Fs(N)40 b Ft(and)30 b(de\014ne)f Fs(\015)i Fq(\021)2341 1491 y Ft(~)2314 1514 y Fs(N)10 b(\017)15 b Ft(log)s(\(1)p Fs(=\017)p Ft(\))22 b(+)e Fs(\017)15 b Ft(log)i Fs(T)3114 1528 y Fr(I)567 1755 y Fs(h)619 1718 y Fr(\025)619 1778 y(L)674 1755 y Ft(\()p Fs(x)p Ft(\))26 b Fq(\021)f Fs(H)1001 1718 y Fr(\025)994 1778 y Fv(\000)1039 1787 y Fl(1)1078 1755 y Ft(1)-23 b(I)1134 1769 y Fr(@)t Fv(\012)1229 1778 y Fl(0)1268 1755 y Fs(f)10 b Ft(\()p Fs(x)p Ft(\))21 b(+)1593 1641 y Fi(1)1563 1669 y Fh(X)1556 1864 y Fr(T)10 b Fv(=1)1765 1669 y Fh(X)1745 1850 y Fk(!)r Fl(:1)p Fg(\030)p Fl(1)1727 1901 y Fg(j)p Fk(!)r Fg(j)p Fl(=2)p Fk(T)1960 1641 y Fv(2)p Fr(T)g Fi(\000)p Fv(1)1993 1669 y Fh(Y)1992 1861 y Fr(t)p Fv(=1)2157 1755 y Fs(H)2240 1718 y Fr(\025)2233 1778 y Fv(\000)2278 1787 y Fk(!)2318 1799 y(t)2358 1755 y Ft(1)-23 b(I)2413 1769 y Fr(@)t Fv(\012)2508 1778 y Fk(!)2548 1793 y(t)p Fl(+1)2665 1755 y Fs(H)2748 1718 y Fr(\025)2741 1778 y Fv(\000)2786 1787 y Fl(1)2825 1755 y Ft(1)g(I)2881 1769 y Fr(@)t Fv(\012)2976 1778 y Fl(0)3016 1755 y Fs(f)10 b Ft(\()p Fs(x)p Ft(\))3477 1778 y(\(3)p Fs(:)p Ft(42\))50 2043 y(F)-8 b(or)26 b(eac)m(h)g(sequence)f Fs(!)j Ft(:)e(1)g Fq(\030)f Ft(1)g(of)g(length)h(2)p Fs(T)38 b(>)25 b Ft(0)h(there)f(are) g(precisely)h Fs(T)38 b Ft(steps)25 b(of)g(the)g(form)f(\()p Fs(k)s(;)15 b(k)d Ft(+)e(1\),)50 2188 y(1)34 b Fq(\024)f Fs(k)k(<)c(L)p Ft(.)55 b(F)-8 b(urthermore,)36 b(for)f(some)g(univ)m (ersal)h(constan)m(t)g Fs(C)42 b Ft(there)35 b(are)h(at)g(most)f(\(2)p Fs(T)13 b Ft(\)!)p Fs(=)p Ft(\(2)p Fs(T)g Ft(!)3527 2155 y Fv(2)3570 2188 y Ft(\))34 b Fq(\024)50 2333 y Fs(C)7 b Ft(4)167 2301 y Fr(T)224 2333 y Fs(=)269 2257 y Fq(p)p 345 2257 66 4 v 76 x Fs(T)47 b Ft(suc)m(h)33 b(sequences.)51 b(Hence,)36 b(com)m(bination)e(of)g(\(3.40\))i(with)e(\(3.28\))i(for)d (\000)e Fq(\021)g Ft(\000)3171 2347 y Fr(k)3215 2333 y Ft(,)k Fs(k)f Fq(\025)d Ft(3,)k(and)50 2479 y(\(3.32\))d(for)f(\000) 25 b Fq(\021)g Ft(\000)687 2493 y Fv(1)727 2479 y Fs(;)15 b Ft(\000)824 2493 y Fv(2)896 2479 y Ft(giv)m(es)616 2636 y(sup)644 2712 y Fv(\000)689 2721 y Fl(1)768 2636 y Fs(h)820 2598 y Fr(\025)820 2658 y(L)900 2636 y Fq(\024)971 2562 y Fh(\000)1013 2636 y Ft(1)20 b(+)g Fs(\025C)7 b(\017)1331 2598 y Fi(\000)p Fr(N)1456 2636 y Ft(\()p Fs(T)1544 2650 y Fr(I)1607 2636 y Ft(+)20 b Fq(jf)p Fs(F)39 b Fq(\024)25 b Fs(F)13 b Ft(\()p Fs(z)2109 2650 y Fv(2)2150 2636 y Ft(\))p Fq(gj)p Ft(\))2290 2562 y Fh(\001)2348 2636 y Ft(sup)2352 2713 y Fr(@)t Fv(\012)2447 2722 y Fl(0)2500 2636 y Fs(f)991 2863 y Ft(+)1082 2790 y Fh(\000)1124 2863 y Ft(1)20 b(+)g Fs(\025C)7 b(\017)1442 2826 y Fi(\000)p Fr(N)1568 2863 y Ft(\()p Fs(T)1656 2877 y Fr(I)1718 2863 y Ft(+)20 b Fq(jf)p Fs(F)39 b Fq(\024)25 b Fs(F)13 b Ft(\()p Fs(z)2220 2877 y Fv(2)2261 2863 y Ft(\))p Fq(gj)p Ft(\))2401 2790 y Fh(\001)2444 2809 y Fv(2)2500 2863 y Ft(sup)2504 2940 y Fr(@)t Fv(\012)2599 2949 y Fl(0)2652 2863 y Fs(f)991 3145 y Fq(\002)1118 3031 y Fi(1)1089 3059 y Fh(X)1082 3254 y Fr(T)d Fv(=1)1242 3145 y Fs(C)d Ft(4)1359 3108 y Fr(T)1416 3145 y Ft(\(1)21 b(+)f Fs(C)7 b(\025e)1775 3108 y Fr(C)t(\015)t(=\017)1946 3145 y Ft(\))1981 3108 y Fv(2\()p Fr(T)j Fi(\000)p Fv(1\))2225 3145 y Fs(C)2297 3108 y Fr(T)2353 3145 y Fs(\017)2390 3108 y Fv(1)p Fi(\000)p Fr(T)2540 3145 y Fs(e)2582 3108 y Fi(\000)p Fr(\015)t Fv(\()p Fr(T)g Fi(\000)p Fv(1\))p Fr(=\017)2956 3145 y Fs(=)3001 3064 y Fq(p)p 3078 3064 V 3078 3145 a Fs(T)3477 2931 y Ft(\(3)p Fs(:)p Ft(43\))50 3399 y(where)41 b(w)m(e)i(ha)m(v)m(e) g(used)e(that)i Fs(\017)1156 3366 y Fi(\000)p Fr(N)1309 3399 y Ft(+)28 b Fq(jf)p Fs(F)13 b Ft(\()p Fs(z)1626 3413 y Fv(1)1668 3399 y Ft(\))45 b Fq(\024)f Fs(F)58 b Fq(\024)44 b Fs(F)13 b Ft(\()p Fs(z)2242 3413 y Fv(2)2284 3399 y Ft(\))p Fq(gj)45 b Fs(<)f(C)7 b(e)2663 3366 y Fr(C)t(\015)t(=\017)2876 3399 y Ft(for)42 b(some)3292 3376 y(~)3266 3399 y Fs(N)55 b Fq(\021)3535 3376 y Ft(~)3509 3399 y Fs(N)10 b Ft(\()p Fs(d)p Ft(\))50 3545 y(and)36 b(some)g(constan)m(t)i Fs(C)k Fq(\021)36 b Fs(C)7 b Ft(\()p Fs(d;)15 b Fq(jf)p Fs(F)50 b Fq(\024)35 b Fs(F)13 b Ft(\()p Fs(z)1680 3559 y Fv(2)1722 3545 y Ft(\))p Fq(gj)p Ft(\).)61 b(Cho)s(osing)36 b Fs(N)46 b Fq(\021)35 b Fs(N)10 b Ft(\()p Fs(d)p Ft(\))38 b(appropriately)-8 b(,)39 b(w)m(e)e(ma)m(y)50 3690 y(estimate)44 b Fs(C)7 b(\025e)594 3657 y Fr(C)t(\015)t(=\017)808 3690 y Ft(b)m(y)43 b(one)h(for)f(all)h(0)j Fq(\024)g Fs(\025)f(<)h(\017)1882 3657 y Fr(N)1951 3690 y Fs(T)2004 3704 y Fr(I)2046 3690 y Ft(.)79 b(\(3.43\))46 b(then)d(tells)h(us)f (for)g(some)f(constan)m(t)50 3835 y Fs(C)32 b Fq(\021)25 b Fs(C)7 b Ft(\()p Fs(d;)15 b Fq(jf)p Fs(F)39 b Fq(\024)25 b Fs(F)13 b Ft(\()p Fs(z)848 3849 y Fv(2)890 3835 y Ft(\))p Fq(gj)p Ft(\))1547 3981 y(sup)1575 4057 y Fv(\000)1620 4066 y Fl(1)1699 3981 y Fs(h)1751 3943 y Fr(\025)1824 3981 y Fq(\024)25 b Fs(C)d Ft(sup)2010 4058 y Fr(@)t Fv(\012)2105 4067 y Fl(0)2159 3981 y Fs(f)1273 b Ft(\(3)p Fs(:)p Ft(44\))50 4197 y(Since)23 b Fs(h)332 4164 y Fr(\025)332 4222 y(L)410 4197 y Ft(is)h(a)f(b)s(ounded,)g(analytic)i(con)m(tin)m (uation)g(of)f Fq(f)p 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Fs(h)2632 5243 y Fv(0)2632 5320 y(\012)2684 5298 y Fk(c)2684 5342 y Fl(2)2720 5320 y Fr(;)p 2742 5265 52 4 v Fv(\012)2794 5329 y Fl(0)2834 5275 y Ft(\()p Fs(x)p Ft(\))1367 5466 y Fq(\024)p Ft(1)h(+)e Fs(C)7 b(e)1708 5429 y Fi(\000)p Fv(\()p Fr(F)j Fv(\()p Fr(z)1912 5438 y Fl(2)1948 5429 y Fv(\))p Fi(\000)p Fr(F)g Fv(\()p Fr(z)2152 5438 y Fl(1)2188 5429 y Fv(\)\))p Fr(=\017)2317 5466 y Fs(=\017)3477 5289 y Ft(\(3)p Fs(:)p Ft(46\))p eop %%Page: 26 26 26 25 bop 50 295 a Fm(26)3270 b Fj(Se)l(ction)27 b Fm(3)50 513 y Ft(where)j(w)m(e)g(ha)m(v)m(e)i(used)e(Corollary)i([4.7)f(in)g (BEGK3])g(in)f(the)g(latter)i(inequalit)m(y)-8 b(.)43 b(Since)30 b(the)g(constan)m(t)i(in)50 658 y(\(3.46\))f(do)s(es)d(not)h (dep)s(end)e(on)i Fs(L)p Ft(,)g Fs(h)1281 625 y Fr(\025)1281 683 y(L)1336 658 y Ft(\()p Fs(x)p Ft(\))g(con)m(v)m(erges)i (increasingly)f(to)f(a)g(\()p Fs(L)2670 672 y Fr(\017)2723 658 y Fq(\000)16 b Fs(\025)p Ft(\))29 b(-)g(harmonic)f(function)50 804 y(on)d(\012)p Fq(n)p 282 731 66 4 v Ft(\012)348 818 y Fv(0)414 804 y Ft(whic)m(h)g(equals)h(1)-23 b(I)995 818 y Fr(@)t Fv(\012)1090 827 y Fl(0)1130 804 y Fs(f)35 b Ft(on)25 b Fs(@)5 b Ft(\012)10 b Fq([)g Fs(@)5 b Ft(\012)1650 818 y Fv(0)1716 804 y Ft(and)24 b(th)m(us)h(equals)h Fs(H)2436 771 y Fr(\025)2429 847 y Fv(\012)p Fi(n)p 2518 792 52 4 v Fv(\012)2570 856 y Fl(0)2610 804 y Ft(1)-23 b(I)2665 818 y Fr(@)t Fv(\012)2760 827 y Fl(0)2800 804 y Fs(f)10 b Ft(.)38 b(T)-8 b(o)26 b(deriv)m(e)g(the)g(result,)50 952 y(it)39 b(remains)e(to)j(compare)e Fs(H)1071 919 y Fr(\025)1064 977 y Fv(\000)1109 986 y Fl(1)1149 