Content-Type: multipart/mixed; boundary="-------------0302031559473" This is a multi-part message in MIME format. ---------------0302031559473 Content-Type: text/plain; name="03-35.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-35.keywords" Euclidean field theory, pde with random coefficients, homogenization ---------------0302031559473 Content-Type: application/postscript; name="eft.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="eft.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.66a (C) 1986-97 Radical Eye Software (www.radicaleye.com) %%Title: eftpub.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -o eftpub.ps eftpub.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.02.03:1603 %%BeginProcSet: texc.pro %! /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul 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y(p)r(de)34 b(metho)r(ds,)i(estimates)e(on)f(correlation)f(functions)j(whic)n(h)f (are)f(as)g(sharp)g(as)g(those)h(ob-)456 4917 y(tained)22 b(using)h(m)n(ulti-scale)f(p)r(erturbation)g(theory)-7 b(.)34 b(Our)22 b(metho)r(d)h(has)f(the)i(added)e(adv)-5 b(an)n(tage)456 5016 y(that)33 b(there)g(is)g(no)g(small)g(parameter)f (as)h(in)g(the)h(m)n(ulti-scale)f(p)r(erturbation)f(theory)-7 b(.)54 b(Nev-)456 5116 y(ertheless,)34 b(the)g(metho)r(d)h(is)e(still)h (based)g(on)f(p)r(erturbation)g(theory)-7 b(.)55 b(W)-7 b(e)34 b(should)g(also)f(note)456 5216 y(that)26 b(m)n(ulti-scale)g(p)r (erturbation)g(theory)f(has)h(yielded)h(results)f(whic)n(h)g(do)g(not)g (app)r(ear)g(to)g(b)r(e)p eop %%Page: 3 3 3 2 bop 1520 226 a Fm(PDE)28 b(AND)h(FIELD)g(THEOR)-5 b(Y)1030 b(3)456 425 y Fr(pro)n(v)-5 b(able)21 b(b)n(y)h(the)h(metho)r (ds)g(in)g(this)g(pap)r(er.)34 b(In)23 b(particular,)g(the)g (analyticit)n(y)e(of)i(the)g(pressure)456 525 y(for)k(the)h 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Fm(+)p Fo(t)1372 2270 y Ff(0)1427 2274 y Fr(and)29 b Fp(\034)1626 2286 y Fm(0)p Fo(;)p Fm(0)1745 2274 y Fr(=)f(iden)n(tit)n(y)-7 b(.)40 b(One)29 b(can)f(also)g(see)g(that)h(the)g(mapping)456 2384 y Fp(\034)492 2396 y Fo(x;t)624 2384 y Fr(:)701 2363 y(^)692 2384 y(\012)45 b Fn(!)935 2363 y Fr(^)925 2384 y(\012)c(is)g(Borel)f(measurable)f(and)i(measure)f(preserving.)75 b(Finally)-7 b(,)44 b(it)d(follo)n(ws)456 2495 y(from)28 b(the)i(fact)f(that)h(the)f(space)1522 2474 y(^)1512 2495 y(\012)h(consists)e(of)h(con)n(tin)n(uous)f(functions,)i(that)f (the)h(mapping)456 2598 y(\()p Fp(t;)14 b(!)s Fr(\))23 b Fn(!)g Fp(\034)807 2610 y Fm(0)p Fo(;t)889 2598 y Fp(!)s(;)37 b(t)23 b Fn(2)g Fu(R)p Fp(;)36 b(!)25 b Fn(2)1431 2577 y Fr(^)1422 2598 y(\012,)e(from)e Fu(R)7 b Fn(\002)1877 2577 y Fr(^)1869 2598 y(\012)21 b(to)2055 2577 y(^)2045 2598 y(\012)h(is)g(measurable.)33 b(Hence)22 b(the)h(probabilit)n(y)456 2709 y(space)h(\()716 2688 y(^)707 2709 y(\012)p Fp(;)827 2688 y Fr(^)804 2709 y Fn(F)8 b Fp(;)928 2688 y Fr(^)909 2709 y Fp(P)k Fr(\))26 b(together)e(with)i(the)f(translation)f(op)r (erators)g Fp(\034)2504 2721 y Fo(x;t)2591 2709 y Fp(;)39 b(x)23 b Fn(2)h Fu(Z)2860 2679 y Fo(d)2899 2709 y Fp(;)39 b(t)23 b Fn(2)g Fu(R)p Fr(,)j(satisfy)456 2809 y(the)g(conditions)g(of) h(Theorem)e(1.2)h(of)g([4].)36 b(W)-7 b(e)27 b(denote)f(b)n(y)g Fp(<)d Fn(\001)g Fp(>)2537 2821 y Fm(^)2530 2836 y(\012)2608 2809 y Fr(exp)r(ectation)j(w.r.)36 b(to)26 b(the)456 2922 y(space)g(\()718 2901 y(^)709 2922 y(\012)q Fp(;)829 2901 y Fr(^)807 2922 y Fn(F)8 b Fp(;)930 2901 y Fr(^)912 2922 y Fp(P)j Fr(\).)555 3030 y(Let)31 b Fu(a)e Fr(:)843 3009 y(^)834 3030 y(\012)f Fn(!)h Fu(R)1106 3000 y Fo(d)p Fm(\()p Fo(d)p Fm(+1\))p Fo(=)p Fm(2)1413 3030 y Fr(b)r(e)i(a)g(b)r (ounded)g(Borel)f(measurable)f(function)j(from)e(\012)h(to)g(the)456 3129 y(space)j(of)h(symmetric)f Fp(d)24 b Fn(\002)f Fp(d)35 b Fr(matrices.)58 b(W)-7 b(e)36 b(assume)e(that)h Fu(a)p Fr(\()p Fp(!)s Fr(\))h(satis\014es)e(an)h(inequalit)n(y)456 3229 y(analogous)25 b(to)i(\(1.1\).)37 b(Th)n(us)1442 3380 y Fp(\025I)1526 3392 y Fo(d)1588 3380 y Fn(\024)23 b Fu(a)p Fr(\()p Fp(!)s Fr(\))h Fn(\024)e Fr(\003)p Fp(I)2046 3392 y Fo(d)2085 3380 y Fp(;)97 b(!)26 b Fn(2)2371 3359 y Fr(^)2361 3380 y(\012)p Fp(;)-1988 b Fr(\(1.5\))456 3528 y(where)24 b(the)h(inequalit)n(y)f(is)g(in)h(the)g(sense)g(of)f (quadratic)g(forms.)35 b(Since)25 b(the)g(mapping)f(\()p Fp(t;)14 b(!)s Fr(\))23 b Fn(!)456 3636 y Fp(\034)492 3648 y Fm(0)p Fo(;t)597 3636 y Fp(!)h Fr(is)c(measurable,)h(it)g(follo) n(ws)f(that)h(for)g(almost)f(ev)n(ery)g Fp(!)25 b Fn(2)2476 3615 y Fr(^)2467 3636 y(\012)c(w)n(e)f(can)g(consider)g(solutions)456 3735 y(to)27 b(the)h(initial)g(v)-5 b(alue)27 b(problem,)765 3861 y Fp(@)5 b(u)p 765 3898 97 4 v 774 3974 a(@)g(t)871 3917 y Fr(\()p Fp(x;)14 b(t;)g(!)s Fr(\))83 b(=)g Fn(\000r)1506 3883 y Fl(\003)1544 3917 y Fu(a)p Fr(\()p Fp(\034)1658 3929 y Fo(x;t)1746 3917 y Fp(!)s Fr(\))p Fn(r)p Fp(u)p 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4591 y Fp(G)1945 4603 y Fi(a)1986 4591 y Fr(\()p Fp(x;)14 b(y)s(;)g(t;)g(!)s Fr(\))p Fp(f)9 b Fr(\()p Fp(y)s(;)14 b(!)s Fr(\))p Fp(;)-2154 b Fr(\(1.7\))456 4830 y(where)34 b Fp(G)768 4842 y Fi(a)809 4830 y Fr(\()p Fp(x;)14 b(y)s(;)g(t;)g(!)s Fr(\))36 b(is)f(the)g (Green's)g(function.)60 b(By)35 b(translation)f(in)n(v)-5 b(ariance)34 b(there)h(is)g(a)456 4929 y(function)28 b Fp(G)846 4941 y Fi(a)887 4929 y Fr(\()p Fp(x;)14 b(t)p Fr(\))p Fp(;)42 b(x)24 b Fn(2)f Fu(Z)1337 4899 y Fo(d)1376 4929 y Fp(;)42 b(t)23 b Fn(\025)f Fr(0,)28 b(suc)n(h)f(that)1397 