952 y Ft(1)-23 b(I)1204 966 y Fr(@)t Fv(\012)1299 975 y Fl(0)1339 952 y Fs(f)48 b Ft(with)38 b Fs(H)1730 919 y Fv(0)1723 977 y(\000)1768 986 y Fl(1)1808 952 y Ft(1)-23 b(I)1863 966 y Fr(@)t Fv(\012)1958 975 y Fl(0)1998 952 y Fs(f)10 b Ft(.)65 b(But)39 b(\(3.32\))i(once)e(more)f(for)g Fs(A)h Fq(\021)p 3418 879 66 4 v 39 w Ft(\012)3484 966 y Fv(0)3563 952 y Ft(and)50 1098 y Fs(B)46 b Fq(\021)c Ft(\012)344 1065 y Fr(c)344 1120 y Fv(2)425 1098 y Ft(sho)m(ws)e(after)h(p)s (ossibly)f(increasing)h Fs(N)50 b Ft(for)40 b(some)g Fs(C)49 b Fq(\021)41 b Fs(C)7 b Ft(\()p Fs(d)p Ft(\),)44 b(some)2974 1075 y(^)2948 1098 y Fs(N)52 b Fq(\021)3212 1075 y Ft(^)3186 1098 y Fs(N)10 b Ft(\()p Fs(d)p Ft(\))41 b(and)f(all)50 1243 y Fs(x)25 b Fq(2)g Ft(\000)270 1257 y Fv(1)1286 1343 y Fs(H)1369 1310 y Fr(\025)1362 1368 y Fv(\000)1407 1377 y Fl(1)1447 1343 y Ft(1)-23 b(I)1502 1357 y Fr(@)t Fv(\012)1597 1366 y Fl(0)1637 1343 y Fs(f)10 b Ft(\()p Fs(x)p Ft(\))p 1286 1392 528 4 v 1286 1475 a Fs(H)1369 1444 y Fv(0)1362 1502 y(\000)1407 1511 y Fl(1)1447 1475 y Ft(1)-23 b(I)1502 1489 y Fr(@)t Fv(\012)1597 1498 y Fl(0)1637 1475 y Fs(f)10 b Ft(\()p Fs(x)p Ft(\))1850 1413 y Fq(\024)25 b Ft(1)c(+)e Fs(C)7 b(\017)2211 1375 y Fi(\000)2288 1359 y Fv(^)2268 1375 y Fr(N)2337 1413 y Fs(\025T)2443 1427 y Fr(I)3477 1413 y Ft(\(3)p Fs(:)p Ft(47\))50 1627 y(again)31 b(b)m(y)g(the)f(condition)h(on)f Fs(\025)h Ft(.)40 b(This)30 b(pro)m(v)m(es)h(the)g(\014rst)e(equation)i (in)f(\(3.35\).)159 1845 y(Di\013eren)m(tiation)j(with)d(resp)s(ect)g (to)i Fs(\025)e Ft(of)g(\(3.45\))j(for)d Fs(f)35 b Fq(\021)25 b Ft(1)-23 b(I)2213 1859 y Fr(B)2268 1868 y Fk(x)2344 1845 y Ft(in)30 b(com)m(bination)g(with)1161 2021 y(_)1127 2044 y Fs(H)1210 2006 y Fr(\025)1203 2066 y Fv(\006)1284 2044 y Ft(=)25 b Fs(G)1451 2006 y Fr(\025)1451 2066 y Fv(\006)1508 2044 y Fs(H)1591 2006 y Fr(\025)1584 2066 y Fv(\006)1640 2044 y Fs(;)197 b Ft(\006)91 b(regular,)31 b(op)s(en)934 b(\(3)p Fs(:)p Ft(48\))50 2242 y(whic)m(h)30 b(follo)m(ws)h(from)e(taking)j(deriv)-5 b(ativ)m(es)31 b(in)f(\(1.3\),)j(yields)179 2494 y Fs(w)246 2456 y Fr(\025)244 2516 y(B)299 2525 y Fk(x)340 2516 y Fr(;B)417 2525 y Fk(I)453 2516 y Fi([)p Fv(\012)554 2494 y Fk(c)554 2540 y(L)p Fl(+1)684 2494 y Ft(\()p Fs(x)p Ft(\))26 b(=)f Fs(w)995 2456 y Fr(\025)993 2516 y(B)1048 2525 y Fk(x)1088 2516 y Fr(;B)1165 2525 y Fk(I)1201 2516 y Fi([)p Fv(\012)1302 2494 y Fk(c)1302 2538 y 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Fk(y)2749 5316 y Fs(\036)p Ft(.)45 b(\(3.35\))35 b(for)c Fs(f)37 b Fq(\021)27 b Ft(1)-23 b(I)3514 5330 y Fr(@)t(B)3612 5339 y Fk(y)3656 5316 y Fs(\036)50 5462 y Ft(is)30 b(the)h(last)g(equation)g(in)f (\(3.55\).)p eop %%Page: 28 28 28 27 bop 50 295 a Fm(28)3270 b Fj(Se)l(ction)27 b Fm(3)159 513 y Ft(F)-8 b(or)35 b(the)f(pro)s(of)f(of)i(\(3.56\))h(w)m(e)f (\014rst)e(note)i(that)f(Corollary)j([4.8)e(in)f(BEGK3])h(in)f(com)m (bination)g(with)50 658 y(\(3.55\))e(and)e(the)h(condition)g(on)f Fs(x)g Ft(implies)g(for)g(all)h Fs(y)d Fq(2)1997 635 y Ft(~)1986 658 y Fs(I)7 b Fq(n)p Fs(x)31 b Ft(and)f(all)h Fs(z)f Fq(2)24 b Ft(\006)h Fq(\021)g Fs(A)2876 614 y Fr(\014)2876 700 y(x;)2947 684 y Fv(~)2939 700 y Fr(I)978 916 y Fq(j)p Fs(\036)1057 930 y Fr(y)1101 916 y Ft(\()p Fs(z)t Ft(\))p Fq(j)h(\024)f Ft(2)p Fs(h)1461 879 y Fv(0)1461 939 y Fr(B)1516 948 y Fk(y)1556 939 y Fr(;B)1640 948 y Fl(~)1633 959 y Fk(I)1674 916 y Ft(\()p Fs(z)t Ft(\))17 b(sup)1805 993 y Fr(@)t(B)1903 1002 y Fk(x)1959 916 y Fq(j)p Fs(\036)p Fq(j)26 b(\024)f Fs(C)7 b(e)2299 879 y Fi(\000)p Fr(\014)s(=\017)2472 916 y Fq(j)p Fs(\036)p Ft(\()p Fs(x)p Ft(\))p Fq(j)p Fs(=\017)697 b Ft(\(3)p Fs(:)p Ft(57\))50 1193 y(where)23 b(w)m(e)h(ha)m(v)m(e)h(used)e(the)h (Harnac)m(k's)h(inequalit)m(y)g(to)g(replace)f(the)g(suprem)m(um)c(in)k (the)g(latter)h(inequalit)m(y)-8 b(.)50 1338 y(F)g(urthermore,)29 b(for)h(all)i Fs(z)d Fq(2)c Ft(\006)967 1593 y Fq(j)p Fs(\036)1046 1607 y Fr(x)1092 1593 y Ft(\()p Fs(z)t Ft(\))p Fq(j)i(\025)e Ft(\(1)p Fs(=)p Ft(2\))p Fs(h)1613 1555 y Fv(0)1613 1615 y Fr(B)1668 1624 y Fk(x)1711 1615 y Fr(;B)1795 1624 y Fl(~)1788 1635 y Fk(I)1828 1593 y Ft(\()p Fs(z)t Ft(\))15 b(inf)1968 1651 y Fr(B)2023 1660 y Fk(x)2086 1593 y Fq(j)p Fs(\036)2165 1607 y Fr(x)2211 1593 y Fq(j)26 b(\025)f Ft(\(1)p Fs(=)p Ft(3\))p Fq(j)p Fs(\036)p Ft(\()p Fs(x)p Ft(\))p Fq(j)688 b Ft(\(3)p Fs(:)p Ft(58\))50 1856 y(In)33 b(particular,)i Fs(\036)e Ft(do)s(es)g(not)h(c)m(hange)h (sign)e(in)h(\006.)49 b(In)33 b(view)h(of)g(\(3.55\))h(again)g(and)e (\(3.57\))j(w)m(e)e(no)m(w)f(ma)m(y)50 2001 y(c)m(ho)s(ose)28 b Fs(r)f Fq(\021)e Fs(C)7 b(\017)608 1968 y Fi(\000)p Fr(N)734 2001 y Ft(\()p Fs(\025T)883 2012 y Fv(~)875 2029 y Fr(I)930 2001 y Ft(+)13 b Fs(e)1056 1968 y Fi(\000)p Fr(\014)s(=\017)1229 2001 y Ft(\))28 b(in)e(\(2.12\))j(applied)e(to)g Fs(\036)p Ft(.)40 b(Since)27 b Fs(\025)g Ft(is)f(smaller)h(than)f(the)h (error)g(terms)50 2147 y(written)j(ab)s(o)m(v)m(e,)i(\(3.56\))h(follo)m (ws)e(from)e(\(2.12\).)43 b Fq(})p eop %%Page: 29 29 29 28 bop 50 295 a Fj(Sp)l(e)l(ctr)l(a)3340 b Fm(29)50 513 y Fw(4.)55 b(Small)40 b(eigen)m(v)-7 b(alues)159 731 y Ft(In)26 b(this)g(c)m(hapter)i(w)m(e)f(deriv)m(e)g(precise)g (asym)m(totics)h(of)f(the)f(exp)s(onen)m(tially)i(small)e(eigen)m(v)-5 b(alues)29 b(of)e Fs(L)3573 745 y Fr(\017)3634 731 y Ft(in)50 876 y(a)33 b(regular,)h(op)s(en)e(set)i(\012)e(indep)s(enden)m (t)g(of)h Fs(\017)g Ft(and)f(con)m(taining)j Fq(f)p Fs(F)43 b(<)29 b(C)2559 890 y Fv(0)2600 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Fq(\024)f Fs(\025)h(<)f(\017)j Ft(with)g(corresp)s(onding)f(eigenfunction)i Fs(\036)p Ft(.)50 1967 y(F)-8 b(or)30 b(eac)m(h)h Fs(x)25 b Fq(2)g(M)30 b Ft(\014x)k(~)-50 b Fs(x)25 b Fq(2)g Fs(B)5 b Ft(\()p Fs(x;)1210 1902 y Fq(p)p 1286 1902 37 4 v 65 x Fs(\017)p Ft(\))29 b(suc)m(h)h(that)g Fs(\036)g Ft(do)s(es)f(not)h(c) m(hange)g(sign)g(in)f Fs(B)5 b Ft(\()h(~)-51 b Fs(x;)15 b(\017)p Ft(\).)41 b(The)29 b(existence)50 2113 y(of)36 b(~)-51 b Fs(x)31 b Ft(is)f(guaran)m(teed)h(b)m(y)g(Lemma)d(2.3.)42 b(Let)1634 2090 y(~)1590 2113 y Fq(M)30 b Ft(b)s(e)g(the)h(collection)i (of)d(all)i(suc)m(h)j(~)-50 b Fs(x)30 b Ft(and)g(put)1286 2363 y Fs(B)1363 2374 y Fv(~)1355 2391 y Fr(I)1422 2363 y Fq(\021)1531 2277 y Fh([)1518 2487 y Fr(x)p Fi(2)1616 2471 y Fv(~)1608 2487 y Fr(I)1660 2363 y Fs(B)5 b Ft(\()p Fs(x;)15 b(\017=)p Ft(4\))p Fs(;)2258 2340 y Ft(~)2247 2363 y Fs(I)7 b Fq(\032)2409 2340 y Ft(~)2365 2363 y Fq(M)1049 b Ft(\(4)p Fs(:)p Ft(1\))50 2683 y(W)-8 b(e)31 b(de\014ne)f Fs(C)542 2694 y Fv(~)534 2711 y Fr(I)576 2683 y Ft(\()p Fs(\025;)15 b(\036)p Ft(\))31 b(to)g(b)s(e)f(the)h (matrix)e(with)h(en)m(tries)596 2949 y Fs(C)669 2960 y Fv(~)661 2977 y