5072 y Fn(h)q Fp(G)1495 5084 y Fi(a)1536 5072 y Fr(\()p Fp(x;)14 b(y)s(;)g(t;)g Fn(\001)p Fr(\))p Fn(i)1894 5084 y Fm(^)1887 5099 y(\012)1962 5072 y Fr(=)23 b Fp(G)2115 5084 y Fi(a)2156 5072 y Fr(\()p Fp(x)c Fn(\000)f Fp(y)s(;)c(t)p Fr(\))p Fp(:)-2047 b Fr(\(1.8\))456 5216 y(W)-7 b(e)28 b(are)e(no)n(w)h(able)g(to)h(state)f(our)g(main)h(result.)p eop %%Page: 4 4 4 3 bop 456 226 a Fm(4)1108 b(JOSEPH)28 b(G.)h(CONLON)456 425 y Fu(Theorem)h(1.2.)40 b Fg(L)l(et)21 b Fp(d)i Fn(\025)g Fr(3)e Fg(and)g Fu(a)j Fr(:)1680 404 y(^)1671 425 y(\012)f Fn(!)g Fu(R)1932 395 y Fo(d)p Fm(\()p Fo(d)p Fm(+1\))p Fo(=)p Fm(2)2229 425 y Fg(b)l(e)e(de\014ne)l(d)h(by)g Fu(a)p Fr(\()p Fp(!)s Fr(\))h(=)g Fp(V)3037 395 y Fl(00)3079 425 y Fr(\()p Fn(r)p Fp(!)s Fr(\(0)p Fp(;)14 b Fr(0\)\))p Fp(;)35 b(!)26 b Fn(2)465 512 y Fr(^)456 533 y(\012)p Fg(,)33 b(wher)l(e)g Fp(V)51 b Fg(satis\014es)32 b(\(1.1\).)48 b(Then)33 b Fu(a)p Fr(\()p Fn(\001)p Fr(\))g Fg(satis\014es)f(\(1.5\))i (and)e(the)h(function)f Fp(G)3082 545 y Fi(a)3123 533 y Fr(\()p Fp(x;)14 b(t)p Fr(\))33 b Fg(de-)456 633 y(\014ne)l(d)c(by)h (\(1.6\),)i(\(1.7\),)f(\(1.8\))g(satis\014es)f(the)g(identity,)1384 814 y Fn(h)p Fp(!)s Fr(\()p Fp(x)p Fr(\))p Fp(!)s Fr(\(0\))p Fn(i)1776 839 y Fm(\012)1850 814 y Fr(=)1938 701 y Fj(Z)2021 722 y Fl(1)1984 890 y Fm(0)2135 814 y Fp(G)2200 826 y Fi(a)2241 814 y Fr(\()p Fp(x;)14 b(t)p Fr(\))p Fp(dt:)-2059 b Fr(\(1.9\))555 1005 y(The)32 b(pro)r(of)f(of)g(Theorem)g(1.2)g(is)g (obtained)g(b)n(y)h(\014rst)f(establishing)g(a)g(\014nite)h (dimensional)456 1105 y(v)n(ersion)25 b(of)h(\(1.9\).)36 b(Then)27 b(the)g(thermo)r(dynamic)f(limit)h(is)g(tak)n(en.)36 b(The)26 b(main)h(tec)n(hnical)f(issue)456 1205 y(in)33 b(the)h(pap)r(er)f(is)g(to)g(pro)n(v)n(e)f(the)i(existence)f(of)g(this) h(limit.)55 b(In)33 b Fn(x)p Fr(2)g(w)n(e)g(giv)n(e)f(a)h(construction) 456 1308 y(of)i(the)h(probabilit)n(y)f(spaces)f(\(\012)p Fp(;)14 b Fn(F)8 b Fp(;)14 b(P)e Fr(\))36 b(and)f(\()1981 1287 y(^)1972 1308 y(\012)q Fp(;)2092 1287 y Fr(^)2070 1308 y Fn(F)7 b Fp(;)2193 1287 y Fr(^)2174 1308 y Fp(P)12 b Fr(\))36 b(b)n(y)f(means)g(of)h(\014nite)g(dimensional)456 1407 y(appro)n(ximations.)55 b(T)-7 b(o)34 b(construct)g(the)g(space)g (\(\012)p Fp(;)14 b Fn(F)8 b Fp(;)14 b(P)e Fr(\))34 b(b)n(y)h(\014nite) g(dimensional)e(appro)n(xi-)456 1507 y(mation)27 b(w)n(e)h(let)g Fp(L)g Fr(b)r(e)g(a)g(p)r(ositiv)n(e)f(ev)n(en)h(in)n(teger)f(and)h Fp(Q)23 b Fr(=)g Fp(Q)2425 1519 y Fo(L)2498 1507 y Fn(\032)h Fu(Z)2645 1477 y Fo(d)2712 1507 y Fr(b)r(e)k(the)g(lattice)h(p)r(oin)n (ts)456 1607 y(con)n(tained)f(in)h(the)h(cub)r(e)f(cen)n(tered)g(at)f (the)i(origin)e(with)h(side)g(of)g(length)g Fp(L)p Fr(.)41 b(Let)29 b(\012)3129 1619 y Fo(Q)3215 