Fr(I)702 2949 y Ft(\()p Fs(\025;)15 b(\036)p Ft(\))919 2963 y Fr(y)r(z)1027 2949 y Fq(\021)25 b Fs(\017)1175 2826 y Fh(Z)1225 3032 y Fr(@)t(B)1323 3041 y Fk(z)1380 2949 y Fs(e)1422 2912 y Fi(\000)p Fr(F)6 b(=\017)1673 2888 y Fs(\036)p 1614 2928 172 4 v 1614 3012 a(\036)p Ft(\()p Fs(z)t Ft(\))1796 2949 y Fs(@)1844 2963 y Fr(n)1894 2949 y Fs(h)1946 2912 y Fr(\025)1946 2985 y(y)r(;)2015 2969 y Fv(~)2007 2985 y Fr(I)2064 2949 y Fs(d\033)24 b Fq(\000)c Fs(\016)2318 2963 y Fr(z)s(y)2399 2949 y Fs(\025)2467 2826 y Fh(Z)2518 3032 y Fr(B)2573 3041 y Fk(y)2632 2949 y Fs(e)2674 2912 y Fi(\000)p Fr(F)6 b(=\017)2925 2888 y Fs(\036)p 2866 2928 173 4 v 2866 3012 a(\036)p Ft(\()p Fs(y)s Ft(\))3065 2949 y Fs(dx)359 b Ft(\(4)p Fs(:)p Ft(2\))50 3248 y(where)30 b Fs(y)s(;)15 b(z)29 b Fq(2)569 3225 y Ft(~)558 3248 y Fs(I)7 b Ft(.)41 b(F)-8 b(or)31 b(the)g(sak)m(e)g(of)g(con)m(v)m(enience)h(w)m(e)f (henceforth)f(write)h(shorthand)e(for)3183 3225 y(~)3172 3248 y Fs(I)7 b(;)3282 3225 y Ft(~)3259 3248 y Fs(J)j Fq(\032)3434 3225 y Ft(~)3389 3248 y Fq(M)459 3498 y Fs(h)511 3461 y Fr(\025)519 3518 y Fv(~)511 3534 y Fr(I)5 b(;)587 3518 y Fv(~)570 3534 y Fr(J)646 3498 y Fq(\021)25 b Fs(h)794 3461 y Fr(\025)794 3521 y(B)856 3530 y Fl(~)849 3541 y Fk(I)886 3521 y Fr(;B)976 3530 y Fl(~)963 3541 y Fk(J)1006 3521 y Fi([)p Fv(\012)1107 3502 y Fk(c)1146 3498 y Fs(;)197 b Ft(cap)1504 3461 y Fr(\025)1552 3498 y Ft(\()1598 3475 y(~)1587 3498 y Fs(I)7 b(;)1697 3475 y Ft(~)1674 3498 y Fs(J)j Ft(\))26 b Fq(\021)f Ft(cap)2027 3461 y Fr(\025)2027 3521 y(B)2089 3530 y Fl(~)2082 3541 y Fk(I)2123 3498 y Ft(\()p Fs(B)2244 3509 y Fv(~)2227 3526 y Fr(J)2298 3498 y Fq([)20 b Ft(\012)2445 3461 y Fr(c)2482 3498 y Ft(\))p Fs(;)197 b(\025)2800 3509 y Fv(~)2792 3526 y Fr(I)2859 3498 y Fq(\021)25 b Fs(\025)p Ft(\(\012)p Fq(n)p Fs(B)3231 3509 y Fv(~)3223 3526 y 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Ft(Lemma)j(4.1)i(can)g(b)s(e)f(used)f(to)j(analyze)g(principal)e (eigen)m(v)-5 b(alues)33 b(leading)f(to)h(the)e(sharp)g(uncertain)m(t)m (y)50 658 y(principle)g(4.2.)45 b(As)31 b(in)g([BEGK2])i(or)e([E2])i (one)e(pro)m(v)m(es)h Fs(\025)2060 669 y Fv(~)2052 686 y Fr(I)2121 658 y Fs(<)27 b(\025)2280 669 y Fv(~)2272 686 y Fr(I)5 b Fi([)p Fr(x)2435 658 y Ft(for)31 b Fs(x)c Fq(2)2786 635 y Ft(~)2742 658 y Fq(Mn)2907 635 y Ft(~)2896 658 y Fs(I)39 b Ft(so)32 b(that)g(from)e(4.1)i(it)50 804 y(follo)m(ws)913 957 y Fs(\017)965 834 y Fh(Z)1015 1040 y Fr(@)t(B)1113 1049 y Fk(x)1173 957 y Fs(e)1215 920 y Fi(\000)p Fr(F)6 b(=\017)1396 957 y Fs(\036)1458 968 y Fv(~)1450 985 y Fr(I)1492 957 y Fs(@)1540 971 y Fr(n)1590 957 y Fs(h)1642 906 y Fr(\025)1692 915 y Fl(~)1685 926 y Fk(I)1642 1000 y Fr(x;)1713 984 y Fv(~)1705 1000 y Fr(I)1762 957 y Fs(d\033)24 b Fq(\000)19 b Fs(\025)2036 968 y Fv(~)2028 985 y Fr(I)2085 834 y Fh(Z)2136 1040 y Fr(B)2191 1049 y Fk(x)2251 957 y Fs(e)2293 920 y Fi(\000)p Fr(F)6 b(=\017)2474 957 y Fs(\036)2536 968 y Fv(~)2528 985 y Fr(I)2585 957 y Fs(dy)29 b Ft(=)c(0)676 b(\(4)p Fs(:)p Ft(5\))50 1207 y(where)30 b Fs(\036)375 1218 y Fv(~)367 1235 y Fr(I)439 1207 y Ft(is)g(the)h(principal)f (eigenfunction)h(of)g Fs(L)1790 1174 y Fv(\012)1790 1229 y Fr(\017)1876 1207 y Ft(suc)m(h)f(that)h Fs(\036)2340 1218 y Fv(~)2332 1235 y Fr(I)2374 1207 y Ft(\()p Fs(x)p Ft(\))26 b(=)f(1.)41 b(This)30 b(equation)h(implies)50 1425 y Fe(Theorem)h(4.2:)48 b Fo(Ther)-5 b(e)31 b(exists)f Fs(N)35 b Fq(\021)25 b Fs(N)10 b Ft(\()p Fs(d)p Ft(\))30 b Fo(and)g Fs(C)i Fq(\021)25 b Fs(C)7 b Ft(\()p Fs(d;)15 b(F)e Ft(\))31 b Fo(with)f(the)g(fol)5 b(lowing)30 b(pr)-5 b(op)g(erties.)44 b(F)-7 b(or)50 1571 y(nonempty,)34 b(pr)-5 b(op)g(erly)35 b(c)-5 b(ontaine)g(d)1262 1548 y Ft(~)1251 1571 y Fs(I)7 b Fq(\032)1413 1548 y Ft(~)1369 1571 y Fq(M)32 b Fo(and)i Fs(x)25 b Fq(2)g(Mn)2015 1548 y Ft(~)2004 1571 y Fs(I)40 b Fo(such)33 b(that)h Fs(T)2530 1598 y Fr(x;)2601 1582 y Fv(~)2593 1598 y Fr(I)2660 1571 y Ft(=)25 b Fs(T)2817 1582 y Fv(~)2809 1598 y Fr(I)2876 1571 y Fs(<)g(\017)3009 1538 y Fr(N)3078 1571 y Fs(T)3139 1582 y Fv(~)3131 1598 y Fr(I)5 b Fi([)p Fr(x)3295 1571 y Fo(we)33 b(have)1063 1882 y Fs(\025)1124 1893 y Fv(~)1116 1910 y Fr(I)1183 1882 y Ft(=)1279 1754 y Fh(\022)1346 1882 y Ft(1)21 b(+)f Fq(O)s Ft(\(1\))p Fs(\017)1730 1845 y Fi(\000)p Fr(N)1867 1819 y Fs(T)1928 1830 y Fv(~)1920 1847 y Fr(I)6 b Fi([)p Fr(x)p 1867 1861 186 4 v 1913 1945 a Fs(T)1974 1956 y Fv(~)1966 1972 y Fr(I)2063 1754 y Fh(\023)2220 1821 y Ft(cap)2356 1788 y Fv(0)2397 1821 y Ft(\()p Fs(x;)2535 1798 y Ft(~)2524 1821 y Fs(I)i Ft(\))p 2141 1861 545 4 v 2141 1873 a Fh(R)2184 1979 y Fr(A)2239 1999 y Fk(x;)2303 1988 y Fl(~)2296 1999 y Fk(I)2352 1946 y Fs(e)2394 1920 y Fi(\000)p Fr(F)e(=\017)2590 1946 y Fs(dy)3523 1882 y Ft(\(4)p Fs(:)p Ft(6\))50 2209 y Fo(wher)-5 b(e)42 b Fs(A)383 2236 y Fr(x;)454 2220 y Fv(~)446 2236 y Fr(I)529 2209 y Fq(\021)f Fs(A)709 2176 y Fv(0)709 2249 y Fr(x;)780 2233 y Fv(~)772 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993 y Fl(2)2427 984 y Fr(=\017)3477 1130 y Ft(\(4)p Fs(:)p Ft(19\))50 1276 y(where)39 b(w)m(e)h(ha)m(v)m(e) h(used)e(cap)1038 1243 y Fv(0)1038 1301 y Fr(B)1093 1310 y Fk(x)1138 1276 y Ft(\()p Fp(R)1233 1243 y Fr(d)1281 1276 y Fq(n)p Fs(B)1400 1184 y Fi(p)p 1460 1184 31 4 v 46 x Fr(\017)1395 1286 y(x)1495 1276 y Ft(\))i Fq(\024)f Fs(C)7 b(e)1796 1243 y Fi(\000)p Fr(F)j Fv(\()p Fr(x)p Fv(\))p Fr(=\017)2079 1276 y Fs(=\017)2161 1243 y Fr(d)p Fi(\000)p Fv(1)2337 1276 y Ft(follo)m(wing)41 b(from)d(Prop)s(osition)i ([4.7)h(in)50 1421 y(BEGK3].)g(Using)31 b(\(4.11\))i(in)d(com)m (bination)g(with)g(\(4.19\))j(in)d(\(4.5\))i(w)m(e)f(no)m(w)g(conclude) 144 1681 y(cap)280 1644 y Fv(0)321 1681 y Ft(\()p Fs(x;)460 1658 y Ft(~)448 1681 y Fs(I)8 b Ft(\))26 b(=)652 1553 y Fh(\022)719 1681 y Ft(1)21 b(+)f Fq(O)s Ft(\(1\))p Fs(\017)1103 1644 y Fi(\000)p Fr(N)1230 1553 y Fh(\022)1308 1620 y Ft(1)g(+)g Fs(e)1506 1587 y Fr(\014)s(=\017)p 1308 1660 316 4 v 1418 1744 a Fs(T)1479 1755 y Fv(~)1471 1771 y 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4349 V Fr(B)2007 4412 y Fl(~)2000 4423 y Fk(I)2032 4424 y Fg([)p Fk(x)2120 4360 y Ft(\()p Fs(B)2232 4371 y Fv(~)2224 4388 y Fr(I)2279 4360 y Fq([)13 b Ft(\012)2419 4327 y Fr(c)2456 4360 y Ft(\))25 b Fq(\024)g Fs(C)7 b(\017)2721 4327 y Fi(\000)p Fr(N)2846 4360 y Fs(T)2907 4371 y Fv(~)2899 4388 y Fr(I)f Fi([)p Fr(x)3044 4360 y Ft(+)13 b Fs(s)3171 4327 y Fv(0)3171 4386 y Fr(@)t(B)3269 4395 y Fk(x)3314 4360 y Ft(\()p Fs(B)3426 4371 y Fv(~)3418 4388 y Fr(I)3473 4360 y Fq([)g Ft(\012)3613 4327 y Fr(c)3649 4360 y Ft(\).)50 4506 y(In)m(v)m(oking)32 b([\(6.1\))g(in)d(BEGK3])i(in)f(com)m (bination)g(with)f(the)h(fact)h(that)g(the)f(nominator)f(in)g(this)h (estimate)50 