1607 y Fr(b)r(e)g(the)456 1706 y(space)22 b(of)h(all)g(p)r(erio)r(dic)g(functions)g Fp(!)j Fr(:)d Fp(Q)g Fn(!)g Fu(R)p Fr(.)35 b(Eviden)n(tly)22 b(\012)2422 1718 y Fo(Q)2479 1706 y Fp(;)37 b(Q)23 b Fr(=)f Fp(Q)2781 1718 y Fo(L)2854 1706 y Fr(can)g(b)r(e)i(iden)n (ti\014ed)456 1809 y(with)f Fu(R)712 1779 y Fo(N)798 1809 y Fr(where)g Fp(N)32 b Fr(=)22 b Fp(L)1277 1779 y Fo(d)1316 1809 y Fr(.)35 b(Let)24 b Fn(F)1579 1821 y Fo(Q)1658 1809 y Fr(b)r(e)g(the)f(Borel)f(algebra)g(for)h(\012)2591 1821 y Fo(Q)2670 1809 y Fr(whic)n(h)h(is)f(generated)f(b)n(y)456 1911 y(the)29 b(op)r(en)f(sets)h(of)f Fu(R)1132 1881 y Fo(N)1195 1911 y Fr(.)40 b(W)-7 b(e)29 b(de\014ne)g(a)f(probabilit)n (y)g(measure)f Fp(P)2516 1923 y Fo(Q;m)2652 1911 y Fp(;)38 b(m)25 b(>)g Fr(0,)j(on)g(\(\012)3202 1923 y Fo(Q)3259 1911 y Fp(;)14 b Fn(F)3356 1923 y Fo(Q)3412 1911 y Fr(\))456 2010 y(b)n(y)811 2240 y(exp)952 2073 y Fj(2)952 2222 y(4)1007 2240 y Fn(\000)1093 2161 y Fj(X)1086 2339 y Fo(x)p Fl(2)p Fo(Q)1234 2123 y Fj(\032)1296 2240 y Fp(V)19 b Fr(\()p Fn(r)p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))h(+)1775 2184 y(1)p 1775 2221 42 4 v 1775 2297 a(2)1849 2240 y Fp(m)1922 2206 y Fm(2)1960 2240 y Fp(!)s Fr(\()p Fp(x)p Fr(\))2126 2206 y Fm(2)2164 2123 y Fj(\033)2226 2073 y(3)2226 2222 y(5)2337 2161 y(Y)2323 2339 y Fo(x)p Fl(2)p Fo(Q)2471 2240 y Fp(d!)s Fr(\()p Fp(x)p Fr(\))2704 2173 y Fj(\016)2780 2240 y Fr(normalization)n Fp(:)-2845 b Fr(\(1.10\))456 2474 y(In)32 b(the)h(de\014nition)g(\(1.10\))f(w)n(e)h (are)e(iden)n(tifying)i(p)r(oin)n(ts)g(on)f(the)h(b)r(oundary)f(of)g Fp(Q)h Fr(since)f(\012)3388 2486 y Fo(Q)456 2574 y Fr(consists)i(of)i (p)r(erio)r(dic)f(functions)h(on)g Fp(Q)p Fr(.)60 b(It)36 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Fl(00)1481 3009 y Fr(\()p Fp(z)t Fr(\))p Fn(j)23 b(\024)g Fp(A)14 b Fr(exp[)p Fp(B)t Fn(j)p Fp(z)t Fn(j)p Fr(])p Fp(;)69 b(z)26 b Fn(2)d Fu(R)2434 2975 y Fo(N)2497 3009 y Fp(;)-2064 b Fr(\(1.11\))456 3144 y(where)27 b Fp(A)h Fr(and)f Fp(B)32 b Fr(are)26 b(constan)n(ts.)36 b(W)-7 b(e)28 b(de\014ne)g(then)512 3295 y Fj(D)562 3388 y Fp(f)612 3295 y Fj(\020)661 3388 y Fp(!)s Fr(\()p Fp(x)795 3400 y Fm(1)833 3388 y Fr(\))p Fp(;)14 b(!)s Fr(\()p Fp(x)1036 3400 y Fm(2)1074 3388 y Fr(\))p Fp(;)g(:::;)g(!)s Fr(\()p Fp(x)1383 3400 y Fo(N)1446 3388 y Fr(\))1478 3295 y Fj(\021E)1579 3446 y Fm(\012)1654 3388 y Fr(=)44 b(lim)1741 3439 y Fo(m)p Fl(!)p Fm(0)1945 3388 y Fr(lim)1913 3441 y Fo(L)p Fl(!1)2105 3295 y Fj(D)2156 3388 y Fp(f)2206 3295 y Fj(\020)2255 3388 y Fp(!)s Fr(\()p Fp(x)2389 3400 y Fm(1)2427 3388 y Fr(\))p Fp(;)14 b(!)s Fr(\()p Fp(x)2630 3400 y Fm(2)2667 3388 y Fr(\))p Fp(;)g(:::;)g(!)s Fr(\()p Fp(x)2976 3400 y Fo(N)3040 3388 y Fr(\))3072 