4651 y(is)g(computed)f(in)h(\(4.18\),)j(w)m(e)e(conclude) 821 4901 y Fs(s)864 4863 y Fv(0)864 4939 y(\006)p Fi(n)p 953 4885 V Fr(B)1017 4948 y Fl(~)1011 4959 y Fk(I)1043 4960 y Fg([)p Fk(x)1131 4901 y Ft(\()p Fs(B)1243 4912 y Fv(~)1235 4928 y Fr(I)1297 4901 y Fq([)20 b Ft(\012)1444 4863 y Fr(c)1480 4901 y Ft(\))26 b Fq(\024)1637 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Fk(x)2453 5404 y Fs(w)2520 5366 y Fv(0)2518 5426 y Fr(I)g Fi([)p Fv(\012)2656 5408 y Fk(c)3477 5404 y Ft(\(4)p Fs(:)p Ft(24\))p eop %%Page: 33 33 33 32 bop 50 295 a Fj(Sp)l(e)l(ctr)l(a)3340 b Fm(33)50 513 y Ft(It)29 b(follo)m(ws)i(from)c(the)j(Green's)f(second)h(form)m (ula)e(and)h(\(4.24\))i(that)f(mo)s(dulo)e(the)h(error)g(term)f(app)s (earing)50 658 y(ab)s(o)m(v)m(e)348 866 y Fs(w)415 828 y Fv(0)413 888 y Fr(I)5 b Fi([)p Fv(\012)551 870 y Fk(c)590 866 y Ft(\()p Fs(x)p Ft(\))15 b(cap)864 828 y Fv(0)905 866 y Ft(\()p Fs(x;)g(I)7 b Ft(\))26 b(=)p Fs(\017)1263 742 y Fh(Z)1314 949 y Fr(@)t(B)1412 958 y Fk(x)1471 866 y Fs(e)1513 828 y Fi(\000)p Fr(F)6 b(=\017)1695 866 y Fs(w)1762 828 y Fv(0)1760 888 y Fr(I)f Fi([)p Fv(\012)1898 870 y Fk(c)1936 866 y Fs(@)1984 880 y Fr(n)2034 866 y Fs(h)2086 828 y Fv(0)2086 888 y Fr(x;I)2206 866 y Fs(d\033)1140 1118 y Ft(=)20 b Fq(\000)g Fs(\017)1374 994 y Fh(Z)1425 1201 y Fr(@)t(B)1523 1210 y Fk(x)1583 1118 y Fs(e)1625 1080 y Fi(\000)p Fr(F)6 b(=\017)1806 1118 y Fs(@)1854 1132 y Fi(\000)p Fr(n)1960 1118 y Fs(w)2027 1080 y Fv(0)2025 1140 y Fr(I)f Fi([)p Fv(\012)2163 1122 y Fk(c)2217 1118 y Fs(d\033)24 b Ft(+)2431 994 y Fh(Z)2481 1201 y Fv(\012)p Fi(n)p 2570 1146 59 4 v Fr(B)2629 1210 y Fk(x)2689 1118 y Fs(e)2731 1080 y Fi(\000)p Fr(F)6 b(=\017)2912 1118 y Fs(h)2964 1080 y Fv(0)2964 1140 y Fr(x;I)3084 1118 y Fs(dy)1140 1377 y Ft(=)1226 1253 y Fh(Z)1277 1460 y Fv(\012)1348 1377 y Fs(e)1390 1339 y Fi(\000)p Fr(F)g(=\017)1571 1377 y Fs(h)1623 1339 y Fv(0)1623 1399 y Fr(x;I)1743 1377 y Fs(dy)3477 1124 y Ft(\(4)p Fs(:)p Ft(25\))50 1612 y(The)30 b(latter)h(in)m(tegral)h(w)m(as)f(computed)e(in)h(\(4.18\))j (so)e(that)g(the)f(assertion)h(follo)m(ws)h(from)d(\(4.6\).)42 b Fq(})159 1830 y Ft(W)-8 b(e)31 b(w)m(an)m(t)h(to)f(analyze)h(the)e (diagonal)i(en)m(tries)f(of)f(the)h(matrix)f Fs(C)2415 1841 y Fv(~)2407 1858 y Fr(I)2448 1830 y Ft(\()p Fs(\025;)15 b(\036)p Ft(\))32 b(in)e(more)g(detail.)50 2048 y Fe(Lemma)e(4.4:)48 b Fo(In)27 b(the)h(situation)g(of)g(the)f(pr)-5 b(evious)29 b(the)-5 b(or)g(em)29 b(ther)-5 b(e)28 b(ar)-5 b(e)29 b Fs(N)35 b Fq(\021)25 b Fs(N)10 b Ft(\()p Fs(d)p Ft(\))28 b Fo(and)g Fs(C)k Fq(\021)25 b Fs(C)7 b Ft(\()p Fs(d;)15 b(F)e Ft(\))50 2194 y Fo(satisfying)33 b(for)g(al)5 b(l)33 b Ft(0)26 b Fq(\024)f Fs(\025)g Fq(\024)g Fs(\025)1123 2208 y Fv(0)1164 2194 y Fo(,)32 b Fs(\025)1277 2208 y Fv(0)1344 2194 y Fq(\021)25 b Fs(\017)1477 2161 y Fi(\000)p Fr(N)1602 2194 y Fs(=T)1708 2205 y Fv(~)1700 2221 y Fr(I)6 b Fi([)p Fr(x)1833 2194 y Fo(,)641 2386 y Fs(C)706 2400 y Fr(x)752 2386 y Ft(\()p Fs(\025;)15 b(\036)942 2397 y Fv(~)934 2414 y Fr(I)977 2386 y Ft(\))1012 2400 y Fr(xx)1124 2386 y Ft(=)20 b Fq(\000)1306 2313 y Fh(\000)1348 2386 y Ft(1)h(+)e Fq(O)s Ft(\(1\))p Fs(\017)1731 2349 y Fi(\000)p Fr(N)1858 2386 y Fs(\025T)1972 2397 y Fv(~)1964 2414 y Fr(I)6 b Fi([)p Fr(x)2096 2313 y Fh(\001)1195 2505 y(\000)1237 2578 y Fs(\025)20 b Fq(\000)g Fs(\025)1462 2589 y Fv(~)1454 2606 y Fr(I)1516 2578 y Fq(\000)g Ft(\()p Fs(\025)g Fq(\000)g Fs(\025)1867 2589 y Fv(~)1859 2606 y Fr(I)1901 2578 y Ft(\))1936 2541 y Fv(2)1977 2578 y Fq(O)s Ft(\(1\))p Fs(\017)2204 2541 y Fi(\000)p Fr(N)2331 2578 y Fs(T)2392 2589 y Fv(~)2384 2606 y Fr(I)5 b Fi([)p Fr(x)2516 2578 y Ft(\))2566 2455 y Fh(Z)2617 2661 y Fr(A)2672 2682 y Fk(x;)2736 2671 y Fl(~)2729 2682 y Fk(I)2785 2578 y Fs(e)2827 2541 y Fi(\000)p Fr(F)h(=\017)3023 2578 y Fs(dy)3477 2521 y Ft(\(4)p Fs(:)p Ft(26\))50 2993 y Fd(Pro)s(of:)53 b Ft(P)m(erforming)31 b(a)i(T)-8 b(a)m(ylor)34 b(expansion)f(at)g Fs(\025)c Fq(\021)g Fs(\025)2005 3004 y Fv(~)1997 3020 y Fr(I)2071 2993 y Ft(to)34 b(second)f(order)f(of)h(the)g(Laplace)g (transform)50 3138 y(on)d(the)h(left-hand)f(side)g(of)h(\(4.26\),)i(w)m (e)e(compute)e(similarly)h(to)h(\(4.10\))i(using)d(\(4.5\))397 3377 y Fs(C)462 3391 y Fr(x)508 3377 y Ft(\()p Fs(\025)604 3388 y Fv(~)596 3404 y Fr(I)638 3377 y Fs(;)15 b(\036)740 3388 y Fv(~)732 3404 y Fr(I)774 3377 y Ft(\))809 3391 y Fr(xx)922 3377 y Ft(=)o(\()p Fs(\025)21 b Fq(\000)f Fs(\025)1253 3388 y Fv(~)1245 3404 y Fr(I)1286 3377 y Ft(\))1321 3249 y Fh(\022)1389 3377 y Fs(\017)1441 3253 y Fh(Z)1491 3460 y Fr(@)t(B)1589 3469 y Fk(x)1649 3377 y Fs(e)1691 3339 y Fi(\000)p Fr(F)6 b(=\017)1872 3377 y Fs(\036)1934 3388 y Fv(~)1926 3404 y Fr(I)1968 3377 y Fs(@)2016 3391 y Fr(n)2066 3377 y Fs(w)2133 3325 y Fr(\025)2183 3334 y Fl(~)2176 3345 y Fk(I)2131 3419 y Fr(x;)2202 3403 y Fv(~)2194 3419 y Fr(I)2251 3377 y Fs(d\033)24 b Fq(\000)2464 3253 y Fh(Z)2515 3460 y Fr(B)2570 3469 y Fk(x)2630 3377 y Fs(e)2672 3339 y Fi(\000)p Fr(F)6 b(=\017)2853 3377 y Fs(\036)2915 3388 y Fv(~)2907 3404 y Fr(I)2964 3377 y Fs(dx)3063 3249 y Fh(\023)1012 3647 y Ft(+)20 b(\()p Fs(\025)h Fq(\000)f Fs(\025)1364 3658 y Fv(~)1356 3675 y Fr(I)1398 3647 y Ft(\))1433 3610 y Fv(2)1474 3647 y Fs(\017)1526 3524 y Fh(Z)1617 3546 y Fv(1)1576 3730 y(0)1673 3647 y Fs(s)15 b(ds)1836 3524 y Fh(Z)1886 3730 y Fr(@)t(B)1984 3739 y Fk(x)2044 3647 y Fs(d\033)s(e)2188 3610 y Fi(\000)p Fr(F)6 b(=\017)2370 3647 y Fs(\036)2432 3658 y Fv(~)2424 3675 y Fr(I)2465 3647 y Fs(@)2513 3661 y Fr(n)2592 3647 y Ft(_)-54 b Fs(w)2630 3596 y Fv(\(1)p Fi(\000)p Fr(s)p Fv(\))p Fr(\025)2863 3605 y Fl(~)2856 3616 y Fk(I)2895 3596 y Fv(+)p Fr(s\025)2628 3689 y(x;)2699 3673 y Fv(~)2691 3689 y Fr(I)3477 3515 y Ft(\(4)p Fs(:)p Ft(27\))50 3891 y(Analogous)31 b(to)g(\(4.9\))i(b)m (y)d(Chauc)m(h)m(y's)g(form)m(ula)g(and)g(\(3.37\))i(on)e Fs(@)5 b(B)2405 3905 y Fr(x)2482 3891 y Ft(w)m(e)31 b(ha)m(v)m(e)g(the) g(estimate)743 4112 y Fs(@)791 4126 y Fr(n)870 4112 y Ft(_)-54 b Fs(w)908 4060 y Fv(\(1)p Fi(\000)p Fr(s)p Fv(\))p Fr(\025)1141 4069 y Fl(~)1134 4080 y Fk(I)1172 4060 y Fv(+)p Fr(s\025)906 4154 y(x;)977 4138 y Fv(~)969 4154 y Fr(I)1335 4112 y Fq(\025)25 b Ft(\()p Fs(C)t(=)p Ft(\()p Fs(\025)1668 4126 y Fv(1)1730 4112 y Fq(\000)20 b Fs(\025)p Ft(\)\))p Fs(@)1992 4126 y Fr(n)2043 4112 y Fs(w)2110 4071 y Fr(\025)2153 4080 y Fl(1)2108 4154 y Fr(x;)2179 4138 y Fv(~)2171 4154 y Fr(I)2238 4112 y Fq(\025)25 b Ft(\(100)p Fs(C)t(=\025)2671 4126 y Fv(0)2714 4112 y Ft(\))p Fs(@)2797 4126 y Fr(n)2847 4112 y Fs(w)2914 4071 y Fr(\025)2957 4080 y Fl(1)2912 4154 y Fr(x;)2983 4138 y Fv(~)2975 4154 y Fr(I)3477 4112 y Ft(\(4)p Fs(:)p Ft(28\))50 4332 y(for)33 b(some)f(univ)m(ersal)h(constan)m(t)i