3295 y Fj(\021E)3173 3446 y Fo(Q)3225 3454 y Fh(L)3269 3446 y Fo(;m)3365 3388 y Fp(:)456 3250 y Fr(\(1.12\))456 3563 y(In)29 b Fn(x)p Fr(2)f(w)n(e)h(sho)n(w)f(that)h(the)g(limit)h(on)e(the)h(RHS)h(of)f (\(1.12\))f(exists)g(pro)n(vided)g Fp(d)e Fn(\025)e Fr(3.)41 b(The)29 b(use)456 3662 y(of)e(the)h(Brascamp-Lieb)e(inequalit)n(y)h ([1)o(])h(is)g(crucial)e(to)i(the)g(pro)r(of.)555 3762 y(An)21 b(alternativ)n(e)e(construction)g(of)h(the)h(space)e(\(\012)p Fp(;)14 b Fn(F)8 b Fp(;)14 b(P)e Fr(\))21 b(is)f(giv)n(en)f(in)h([7])g (\(see)g(also)f([13)o(]\).)35 b(In)456 3861 y(that)22 b(construction)f(the)i(elemen)n(ts)f(of)g(the)g(probabilit)n(y)f(space) h(are)f(gradien)n(ts)g Fn(r)p Fp(!)k Fr(of)d(the)g(\014eld)456 3961 y Fp(!)s Fr(.)35 b(Hence)24 b(the)f(construction)g(is)h(v)-5 b(alid)23 b(in)h(all)f(dimensions)g Fp(d)h Fn(\025)e Fr(1)h(whereas)g(our)f(construction)456 4061 y(b)n(y)36 b(means)h(of)g(\(1.12\))f(is)h(restricted)f(to)h Fp(d)h Fn(\025)h Fr(3.)64 b(One)37 b(migh)n(t)g(therefore)f(exp)r(ect)h(that)g (for)456 4160 y Fp(f)31 b Fr(:)23 b Fu(R)646 4130 y Fo(N)6 b(d)767 4160 y Fn(!)23 b Fu(C)k Fr(satisfying)g(\(1.11\))g(the)h (limit,)1056 4320 y(lim)1034 4371 y Fo(m)p Fl(!)p Fm(0)1238 4320 y Fr(lim)1206 4373 y Fo(L)p Fl(!1)1398 4228 y Fj(D)1449 4320 y Fp(f)1499 4228 y Fj(\020)1548 4320 y Fn(r)p Fp(!)s Fr(\()p Fp(x)1751 4332 y Fm(1)1789 4320 y Fr(\))p Fp(;)14 b Fn(r)p Fp(!)s Fr(\()p Fp(x)2061 4332 y Fm(2)2099 4320 y Fr(\))p Fp(;)g(:::;)g Fn(r)p Fp(!)s Fr(\()p Fp(x)2477 4332 y Fo(N)2541 4320 y Fr(\))2573 4228 y Fj(\021)o(E)2673 4378 y Fo(Q)2725 4386 y Fh(L)2769 4378 y Fo(;m)456 4320 y Fr(\(1.13\))456 4495 y(exists)28 b(for)h(all)g Fp(d)d Fn(\025)f Fr(1.)42 b(Eviden)n(tly)28 b(b)n(y)h(virtue)g(of)g(\(1.12\))g (the)h(limit)g(\(1.13\))e(exists)h(for)g Fp(d)c Fn(\025)h Fr(3.)456 4594 y(Our)g(pro)r(of)i(ho)n(w)n(ev)n(er)d(breaks)h(do)n(wn)h (if)h Fp(d)c Fn(\024)e Fr(2.)561 4767 y(2.)77 b Fq(Constr)n(uction)31 b(of)h(the)g(pr)n(obability)f(sp)-6 b(a)n(ces)33 b Fr(\(\012)p Fp(;)14 b Fn(F)8 b Fp(;)14 b(P)e Fr(\))31 b Fq(and)g Fr(\()3022 4746 y(^)3013 4767 y(\012)q Fp(;)3133 4746 y Fr(^)3111 4767 y Fn(F)7 b Fp(;)3234 4746 y Fr(^)3215 4767 y Fp(P)12 b Fr(\))p Fq(.)555 4917 y Fr(W)-7 b(e)42 b(turn)f(to)g(the)g(construction)g(of)g(the)g(space)g(\(\012)p Fp(;)14 b Fn(F)8 b Fp(;)14 b(P)e Fr(\))41 b(as)f(a)h(limit)h(of)f(the)g (spaces)456 5016 y(\(\012)548 5028 y Fo(Q)604 5016 y Fp(;)14 b Fn(F)701 5028 y Fo(Q)757 5016 y Fp(;)g(P)847 5028 y Fo(Q;m)983 5016 y Fr(\))23 b(with)g(probabilit)n(y)f(measure)f (\(1.10\).)35 b(Our)22 b(main)h(task)f(will)h(b)r(e)g(to)g(establish) 456 5116 y(the)e(existence)g(of)g(the)h(limit)g(on)f(the)g(R.H.S.)h(of) f(\(1.12\).)34 b(Let)22 b Fp(L)2419 