Fs(C)7 b Ft(,)33 b(\(1)p Fs(=)p Ft(100\))p Fs(\025)1657 4346 y Fv(0)1731 4332 y Fq(\024)c Fs(\025)1884 4346 y Fv(1)1955 4332 y Fs(<)g(\025)2108 4346 y Fv(0)2182 4332 y Ft(and)k(all)h(0)c Fq(\024)f Fs(\025)h(<)g Ft(\(1)p Fs(=)p Ft(200\))p Fs(\025)3198 4346 y Fv(0)3242 4332 y Ft(.)48 b(Moreo)m(v)m(er,)50 4478 y(\(4.19\))37 b(remains)c(v)-5 b(alid)35 b(for)f Fs(\025)f Fq(\021)e Fs(\025)1268 4492 y Fr(I)1310 4478 y Fs(;)15 b(\025)1403 4492 y Fv(1)1479 4478 y Ft(so)35 b(that)g(analogously)h(to)f(\(4.20\))i(for)e Fs(C)2884 4492 y Fv(2)2959 4478 y Ft(large)g(the)g(last)h(term)50 4623 y(on)30 b(the)h(righ)m(t-hand)f(side)g(of)h(\(4.27\))h(is)f(of)f (order)1171 4854 y(\()p Fs(\025)21 b Fq(\000)e Fs(\025)1431 4865 y Fv(~)1423 4881 y Fr(I)1465 4854 y Ft(\))1500 4816 y Fv(2)1541 4854 y Fs(\017)1578 4816 y Fi(\000)p Fr(N)1704 4854 y Fs(T)1765 4865 y Fv(~)1757 4881 y Fr(I)5 b Fi([)p Fr(x)1904 4730 y Fh(Z)1955 4937 y Fr(A)2010 4906 y Fk(\014)2010 4972 y(x;)2074 4961 y Fl(~)2067 4972 y Fk(I)2103 4937 y Fi(n)p Fr(B)2195 4946 y Fk(x)2255 4854 y Fs(e)2297 4816 y Fi(\000)p Fr(F)h(=\017)2494 4854 y Fs(dy)891 b Ft(\(4)p Fs(:)p Ft(29\))50 5128 y(while)30 b(the)h(\014rst)e(term)h (can)g(b)s(e)g(estimated)h(b)m(y)676 5371 y(\()p Fs(\025)21 b Fq(\000)f Fs(\025)937 5382 y Fv(~)929 5399 y Fr(I)971 5371 y Ft(\))1006 5243 y Fh(\022)1073 5371 y Ft(1)h(+)e Fq(O)s Ft(\(1\))p Fs(\017)1456 5334 y Fi(\000)p Fr(N)1583 5243 y Fh(\022)1650 5371 y Fs(\025)1711 5382 y Fv(~)1703 5399 y Fr(I)1745 5371 y Fs(T)1806 5382 y Fv(~)1798 5399 y Fr(I)5 b Fi([)p Fr(x)1950 5371 y Ft(+)2052 5310 y(1)21 b(+)f Fs(e)2251 5277 y Fr(\014)s(=\017)p 2052 5351 316 4 v 2162 5434 a Fs(T)2223 5445 y Fv(~)2215 5461 y Fr(I)2378 5243 y Fh(\023\023)2527 5248 y(Z)2578 5454 y Fr(A)2633 5424 y Fk(\014)2633 5489 y(x;)2697 5478 y Fl(~)2690 5489 y Fk(I)2746 5371 y Fs(e)2788 5334 y Fi(\000)p Fr(F)6 b(=\017)2984 5371 y Fs(dx)394 b Ft(\(4)p Fs(:)p Ft(30\))p eop %%Page: 34 34 34 33 bop 50 295 a Fm(34)3270 b Fj(Se)l(ction)27 b Fm(4)50 513 y Ft(In)c(view)h(of)g(\(3.37\))i(from)d(\(4.30\))j(and)d(\(4.29\))j (for)e(some)f Fs(N)36 b Fq(\021)25 b Fs(N)10 b Ft(\()p Fs(d)p Ft(\))24 b(and)g Fs(\014)30 b Fq(\021)25 b Fs(N)10 b(\017)15 b Ft(log)s(\(1)p Fs(=\017)p Ft(\))25 b(the)f(assertion)50 658 y(follo)m(ws.)42 b Fq(})50 876 y Fu(Small)24 b(eigen)m(v)-6 b(alues:)35 b Ft(W)-8 b(e)24 b(no)m(w)e(turn)g(to)h(the)g(main)e (result)i(of)f(this)h(pap)s(er.)37 b(Namely)-8 b(,)24 b(w)m(e)f(will)g(sho)m(w)g(ho)m(w)50 1022 y(the)k(capacit)m(y)i(matrix) d(in)m(tro)s(duced)g(in)g(\(4.2\))j(can)e(b)s(e)f(used)f(to)j(analyze)g (the)f(sp)s(ectrum)d(of)j Fs(L)3241 989 y Fv(\012)3241 1044 y Fr(\017)3324 1022 y Ft(near)g(zero.)50 1167 y(The)i(pro)s(of)h (of)g(Theorem)f(4.6)i(pro)s(ceeds)f(close)h(to)g(the)g(line)f(of)g (argumen)m(ts)g(of)g(the)h(analogous)g(assertion)50 1313 y(in)j([E2])h(or)f(in)g([BEGK2].)54 b(In)34 b(particular,)i(for)e(pro)s (ofs,)h(whic)m(h)f(are)h(straigh)m(tforw)m(ard)g(generalizations,)50 1458 y(w)m(e)k(refer)f(the)h(reader)f(to)i(the)f(coun)m(terparts)g (therein.)65 b(Let)39 b(us)f(brie\015y)g(outline)h(the)g(strategy)h(of) f(the)50 1604 y(pro)s(of.)g(F)-8 b(or)29 b(eac)m(h)h Fs(I)i Fq(\021)p 854 1531 110 4 v 25 w(M)963 1618 y Fr(x)1009 1604 y Ft(,)d Fs(x)c Fq(2)g(M)k Ft(w)m(e)g(lo)s(ok)g(for)f(a)h (solution)g(of)f(the)h(equation)g(app)s(earing)f(in)g(4.1)i(near)50 1749 y(the)37 b(ground)f(state)i(energy)f(of)h(the)f(asso)s(ciated)h (Diric)m(hlet)h(op)s(erator)e Fs(L)2584 1688 y Fv(\012)p Fi(n)p Fr(B)2756 1697 y Fl(~)2728 1708 y Fg(M)2799 1717 y Fk(x)2584 1760 y Fr(\017)2849 1749 y Ft(.)60 b(W)-8 b(e)39 b(then)d(sho)m(w)h(that)50 1895 y(these)e(solutions)f(are)h(the) f(only)h(candidates)g(for)f(eigen)m(v)-5 b(alues)36 b(b)s(elo)m(w)e Fs(\017)2559 1862 y Fr(N)2662 1895 y Ft(for)g(some)g Fs(N)44 b Ft(and)34 b(that)h(they)50 2040 y(ha)m(v)m(e)29 b(to)g(b)s(e)f(simple)g(in)g(case.)41 b(Since)28 b(from)f(e.g.)41 b([S])28 b(or)h([CFKS])f(w)m(e)h(already)f(kno)m(w)h(that)g(there)f (are)h Fq(jMj)50 2185 y Ft(man)m(y)g(eigen)m(v)-5 b(alues,)33 b(these)e(candidates)g(in)f(fact)h(constitute)h(all)f(eigen)m(v)-5 b(alues)32 b(b)s(elo)m(w)f Fs(\017)3077 2152 y Fr(N)3146 2185 y Ft(.)159 2404 y(In)f(addition)g(to)h(the)g(notation)g(in)m(tro)s (duced)f(in)g(\(3.1\))i(let)g(us)d(de\014ne)623 2628 y Fq(M)p Ft(\()p Fs(x)p Ft(\))d Fq(\021)f(f)p Fs(y)k Fq(2)c(M)15 b(j)g Fs(F)e Ft(\()p Fs(y)s Ft(\))26 b Fs(<)f(F)13 b Ft(\()p Fs(x)p Ft(\))p Fq(g)p Fs(;)199 b(T)2171 2642 y Fr(x)2242 2628 y Fq(\021)25 b Fs(T)2391 2645 y Fr(x;)p Fi(M)p Fv(\()p Fr(x)p Fv(\))2642 2628 y Fs(;)197 b(x)25 b Fq(2)g(M)341 b Ft(\(4)p Fs(:)p Ft(31\))50 2853 y(in)30 b(case)h(that)g Fq(M)p Ft(\()p Fs(x)p Ft(\))26 b Fq(6)p Ft(=)f Fq(;)p Ft(.)41 b(W)-8 b(e)32 b(us)e(the)g(con)m(v)m(en)m(tion)j Fs(T)1947 2867 y Fr(x)2018 2853 y Fq(\021)25 b(1)30 b Ft(for)g Fq(M)p Ft(\()p Fs(x)p Ft(\))c Fq(\021)f(;)p Ft(.)50 3072 y Fe(Assumption)42 b(4.5:)109 b Fs(F)49 b Fo(is)36 b(generic)g(in)g(the)h(sense)f(that)h Fs(\032)32 b(>)g(N)10 b(\017)15 b Ft(log)r(\(1)p Fs(=\017)p Ft(\))38 b Fo(for)f(some)g Fs(N)k Fq(\021)32 b Fs(N)10 b Ft(\()p Fs(d)p Ft(\))p Fo(,)50 3217 y(wher)-5 b(e)34 b Fs(\032)e Fo(was)i(intr)-5 b(o)g(duc)g(e)g(d)35 b(in)d(\(1.14\).)159 3435 y Ft(In)h(particular,)j(it)e(follo)m(ws)i(as)e(in)g([BEGK2])h(or)f ([E2])h(that)f Fq(M)e(3)f Fs(x)g Fq(7!)h Fs(T)2745 3449 y Fr(x)2825 3435 y Ft(is)i(injectiv)m(e.)53 b(W)-8 b(e)36 b(hence)50 3581 y(obtain)31 b(an)f(ordering)g(of)h Fq(M)f Ft(via)1264 3806 y Fs(x)25 b(<)g(y)94 b Ft(if)30 b(and)g(only)g(if)91 b Fs(T)2232 3820 y Fr(x)2303 3806 y Fs(>)25 b(T)2452 3820 y Fr(y)3477 3806 y Ft(\(4)p Fs(:)p Ft(32\))50 4030 y(Let)31 b(us)e(de\014ne)857 4176 y Fq(M)966 4190 y Fr(x)1037 4176 y Fq(\021)c(f)p Fs(y)k Fq(2)c(M)15 b(j)g Fs(y)28 b(<)d(x)p Fq(g)p Fs(;)p 1991 4103 V 198 w Fq(M)2100 4190 y Fr(x)2171 4176 y Fq(\021)g(f)p Fs(y)k Fq(2)24 b(M)15 b(j)g Fs(y)29 b Fq(\024)c Fs(x)p Fq(g)575 b Ft(\(4)p Fs(:)p Ft(33\))50 4372 y(W)-8 b(e)31 b(also)h(shall)e(need)1427 4517 y Fq(T)1477 4531 y Fr(x)1548 4517 y Fq(\021)56 b Ft(min)1644 4576 y Fr(y)r Fi(2M)1817 4585 y Fk(x)1941 4517 y Ft(min)1873 4580 y Fr(z)s Fi(2M)2045 4589 y Fk(x)2085 4580 y Fi(n)p Fr(y)2177 4517 y Fs(T)2230 4531 y Fr(y)r(;z)3477 4517 y Ft(\(4)p Fs(:)p Ft(34\))50 4728 y(The)30 b(result)g(states)50 4946 y Fe(Theorem)41 b(4.6:)61 b Fo(Set)37 b Fs(j)977 4960 y Fv(0)1050 4946 y Fq(\021)31 b(jMj)p Fo(.)54 b(F)-7 b(or)37 b(some)g Fs(N)42 b Fq(\021)32 b Fs(N)10 b Ft(\()p Fs(d)p Ft(\))37 b Fo(ther)-5 b(e)38 b(is)e Fs(C)i Fq(\021)32 b Fs(C)7 b Ft(\()p Fs(d;)15 b(F)e Ft(\))37 b Fo(dominating)h(al)5 b(l)50 5091 y(mo)-5 b(duli)41 b(of)f(the)f(L)-5 b(andau)42 b(symb)-5 b(ols)41 b(app)-5 b(e)g(aring)42 b(b)-5 b(elow.)63 b(Assume)40 b(1.2)g(and)g(4.5.)64 b(Ther)-5 b(e)40 b(ar)-5 b(e)40 b Fs(j)3378 5105 y Fv(0)3459 5091 y Fo(simple)50 5237 y(eigenvalues)32 b Fs(\025)579 5251 y Fr(x)650 5237 y Fs(<)25 b(\025)799 5251 y Fr(y)843 5237 y Fo(,)32 b Fs(x;)15 b(y)29 b Fq(2)c(M)p Fo(,)32 b Fs(x)25 b(<)g(y)s Fo(,)33 b(satisfying)1262 5462 y Fs(\033)s Ft(\()p Fs(L)1414 5424 y Fv(\012)1414 5484 y Fr(\017)1471 5462 y Ft(\))20 b Fq(\\)g Ft([0)p Fs(;)15 b(\017)1754 5424 y Fr(N)1824 5462 y Ft(\))25 b(=)g Fq(f)p Fs(\025)2078 5476 y Fr(x)2140 5462 y Fq(j)15 b Fs(x)25 b Fq(2)g(Mg)980 b Ft(\(4)p Fs(:)p Ft(35\))p eop %%Page: 35 35 35 34 bop 50 295 a Fj(Sp)l(e)l(ctr)l(a)3340 b Fm(35)50 513 y Fo(F)-7 b(or)25 b(every)g Fs(x)g Fq(2)g(M)p Fo(,)h Fs(x)f Fq(6)p Ft(=)g(min)14 b Fq(M)p Fo(,)26 b(we)e(have)h Fq(T)1650 527 y Fr(x)1721 513 y Fq(\025)g Fs(e)1859 480 y Fr(\032=\017)1969 513 y Fs(T)2022 527 y Fr(x)2092 513 y Fo(and)h Fs(T)2314 527 y Fr(x)2385 513 y Fq(\025)f Fs(e)2523 480 y Fr(\032=\017)2648 513 y Ft(max)2817 543 y Fr(y)r Fi(2Mn)p 3027 489 86 4 v(M)3114 552 y Fk(x)3174 513 y Fs(T)3227 527 y Fr(y)3271 513 y Fo(,)h(wher)-5 b(e)25 b Fs(\032)g Fo(is)50 658 y(de\014ne)-5 b(d)25 b(in)g(\(1.14\).)41 b(F)-7 b(urthermor)i(e,)28 b(ther)-5 b(e)26 b(exists)f(a)g(set)2016 635 y Ft(~)1971 658 y Fq(M)2080 672 y Fr(x)2151 658 y Fo(of)g Fq(jM)2384 672 y Fr(x)2430 658 y Fq(j)g Fo(p)-5 b(oints)26 b(such)f(that)3160 635 y Ft(~)3116 658 y Fq(M)3225 672 y Fr(x)3273 658 y Fq(\\)s Fs(B)5 b Ft(\()p Fs(y)s(;)3534 593 y Fq(p)p 3610 593 37 4 v 65 x Fs(\017)p Ft(\))p Fo(,)50 804 y Fs(y)28 b Fq(2)d(M)318 818 y Fr(x)364 804 y Fo(,)32 b(is)h(a)g(singleton,)g (and)864 1081 y Fs(\025)917 1095 y Fr(x)988 1081 y Ft(=)1058 953 y Fh(\022)1125 1081 y Ft(1)21 b(+)f Fq(O)s Ft(\(1\))p Fs(\017)1509 1043 y Fi(\000)p Fr(N)1636 953 y Fh(\022)1713 1019 y Fs(T)1766 1033 y Fr(x)p 1713 1060 99 4 v 1715 1143 a Fq(T)1765 1157 y Fr(x)1843 1081 y Ft(+)1945 999 y(max)2114 1030 y Fr(y)r Fi(2Mn)p 2324 975 86 4 v(M)2411 1039 y Fk(x)2471 999 y Fs(T)2524 1013 y Fr(y)p 1945 1060 623 4 v 2207 1143 a Fs(T)2260 1157 y Fr(x)2579 953 y Fh(\023\023)2712 1081 y Fs(\025)2800 1092 y Fv(~)2765 1109 y Fi(M)2850 1118 y Fk(x)3477 1075 y Ft(\(4)p Fs(:)p Ft(36\))50 1351 y Fo(wher)-5 b(e)47 b(we)e(use)h(the)g(notation)h(intr) -5 b(o)g(duc)g(e)g(d)49 b(in)c(\(4.1\))i(and)f(\(4.3\))h(for)f(the)g (princip)-5 b(al)48 b(eigenvalue)d Fs(\025)3648 1362 y Fv(~)3640 1379 y Fr(I)3682 1351 y Fo(.)50 1497 y(Mor)-5 b(e)g(over,)33 b(every)g(eigenfunction)f Fs(\036)1334 1511 y Fr(x)1413 1497 y Fo(c)-5 b(orr)g(esp)g(onding)35 b(to)f Fs(\025)2149 1511 y Fr(x)2227 1497 y Fo(satis\014es)f(for)h(al)5 b(l)33 b Fs(z)c Fq(2)c(f)p Fs(F)39 b(<)25 b(C)3304 1511 y Fv(0)3345 1497 y Fq(g)20 b(\\)g Ft(\012)354 1759 y Fs(\036)408 1773 y Fr(x)454 1759 y Ft(\()p Fs(z)t Ft(\))26 b(=)f(\(1)c(+)f Fq(O)s Ft(\(1\))p Fs(\017)1111 1721 y Fi(\000)p Fr(N)1238 1759 y Fs(e)1280 1721 y Fi(\000)p Fr(\032=\017)1447 1759 y Ft(\))p Fs(h)1534 1721 y Fv(0)1534 1795 y Fr(x;)1632 1779 y Fv(~)1597 1795 y Fi(M)1682 1804 y Fk(x)1728 1759 y Ft(\()p Fs(y)s Ft(\))h(+)1999 1673 y Fh(X)1958 1883 y Fr(y)r Fi(2)2080 1867 y Fv(~)2046 1883 y Fi(M)2131 1892 y Fk(x)2187 1759 y Fq(O)s Ft(\(1\))p Fs(\017)2414 1721 y Fi(\000)p Fr(N)2582 1697 y Fs(T)2635 1711 y Fr(x)p 2552 1738 160 4 v 2552 1821 a Fs(T)2605 1835 y Fr(y)r(;x)2722 1759 y Fs(h)2774 1721 y Fv(0)2774 1795 y Fr(y)r(;)2870 1779 y Fv(~)2835 1795 y Fi(M)2920 1804 y Fk(x)2961 1795 y Fi([)p Fr(x)3056 1759 y Ft(\()p Fs(z)t Ft(\))305 b(\(4)p Fs(:)p Ft(37\))159 2238 y(Com)m(bination)30 b(of)g(4.2)i(with)e(\(4.36\))j(and)c(\(1.9\))j(yields)f(the)g(follo)m (wing)50 2457 y Fe(Corollary)40 b(4.7:)51 b Fo(In)33 b(the)g(situation)g(of)g(4.6)h(we)e(have)h(for)h(al)5 b(l)33 b Fs(x)25 b Fq(2)g(M)1243 2712 y Fs(\025)1296 2726 y Fr(x)1367 2712 y Ft(=)g(\(1)c(+)f Fq(O)s Ft(\(1\))p Fs(\017)1882 2674 y Fi(\000)p Fr(N)2009 2712 y Fs(e)2051 2674 y Fi(\000)p Fr(\032=\017)2218 2712 y Ft(\))2362 2650 y(1)p 2264 2691 242 4 v 2264 2774 a Fs(T)2352 2785 y Fv(~)2317 2802 y Fi(M)q Fv(\()p Fr(x)p Fv(\))3477 2712 y Ft(\(4)p Fs(:)p Ft(38\))50 2997 y Fo(In)32 b(p)-5 b(articular,)35 b(under)e(the)g(c)-5 b(onditions)35 b(of)e(1.1)g(it)g(fol)5 b(lows)606 3301 y Fs(\025)659 3315 y Fr(x)730 3301 y Ft(=)25 b(\(1)c(+)f Fq(O)s Ft(\(1\))p Fs(\017)15 b Ft(log)s(\(1)p Fs(=\017)p Ft(\)\))p Fq(j)p Fs(\025)1688 3263 y Fi(\003)1732 3301 y Fq(j)1757 3137 y Fh(s)p 1848 3137 655 4 v 1910 3239 a Ft(det)g(Hess)h Fs(F)d Ft(\()p Fs(x)p Ft(\))p 1859 3280 633 4 v 1859 3363 a Fq(j)i Ft(det)h(Hess)f Fs(F)e Ft(\()p Fs(z)2388 3337 y Fi(\003)2431 3363 y Ft(\))p Fq(j)2502 3301 y Fs(e)2544 3263 y Fi(\000)p Fv(\()2646 3247 y(^)2629 3263 y Fr(F)e Fv(\()p Fr(x;I)5 b Fv(\))p Fi(\000)p Fr(F)10 b Fv(\()p Fr(x)p Fv(\)\))p Fr(=\017)3477 3301 y Ft(\(4)p Fs(:)p Ft(39\))159 3771 y(W)-8 b(e)42 b(ha)m(v)m(e)g(to)f(in)m(tro)s(duce)g(some)f(more)g(notation.)72 b(Fix)42 b(a)f(set)2443 3748 y(~)2399 3771 y Fq(M)g Ft(of)f(cardinalit) m(y)j Fq(jMj)e Ft(suc)m(h)f(that)50 3917 y Fs(B)5 b Ft(\()p Fs(x;)251 3851 y Fq(p)p 327 3851 37 4 v 66 x Fs(\017)p Ft(\))17 b Fq(\\)538 3894 y Ft(~)494 3917 y Fq(M)29 b Ft(is)g(a)g(singleton.)1260 3894 y(~)1215 3917 y Fq(M)g Ft(inherits)f(the)i(ordering)e(of)h Fq(M)g Ft(in)f(an)h(ob)m(vious)g(w) m(a)m(y)-8 b(.)42 b(Let)29 b(us)f(de\014ne)65 4039 y(~)50 4062 y Fq(E)98 4076 y Fr(x)169 4062 y Fq(\021)d(1)30 b Ft(for)g Fs(x)25 b Fq(\021)g Ft(min)909 4039 y(~)865 4062 y Fq(M)30 b Ft(and)g(for)g Fs(x)c Fq(2)1528 4039 y Ft(~)1483 4062 y Fq(M)q(n)15 b Ft(min)1864 4039 y(~)1820 4062 y Fq(M)30 b Ft(set)1513 4291 y(~)1497 4314 y Fq(E)1545 4328 y Fr(x)1616 4314 y Fq(\021)57 b Ft(min)1712 4388 y Fr(y)r Fi(2)1835 4372 y Fv(~)1800 4388 y Fi(M)1885 4397 y Fk(x)1942 4314 y Fs(T)1995 4360 y Fr(y)r(;)p 2056 4290 86 4 v 2090 4344 a Fv(~)2056 4360 y Fi(M)2141 4369 y Fk(x)2181 4360 y Fi(n)p Fr(y)3477 4314 y Ft(\(4)p Fs(:)p Ft(40\))50 4594 y(The)30 b(\014rst)f(lemma)g(actually)j(is)f(a)f(sp)s (ecial)h(case)h(of)e(Lemma)f([4.5)j(in)e(E2])g(or)h(Lemma)d([5.3)k(in)e (BEGK2].)50 4812 y Fe(Lemma)36 b(4.8:)51 b Fo(F)-7 b(or)33 b(al)5 b(l)33 b Fs(x)26 b Fq(2)1192 4789 y Ft(~)1148 4812 y Fq(M)32 b Fo(it)h(fol)5 b(lows)1494 5064 y Fs(T)1547 5078 y Fr(x)1618 5064 y Ft(=)25 b Fs(T)1802 5075 y Fv(~)1767 5092 y Fi(M)1852 5101 y Fk(x)1923 5064 y Ft(=)g Fs(T)2072 5092 y Fr(x;)2169 5076 y Fv(~)2135 5092 y Fi(M)2220 5101 y Fk(x)3477 5064 y Ft(\(4)p Fs(:)p Ft(41\))50 5316 y Fo(F)-7 b(or)34 b Fs(x)25 b Fq(2)427 5293 y Ft(~)383 5316 y Fq(Mn)15 b Ft(min)764 5293 y(~)719 5316 y Fq(M)33 b Fo(we)f(have)1555 5439 y Ft(~)1539 5462 y Fq(T)1589 5476 y Fr(x)1659 5462 y Fq(\025)1770 5439 y Ft(~)1755 5462 y