5086 y Fm(2)2455 5116 y Fr(\()p Fp(Q)p Fr(\))g(b)r(e)f(the)h(space)e(of)i(p)r(erio)r (dic)456 5216 y(functions)38 b Fp(h)i Fr(:)h Fp(Q)f Fn(!)h Fu(C)d Fr(with)h(the)f(standard)f(Euclidean)h(inner)g(pro)r(duct.)68 b(F)-7 b(or)38 b(a)f(giv)n(en)p eop %%Page: 5 5 5 4 bop 1520 226 a Fm(PDE)28 b(AND)h(FIELD)g(THEOR)-5 b(Y)1030 b(5)456 425 y Fp(!)25 b Fn(2)f Fr(\012)672 437 y Fo(Q)755 425 y Fr(w)n(e)j(consider)g(the)g(op)r(erator)f([)p Fn(r)1770 395 y Fl(\003)1808 425 y Fp(V)1875 395 y Fl(00)1917 425 y Fr(\()p Fn(r)p Fp(!)s Fr(\))p Fn(r)19 b Fr(+)e Fp(m)2348 395 y Fm(2)2385 425 y Fr(])28 b(acting)e(on)h Fp(L)2856 395 y Fm(2)2893 425 y Fr(\()p Fp(Q)p Fr(\).)37 b(In)27 b(view)h(of)456 525 y(\(1.1\))19 b(the)i(op)r(erator)d([)p Fn(r)1201 495 y Fl(\003)1239 525 y Fp(V)1306 495 y Fl(00)1349 525 y Fr(\()p Fn(r)p Fp(!)s Fr(\))p Fn(r)s Fr(+)s Fp(m)1750 495 y Fm(2)1787 525 y Fr(])i(is)g(b)r(ounded)g(b)r(elo)n(w)g(b)n(y)g (the)g(op)r(erator)e([)p Fn(\000)p Fp(\025)p Fr(\001)s(+)s Fp(m)3384 495 y Fm(2)3421 525 y Fr(])456 624 y(on)33 b Fp(L)634 594 y Fm(2)670 624 y Fr(\()p Fp(Q)p Fr(\),)j(where)c(\001)i (is)f(the)h(standard)e(lattice)i(Laplacian.)53 b(The)33 b(eigenfunctions)g(of)h Fn(\000)p Fr(\001)456 732 y(are)e(exp[)p Fp(i\030)26 b Fn(\001)d Fp(x)p Fr(])p Fp(;)48 b(x)33 b Fn(2)h Fp(Q)p Fr(,)h(where)e Fp(\030)38 b Fr(lies)33 b(in)h(the)g(dual)2251 711 y(^)2232 732 y Fp(Q)g Fr(of)f Fp(Q)p Fr(.)55 b(In)34 b(fact)f(if)i Fp(Q)d Fr(=)h Fp(Q)3200 744 y Fo(L)3283 732 y Fr(then)475 819 y(^)456 840 y Fp(Q)e Fr(is)h(the)h(set)f(of)g(lattice)g(p)r(oin)n(ts)g(of)g(\(2)p Fp(\031)s(=L)p Fr(\))p Fu(Z)1946 810 y Fo(d)2017 840 y Fr(whic)n(h)g(lie)g(in)h(the)f(cub)r(e)h([)p Fn(\000)p Fp(\031)s(;)14 b(\031)s Fr(])3072 810 y Fo(d)3111 840 y Fr(.)50 b(Just)32 b(as)456 939 y(with)g Fp(Q)p Fr(,)h(the)f(b)r (oundary)f(p)r(oin)n(ts)g(of)h([)p Fn(\000)p Fp(\031)s(;)14 b(\031)s Fr(])1896 909 y Fo(d)1967 939 y Fr(are)31 b(iden)n(ti\014ed)h (,)h(whence)f(there)f(are)g(exactly)456 1047 y Fp(L)513 1017 y Fo(d)578 1047 y Fr(distinct)e(p)r(oin)n(ts)e(in)1244 1026 y(^)1225 1047 y Fp(Q)p Fr(.)37 b(Since)28 b(the)f(eigen)n(v)-5 b(alues)27 b(of)g Fn(\000)p Fr(\001)h(are)e(giv)n(en)h(b)n(y)1282 1259 y Fn(\000)p Fr(\001[)p Fp(e)1478 1225 y Fo(i\030)r Fl(\001)p Fo(x)1594 1259 y Fr(])c(=)g(2)1826 1155 y Fo(d)1783 1180 y Fj(X)1789 1357 y Fo(i)p 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y Fo(Q)2547 2156 y Fp(:)-2114 b Fr(\(2.1\))456 2336 y(Let)32 b Fp(<)d Fn(\001)h Fp(>)821 2348 y Fo(Q;m)988 2336 y Fr(denote)h(exp)r(ectation)h(on)f(\012)1890 2348 y Fo(Q)1978 2336 y Fr(with)i(resp)r(ect)e(to)h(the)g(measure)f Fp(P)3094 2348 y Fo(Q;m)3229 2336 y Fr(.)49 b(The)456 2441 y(Brascamp-Lieb)27 b(inequalit)n(y)h(states)h(that)g(if)g Fp(J)38 b Fr(satis\014es)28 b(\(2.1\))g(then)i Fp(J)37 b Fr(and)2950 2408 y Fo(@)t(J)p 2949 2422 83 4 v 2949 2469 a(@)t(!)3071 2441 y Fr(are)28 b(square)456 2540 y(in)n(tegrable.)35 b(F)-7 b(urther,)28 b(there)f(is)h(the)g(inequalit)n(y)-7 b(,)686 2653 y Fj(\012)725 2720 y Fn(j)p Fp(J)8 b Fr(\()p Fp(!)s Fr(\))p Fn(\000)23 b Fp(<)g(J)8 b Fr(\()p Fn(\001)p Fr(\))23 b Fp(>)1326 2732 y Fo(Q;m)1484 2720 y Fn(j)1507 2686 y Fm(2)1544 2653 y Fj(\013)1584 2753 y Fo(Q;m)1742 2720 y Fn(\024)1829 2603 y Fj(\034)q(\022)1963 2664 y Fp(@)5 b(J)p 1963 2701 104 4 v 1963 2777 a(@)g(!)2076 2720 y(;)2113 2628 y Fj(h)2152 2720 y Fn(r)2221 2686 y Fl(\003)2259 2720 y Fp(V)2326 2686 y Fl(00)2369 2720 y Fr(\()p Fn(r)p Fp(!)s Fr(\))p Fn(r)19 b Fr(+)f Fp(m)2801 2686 y Fm(2)2838 2628 y Fj(i)2877 2645 y Fl(\000)p Fm(1)2977 2664 y Fp(@)5 b(J)p 2976 2701 V 2976 2777 a(@)g(!)3090 2603 y Fj(\023\035)3213 2803 y Fo(Q;m)3362 2720 y Fp(:)-2929 b Fr(\(2.2\))456 2920 y(One)27 b(also)f(has)h(in)h(view)g(of)f(\(1.1\))h(the)g (inequalit)n(y)-7 b(,)1048 3104 y Fn(h)p Fr(exp[\()p Fp(h;)14 b(!)s Fr(\)])p Fn(i)1489 3129 y Fo(Q;m)1647 3104 y Fn(\024)23 b Fr(exp)1876 2987 y Fj(\024)1929 3048 y Fr(1)p 1929 3085 42 4 v 1929 3161 a(2)2009 3104 y(\()p Fp(h;)14 b Fr([)p Fn(\000)p Fp(\025)p Fr(\001)k(+)g Fp(m)2505 3070 y Fm(2)2542 3104 y Fr(])2565 3070 y Fl(\000)p Fm(1)2678 3104 y Fp(h)p Fr(\))2758 2987 y Fj(\025)2815 3104 y Fp(;)-2382 b Fr(\(2.3\))456 3294 y(for)27 b(an)n(y)f Fp(h)d Fn(2)h Fp(L)946 3264 y Fm(2)983 3294 y Fr(\()p Fp(Q)p Fr(\).)456 3441 y Fu(Lemma)29 b(2.1.)40 b Fg(Supp)l(ose)29 b Fp(Q)23 b Fr(=)f Fp(Q)1541 3453 y Fo(L)1619 3441 y Fg(and)29 b Fp(m)f 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980 5123 244 4 v 980 5199 a(\(2)p Fp(\031)s Fr(\))1136 5175 y Fo(d)1175 5199 y Fp(\025)1247 5030 y Fj(Z)1308 5203 y Fm(^)1293 5218 y Fo(Q)1364 5143 y Fp(d\030)1461 5050 y Fj(D)1512 5143 y Fn(j)1552 5121 y Fr(^)1535 5143 y Fp(f)1576 5155 y Fm(1)1612 5143 y Fr(\()p Fp(!)s(;)14 b(\030)t Fr(\))p Fn(j)1831 5108 y Fm(2)1869 5050 y Fj(E)1920 5201 y Fo(Q;m)2069 5050 y Fj(.)2150 5001 y(")2198 5143 y Fp(m)2271 5108 y Fm(2)2308 5143 y Fp(=\025)19 b Fr(+)f(2)2598 5039 y Fo(d)2555 5064 y Fj(X)2561 5240 y Fo(i)p Fm(=1)2689 5143 y Fn(f)o Fr(1)g Fn(\000)g Fr(cos\()p Fp(\030)23 b Fn(\001)18 b Fu(e)3161 5155 y Fo(i)3188 5143 y Fr(\))p Fn(g)3262 5001 y Fj(#)3324 5143 y Fp(;)p eop %%Page: 7 7 7 6 bop 1520 226 a Fm(PDE)28 b(AND)h(FIELD)g(THEOR)-5 b(Y)1030 b(7)456 425 y Fr(where)1446 541 y(^)1428 563 y Fp(f)1469 575 y Fm(1)1506 563 y Fr(\()p Fp(!)s(;)14 b(\030)t Fr(\))23 b(=)1820 484 y Fj(X)1813 662 y Fo(x)p Fl(2)p Fo(Q)1989 563 y Fp(f)2039 528 y Fl(0)2062 563 y Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))p 