Fq(E)1803 5476 y Fr(x)1874 5462 y Fq(\025)25 b Fs(e)2012 5424 y Fr(\032=\017)2122 5462 y Fs(T)2175 5476 y Fr(x)3477 5462 y Ft(\(4)p Fs(:)p Ft(42\))p eop %%Page: 36 36 36 35 bop 50 295 a Fm(36)3270 b Fj(Se)l(ction)27 b Fm(4)50 513 y Fo(Mor)-5 b(e)g(over,)33 b(for)h Fs(x;)15 b(y)28 b Fq(2)918 490 y Ft(~)874 513 y Fq(M)p Fo(,)33 b Fs(y)28 b(<)d(x)p Fo(,)1303 706 y Ft(max)1220 798 y Fr(z)s Fi(2)1340 782 y Fv(~)1307 798 y Fi(Mn)p 1429 728 86 4 v 1463 782 a Fv(~)1429 798 y Fi(M)1514 807 y Fk(x)1570 706 y Fs(T)1623 752 y Fr(z)s(;)p 1683 682 V 1716 736 a Fv(~)1683 752 y Fi(M)1767 761 y Fk(x)1808 752 y Fi(n)p Fr(y)1914 706 y Fq(\024)g Fs(e)2052 668 y Fi(\000)p Fr(\032=\017)2219 706 y Fs(T)2272 752 y Fr(y)r(;)p 2333 682 V 2367 736 a Fv(~)2333 752 y Fi(M)2418 761 y Fk(x)2459 752 y Fi(n)p Fr(y)3477 706 y Ft(\(4)p Fs(:)p Ft(43\))50 964 y Fo(In)32 b(p)-5 b(articular,)35 b(for)e Fs(y)c Fq(2)961 941 y Ft(~)916 964 y Fq(M)1025 978 y Fr(x)1529 1109 y Fs(T)1582 1155 y Fr(y)r(;)p 1643 1085 V 1677 1139 a Fv(~)1643 1155 y Fi(M)1728 1164 y Fk(x)1769 1155 y Fi(n)p Fr(y)1875 1109 y Ft(=)c Fs(T)p 2024 1085 V 2058 1139 a Fv(~)2024 1155 y Fi(M)2109 1164 y Fk(x)2150 1155 y Fi(n)p Fr(y)3477 1109 y Ft(\(4)p Fs(:)p Ft(44\))159 1509 y(W)-8 b(e)31 b(no)m(w)g(analyze)h(the)e(p)s(ossible)g(solutions)h(of)f(the)h (capacit)m(y)i(matrix)c(in)m(tro)s(duced)h(in)g(\(4.2\).)159 1727 y Fo(Henc)-5 b(eforth,)32 b(we)f(assume)g(for)g(some)h Fs(x)25 b Fq(2)g(Mn)15 b Ft(min)f Fq(M)31 b Fo(that)h Fs(\025)2346 1741 y Fr(x)2423 1727 y Fo(is)f(an)g(eigenvalue)f(of)h Fs(L)3251 1694 y Fv(\012)3251 1750 y Fr(\017)3338 1727 y Fo(satisfying)50 1873 y(for)i(some)g Fs(\013)26 b(>)f Ft(0)1358 2018 y Fs(e)1400 1981 y Fr(\013=\017)1519 2018 y Fs(=)p Fq(E)1612 2032 y Fr(x)1684 2018 y Fs(<)g(\025)1833 2032 y Fr(x)1904 2018 y Fs(<)g(e)2042 1981 y Fi(\000)p Fr(\013=\017)2218 2018 y Fs(T)p 2271 1994 V 31 x Fi(M)2357 2058 y Fk(x)3477 2018 y Ft(\(4)p Fs(:)p Ft(45\))50 2200 y Fo(F)-7 b(urthermor)i(e,)35 b Fs(\036)657 2214 y Fr(x)735 2200 y Fo(denotes)f(a)f(c)-5 b(orr)g(esp)g(onding)36 b(eigenfunction.)159 2418 y Ft(F)-8 b(or)36 b Fs(y)g Fq(2)p 502 2345 110 4 v 33 w(M)611 2432 y Fr(x)692 2418 y Ft(according)g(to)g(2.3)g(w)m(e)g(c)m(ho)s(ose)g(a)g(set)p 2033 2324 V 2077 2395 a(~)2033 2418 y Fq(M)2142 2432 y Fr(x)2223 2418 y Ft(of)f Fq(j)p 2356 2345 V(M)2465 2432 y Fr(x)2511 2418 y Fq(j)h Ft(p)s(oin)m(ts)f(eac)m(h)h(of)f(them)f (lying)i(in)50 2564 y(one)c(ball)g Fs(B)5 b Ft(\()p Fs(y)s(;)593 2498 y Fq(p)p 669 2498 37 4 v 66 x Fs(\017)21 b Fq(\000)g Fs(\017)p Ft(\),)32 b Fs(y)f Fq(2)c(M)1221 2578 y Fr(x)1267 2564 y Ft(,)32 b(suc)m(h)g(that)g Fs(\036)1783 2578 y Fr(x)1861 2564 y Ft(do)s(es)f(not)h(c)m(hange)h(sign)f(in)f Fs(B)5 b Ft(\()h(~)-51 b Fs(y)s(;)15 b(\017)p Ft(\).)45 b(\(3.56\))35 b(implies)50 2709 y(that)h Fs(\036)306 2723 y Fr(x)388 2709 y Ft(do)s(es)f(not)h(c)m(hange)h(sign)f(in)f Fs(B)5 b Ft(\()p Fs(x;)15 b(\017=)p Ft(4\).)58 b(Indeed,)37 b(assume)d(that)i Fq(j)p Fs(\036)2749 2723 y Fr(x)2796 2709 y Fq(j)f Ft(attains)i(its)f(suprem)m(um)50 2855 y(in)41 b(some)f(ball)i Fs(B)5 b Ft(\()p Fs(y)s(;)15 b(\017=)p Ft(4\),)45 b(where)40 b Fs(y)46 b Fq(2)1501 2832 y Ft(~)1490 2855 y Fs(I)k Fq(\021)1738 2832 y Ft(~)1694 2855 y Fq(M)1803 2869 y Fr(x)1849 2855 y Ft(,)44 b Fs(y)56 b(=)-55 b Fq(2)43 b Fs(B)5 b Ft(\()p Fs(x;)2314 2789 y Fq(p)p 2390 2789 V 66 x Fs(\017)o Ft(\))42 b(in)f(3.9.)73 b(It)42 b(follo)m(ws)g(from)d(\(3.56\))50 3000 y(in)c(com)m(bination)g (with)g(the)g(upp)s(er)f(b)s(ound)f(in)i(\(4.45\))i(that)f Fs(\036)2241 3014 y Fr(x)2322 3000 y Ft(do)s(es)f(not)g(c)m(hange)h (sign)g(in)e Fs(A)3374 2956 y Fr(\014)3374 3060 y(y)r(;)p 3435 2991 86 4 v 3470 3044 a Fv(~)3435 3060 y Fi(M)3521 3069 y Fk(x)3601 3000 y Ft(for)50 3184 y(arbitrary)d(but)f(\014xed)g Fs(\014)i(>)26 b Ft(0.)43 b(Cho)s(osing)30 b Fs(u)d Fq(\021)f(j)p Fs(\036)1770 3198 y Fr(x)1816 3184 y Fq(j)31 b Ft(restricted)h(to)p 2390 3111 69 4 v 32 w Fs(A)2458 3126 y Fr(\014)2458 3230 y(y)r(;)p 2519 3160 86 4 v 2553 3214 a Fv(~)2519 3230 y Fi(M)2604 3239 y Fk(x)2680 3184 y Ft(in)f(\(2.8\))i(for)d(\006)c Fq(\021)g Fs(A)3401 3140 y Fr(\014)3401 3244 y(y)r(;)p 3462 3174 V 3497 3228 a Fv(~)3462 3244 y Fi(M)3548 3253 y Fk(x)3593 3184 y Ft(,)31 b(it)50 3334 y(follo)m(ws)e Fs(\025)p Ft(\(\006\))d Fq(\025)f Fs(\025)710 3348 y Fr(x)756 3334 y Ft(.)40 b(On)27 b(the)i(other)f(hand)f(w)m(e)i(obtain)g (from)e(\(2.9\))j(in)e(com)m(bination)h(with)f(Prop)s(osition)50 3479 y([4.7)42 b(in)f(BEGK3])h(that)f Fs(\025)p Ft(\(\006\))j Fq(\024)e Fs(\017)1319 3446 y Fi(\000)p Fr(N)1445 3479 y Fs(\017)1482 3446 y Fr(\014)s(=\017)1599 3479 y Fs(T)1652 3525 y Fr(y)r(;)p 1713 3455 V 1747 3509 a Fv(~)1713 3525 y Fi(M)1798 3534 y Fk(x)1838 3525 y Fi(n)p Fr(y)1962 3479 y Fq(\024)h Fs(\017)2113 3446 y Fi(\000)p Fr(N)2239 3479 y Fs(\017)2276 3446 y Fr(\014)s(=\017)2392 3479 y Fs(=)2453 3456 y Ft(~)2437 3479 y Fq(E)2485 3493 y Fr(x)2531 3479 y Ft(.)73 b(Since)2892 3456 y(~)2877 3479 y Fq(E)2925 3493 y Fr(x)3014 3479 y Fq(\024)42 b Fs(\017)3164 3446 y Fr(N)3233 3479 y Fq(E)3281 3493 y Fr(x)3327 3479 y Ft(,)i(w)m(e)d(th)m(us)50 3625 y(ha)m(v)m(e)c(deriv)m(ed)f(a)g(con)m (tradiction)h(to)g(the)f(lo)m(w)m(er)h(b)s(ound)d(in)h(\(4.45\))j(for)e Fs(\014)j(<)34 b(\013)p Ft(.)57 b(In)35 b(view)h(of)g(\(3.56\))i(w)m(e) 50 3770 y(no)m(w)e(ma)m(y)g(assume)f(that)p 971 3676 110 4 v 1016 3747 a(~)971 3770 y Fq(M)1080 3784 y Fr(x)1150 3770 y Fq(\\)24 b Fs(B)5 b Ft(\()p Fs(x;)1436 3705 y Fq(p)p 1512 3705 37 4 v 65 x Fs(\017)p Ft(\))36 b(=)f Fs(x)p Ft(.)58 b(Note)38 b(that)f(the)g(v)-5 b(arious)36 b(time)g(scales)i(in)e(the)g(error)50 3916 y(estimates)f(app)s(earing)e (in)h(4.6)i(ma)m(y)d(b)s(e)h(replaced)h(without)f(harm)e(b)m(y)i(those) h(de\014ned)e(with)h(resp)s(ect)g(to)94 4038 y(~)50 4061 y Fq(M)159 4075 y Fr(x)240 4061 y Ft(instead.)54 b(In)34 b(addition,)j(the)e(replacemen)m(t)g(of)p 1891 3988 110 4 v 35 w Fq(M)2000 4075 y Fr(x)2081 4061 y Ft(b)m(y)p 2212 3967 V 2256 4038 a(~)2212 4061 y Fq(M)2321 4075 y Fr(x)2402 4061 y Ft(c)m(hanges)g Fs(\013)h Ft(only)f(b)m(y)f(an)h (amoun)m(t)g(of)50 4207 y(order)30 b Fs(N)10 b(\017)15 b Ft(log)r(\(1)p Fs(=\017)p Ft(\).)42 b(Therefore,)159 4425 y Fo(In)33 b(the)g(se)-5 b(quel)32 b(we)h(identify)g Fq(M)1265 4439 y Fr(x)1343 4425 y Fo(with)h(the)f(set)1878 4402 y Ft(~)1833 4425 y Fq(M)1942 4439 y Fr(x)1988 4425 y Fo(.)159 4643 y Ft(Let)f Fs(C)388 4657 y Fr(x)461 4643 y Fq(\021)27 b Ft(diag)r(\()p Fs(e)804 4610 y Fr(F)10 b Fv(\()p Fr(y)r Fv(\))p Fr(=\017)1028 4643 y Ft(\))1063 4673 y Fr(y)r Fi(2)p 1151 4619 86 4 v(M)1237 4682 y Fk(x)1282 4643 y Fs(C)p 1347 4619 V 30 x Fi(M)1433 4682 y Fk(x)1478 4643 y Ft(\()p Fs(\025;)15 b(\036)1660 4657 y Fr(x)1707 4643 y Ft(\),)32 b(where)f Fs(C)p 2128 4619 V 30 x Fi(M)2214 4682 y Fk(x)2259 4643 y Ft(\()p Fs(\025;)15 b(\036)2441 4657 y Fr(x)2487 4643 y Ft(\))32 b(is)g(the)f(matrix)g(de\014ned)g(in)g (\(4.2\))50 4788 y(for)201 4765 y(~)190 4788 y Fs(I)k Fq(\021)p 363 4715 110 4 v 27 w(M)472 4802 y Fr(x)518 4788 y Ft(.)45 b(Note)33 b(that)f(b)m(y)g(Green's)g(second)g(form)m (ula)f(and)g Fs(h)2326 4755 y Fr(\025)2326 4832 y(z)s(;)p 2386 4777 86 4 v Fi(M)2470 4841 y Fk(x)2515 4788 y Fq(j)p Fs(@)5 b(B)2662 4802 y Fr(z)2732 4788 y Ft(=)28 b(1,)k Fs(B)3002 4802 y Fr(z)3076 4788 y Ft(de\014ned)e(in)i(\(4.1\),)50 4934 y(it)f(follo)m(ws)g(that)g(the)g(latter)g(matrix)f(is)h (symmetric.)39 b(Hence)276 5012 y Fh(\022)889 5080 y Fq(K)958 5094 y Fr(x)1004 5080 y Ft(\()p Fs(\025)p Ft(\))888 b Fq(\000)l Fs(~)-41 b(g)2129 5094 y Fr(x)2175 5080 y Ft(\()p Fs(\025)p Ft(\))358 5193 y Fq(\000)p Ft(\(diag)r(\()p Fs(e)709 5160 y Fv(\()p Fr(F)10 b Fv(\()p Fr(x)p Fv(\))p Fi(\000)p Fr(F)g Fv(\()p Fr(y)r Fv(\)\))p Fr(=\017)1202 5193 y Ft(\))1237 5207 y Fr(y)r Fi(\024)p Fr(x)1375 5193 y Fs(~)-41 b(g)1422 5207 y Fr(x)1468 5193 y Ft(\()p Fs(\025)p Ft(\)\))1626 5160 y Fr(t)1750 5193 y Fs(e)1792 5160 y Fr(F)10 b Fv(\()p Fr(x)p Fv(\))p Fr(=\017)2018 5193 y Fs(C)p 2083 5169 V 30 x Fi(M)2169 5232 y Fk(x)2214 5193 y Ft(\()p Fs(\025;)15 b(\036)2396 5207 y Fr(x)2442 5193 y Ft(\))2477 5207 y Fr(xx)2580 5012 y Fh(\023)2672 5140 y Fq(\021)25 b Fs(C)2833 5154 y Fr(x)2879 5140 y Ft(\()p Fs(\025)p Ft(\))h(=)261 5267 y Fh(\022)328 5395 y Fs(\017)380 5271 y Fh(Z)431 5478 y Fr(@)t(B)529 5487 y Fk(z)586 5395 y Fs(e)628 5357 y Fi(\000)p Fv(\()p Fr(F)10 b Fi(\000)p Fr(F)g Fv(\()p Fr(z)s Fv(\)\))p Fr(=\017)1146 5333 y Fs(\036)1200 5347 y Fr(x)p 1087 5374 217 4 v 1087 5457 a Fs(\036)1141 5471 y Fr(x)1187 5457 y Ft(\()p Fs(z)t Ft(\))1315 5395 y Fs(@)1363 5409 y Fr(n)1413 5395 y Fs(h)1465 5357 y Fr(\025)1465 5417 y(y)1528 5395 y Fs(d\033)24 b Fq(\000)c Fs(\016)1782 5409 y Fr(z)s(y)1863 5395 y Fs(\025)1931 5271 y Fh(Z)1982 5478 y Fr(B)2037 5487 y Fk(y)2096 5395 y Fs(e)2138 5357 y Fi(\000)p Fv(\()p Fr(F)10 b Fi(\000)p Fr(F)g Fv(\()p Fr(y)r Fv(\)\))p Fr(=\017)2659 5333 y Fs(\036)2713 5347 y Fr(x)p 2599 5374 219 4 v 2599 5457 a Fs(\036)2653 5471 y Fr(x)2699 5457 y Ft(\()p Fs(y)s Ft(\))2844 5395 y Fs(du)2943 5267 y Fh(\023)3010 5486 y Fr(z)s(y)r Fi(2)p 3136 5431 86 4 v(M)3221 5495 y Fk(x)3477 5276 y Ft(\(4)p Fs(:)p Ft(46\))p eop %%Page: 37 37 37 36 bop 50 295 a Fj(Sp)l(e)l(ctr)l(a)3340 b Fm(37)50 513 y Ft(During)30 b(the)g(rest)h(of)g(the)f(c)m(hapter)h(w)m(e)g (write)g(shorthand)e Fs(h)2126 480 y Fr(\025)2126 535 y(y)2199 513 y Fq(\021)c Fs(h)2347 480 y Fr(\025)2347 556 y(y)r(;)p 2408 502 86 4 v Fi(M)2493 565 y Fk(x)2538 513 y Ft(.)41 b(Let)31 b(us)f(furthermore)e(de\014ne)1588 775 y Fq(N)1663 789 y Fr(x)1734 775 y Fq(\021)d(D)1900 789 y Fr(x)1966 775 y Fq(\000)20 b(K)2126 789 y Fr(x)3477 775 y Ft(\(4)p Fs(:)p Ft(47\))50 1038 y(where)174 1301 y Fq(D)244 1315 y Fr(x)315 1301 y Fq(\021)25 b Ft(diag)579 1173 y Fh(\022)646 1301 y Fs(\017)698 1177 y Fh(Z)748 1384 y Fr(@)t(B)846 1393 y Fk(y)905 1301 y Fs(e)947 1263 y Fi(\000)p Fv(\()p Fr(F)10 b Fi(\000)p Fr(F)g Fv(\()p Fr(y)r Fv(\)\))p Fr(=\017)1468 1239 y Fs(\036)1522 1253 y Fr(x)p 1409 1280 219 4 v 1409 1363 a Fs(\036)1463 1377 y Fr(x)1509 1363 y Ft(\()p Fs(y)s Ft(\))1638 1301 y Fs(@)1686 1315 y Fr(n)1736 1301 y Fs(h)1788 1263 y Fr(\025)1788 1323 y(y)1851 1301 y Fs(d\033)24 b Fq(\000)c Fs(\025)2133 1177 y Fh(Z)2183 1384 y Fr(B)2238 1393 y Fk(y)2297 1301 y Fs(e)2339 1263 y Fi(\000)p Fv(\()p Fr(F)10 b Fi(\000)p Fr(F)g Fv(\()p Fr(y)r Fv(\)\))p Fr(=\017)2860 1239 y Fs(\036)2914 1253 y Fr(x)p 2801 1280 V 2801 1363 a Fs(\036)2855 1377 y Fr(x)2901 1363 y Ft(\()p Fs(y)s Ft(\))3045 1301 y Fs(du)3144 1173 y Fh(\023)3212 1392 y Fr(y)r()f Ft(0)g Fo(and)h(some)f Fs(C)50 b Fq(\021)43 b Fs(C)7 b Ft(\()p Fs(d;)15 b(F)s(;)g(\013)p Ft(\))50 4018 y Fo(dominating)34 b(the)f(supr)-5 b(emum)34 b(norms)g(of)f(the)g(L)-5 b(andau)34 b(symb)-5 b(ols)34 b(app)-5 b(e)g(aring)35 b(b)-5 b(elow)34 b(and)f(al)5 b(l)1358 4281 y Fs(e)1400 4244 y Fr(\013=\017)1520 4281 y Fs(=)p Fq(E)1613 4295 y Fr(x)1684 4281 y Fs(<)25 b(\025)g(<)g(e)1996 4244 y Fi(\000)p Fr(\013=\017)2173 4281 y Fs(=T)p 2271 4257 86 4 v 31 x Fi(M)2357 4321 y Fk(x)3477 4281 y Ft(\(4)p Fs(:)p Ft(51\))50 4544 y Fo(the)33 b(inverse)f(of)h Fq(K)689 4558 y Fr(x)735 4544 y Ft(\()p Fs(\025)p Ft(\))g Fo(exists.)43 b(Mor)-5 b(e)33 b(pr)-5 b(e)g(cisely,)33 b(uniformly)h(in)f Fs(\025)995 4813 y Ft(\()p Fq(K)1099 4827 y Fr(x)1145 4813 y Ft(\()p Fs(\025)p Ft(\))1268 4776 y Fi(\000)p Fv(1)1363 4813 y Fs(~)-41 b(g)1410 4827 y Fr(x)1456 4813 y Ft(\()p Fs(\025)p Ft(\)\))1614 4827 y Fr(y)1684 4813 y Ft(=)25 b Fq(O)1855 4739 y Fh(\000)1896 4813 y Ft(1\))2125 4752 y(1)p 1987 4792 319 4 v 1987 4875 a Fs(\017)2024 4849 y Fr(N)2094 4875 y Fs(\025T)2200 4889 y Fr(y)r(;x)2317 4813 y Fs(;)202 b(y)28 b Fq(\024)d Fs(x)712 b Ft(\(4)p Fs(:)p Ft(52\))50 5091 y Fo(Mor)-5 b(e)g(over,)33 b(we)g(obtain)1188 5245 y Fs(\025)25 b Fq(2)g Fs(\033)s Ft(\()p Fs(L)1504 5259 y Fr(\017)1540 5245 y Ft(\))211 b Fq(\()-15 b(\))211 b Fs(G)2235 5259 y Fr(x)2281 5245 y Ft(\()p Fs(\025)p Ft(\))26 b(=)f(0)906 b(\(4)p Fs(:)p Ft(53\))50 5462 y Fo(wher)-5 b(e)34 b Fs(G)378 5476 y Fr(x)424 5462 y Ft(\()p Fs(\025)p Ft(\))f Fo(is)g(de\014ne)-5 b(d)33 b(in)g(\(4.50\).)p eop %%Page: 38 38 38 37 bop 50 295 a Fm(38)3270 b Fj(Se)l(ction)27 b Fm(4)50 513 y Fd(Pro)s(of:)47 b Ft(F)-8 b(ormally)31 b(w)m(e)g(obtain)261 768 y Fq(K)330 782 y Fr(x)376 768 y Ft(\()p Fs(\025)p Ft(\))499 731 y Fi(\000)p Fv(1)623 768 y Ft(=)719 695 y Fh(\000)761 768 y Ft(1)-23 b(I)20 b Fq(\000)g(D)997 782 y Fr(x)1043 768 y Ft(\()p Fs(\025)p Ft(\))1166 731 y Fi(\000)p Fv(1)1265 768 y Fq(N)1340 782 y Fr(x)1385 768 y Ft(\()p Fs(\025)p Ft(\))1508 695 y Fh(\001)1550 714 y Fi(\000)p Fv(1)1648 768 y Fq(D)1718 782 y Fr(x)1764 768 y Ft(\()p Fs(\025)p Ft(\))1887 731 y Fi(\000)p Fv(1)2010 768 y Ft(=)2136 655 y Fi(1)2106 682 y Fh(X)2109 875 y Fr(s)p Fv(=0)2238 768 y Ft(\()p Fq(D)2343 782 y Fr(x)2389 768 y Ft(\()p Fs(\025)p Ft(\))2512 731 y Fi(\000)p Fv(1)2610 768 y Fq(N)2685 782 y Fr(x)2731 768 y Ft(\()p Fs(\025)p Ft(\)\))2889 731 y Fr(s)2929 768 y Fq(D)2999 782 y Fr(x)3045 768 y Ft(\()p Fs(\025)p Ft(\))3168 731 y Fi(\000)p Fv(1)3477 768 y Ft(\(4)p Fs(:)p Ft(54\))50 1035 y(In)29 b(order)h(to)h(mak)m(e)f (sense)g(out)g(of)g(this)g(calculation)j(and)c(to)i(extract)g(the)g (exp)s(onen)m(tial)g(deca)m(y)g(estimate)50 1180 y(written)c(in)f (\(4.52\))j(out)e(of)g(the)g(sum)e(a)i(straigh)m(tforw)m(ard)h (computation)f(for)f Fs(s)f Fq(2)g Fp(N)7 b Fq(n)p Ft(0)33 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