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Fp(G)p Fr(\()p Fp(x)p Fr(\))i(is)e(the)h (Green's)g(function)g(satisfying)1283 1664 y([)p Fn(\000)p Fr(\001)18 b(+)g Fp(m)1614 1630 y Fm(2)1651 1664 y Fp(=\025)p Fr(])28 b Fp(G)p Fr(\()p Fp(x)p Fr(\))c(=)f Fp(\016)s Fr(\()p Fp(x)p Fr(\))p Fp(;)126 b(x)23 b Fn(2)g Fp(Q;)456 1801 y Fr(with)29 b Fp(\016)j Fr(b)r(eing)d(the)g(Kronec)n(k)n(er)e Fp(\016)32 b Fr(function.)41 b(By)29 b(the)g(Planc)n(herel)e(theorem)i (it)g(follo)n(ws)f(from)456 1901 y(\(2.9\),)f(\(2.10\))g(that)758 1971 y Fj(*)825 1993 y(\014)825 2043 y(\014)825 2092 y(\014)825 2142 y(\014)873 2034 y(X)866 2212 y Fo(x)p Fl(2)p Fo(Q)1015 2113 y Fr([)p Fp(f)9 b Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))p Fn(\000)23 b Fp(<)g(f)9 b Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))24 b Fp(>)1863 2125 y Fo(Q;m)1997 2113 y Fr(])2034 1993 y Fj(\014)2034 2043 y(\014)2034 2092 y(\014)2034 2142 y(\014)2062 2013 y Fm(2)2099 1971 y Fj(+)2166 2236 y Fo(Q;m)2324 2113 y Fn(\024)2422 2057 y Fp(L)2479 2027 y Fo(d)p 2422 2094 96 4 v 2427 2170 a Fp(\025)2475 2146 y Fm(2)2548 2034 y Fj(X)2541 2212 y Fo(x)p Fl(2)p Fo(Q)2717 2113 y Fp(H)7 b Fr(\()p Fp(x)p Fr(\))p Fp(G)p Fr(\()p Fp(x)p Fr(\))3080 2079 y Fm(2)3119 2113 y Fp(:)456 2371 y Fr(Since)27 b Fp(m=)787 2300 y Fn(p)p 856 2300 49 4 v 71 x Fp(\025)c Fn(\025)g Fr(1)p Fp(=L)j Fr(it)i(follo)n(ws)f(from)g(\(2.3\),)h(\(2.4\))f(that)1310 2510 y Fn(j)p Fp(H)7 b Fr(\()p Fp(x)p Fr(\))p Fn(j)24 b(\024)e Fp(H)7 b Fr(\(0\))23 b Fn(\024)g Fp(A)2009 2475 y Fm(2)2060 2510 y Fr(exp[2)p Fp(B)2319 2475 y Fm(2)2356 2510 y Fp(C)2415 2522 y Fo(d)2454 2510 y Fp(=\025)p Fr(])p Fp(;)456 2647 y Fr(for)31 b(some)g(constan)n(t)g Fp(C)1197 2659 y Fo(d)1267 2647 y Fr(dep)r(ending)h(only)f(on)h Fp(d)e Fn(\025)f Fr(3.)48 b(The)32 b(inequalit)n(y)f(\(2.5\))h(follo)n (ws)e(no)n(w)456 2747 y(on)d(using)g(the)h(iden)n(tit)n(y)-7 b(,)1508 2817 y Fj(X)1500 2995 y Fo(x)p Fl(2)p Fo(Q)1676 2896 y Fp(G)p Fr(\()p Fp(x)p Fr(\))1852 2862 y Fm(2)1914 2896 y Fr(=)23 b Fp(g)2042 2908 y Fo(d)2080 2896 y Fr(\()p Fp(m=)2227 2821 y Fn(p)p 2296 2821 V 75 x Fp(\025)p Fr(\))p Fp(;)456 3123 y Fr(where)k Fp(g)736 3135 y Fo(d)802 3123 y Fr(is)g(giv)n(en)g(b)n(y)g(\(2.7\).)p 3384 3123 4 57 v 3388 3071 50 4 v 3388 3123 V 3437 3123 4 57 v 456 3277 a Fu(Lemma)i(2.2.)40 b Fg(L)l(et)34 b Fp(m)d(>)f Fr(0)p Fg(.)52 b(Then)34 b(for)h(any)g Fp(C)2036 3247 y Fm(2)2107 3277 y Fg(function)f Fp(f)40 b Fr(:)31 b Fu(R)g Fn(!)g Fu(C)j Fg(satisfying)h(\(2.4\))456 3377 y(the)30 b(limit,)g Fr(lim)932 3389 y Fo(L)p Fl(!1)1166 3377 y Fp(<)23 b(f)9 b Fr(\()p Fp(!)s Fr(\(0\)\))23 b Fp(>)1617 3389 y Fo(Q)1669 3397 y Fh(L;m)1817 3377 y Fg(exists)29 b(and)h(is)g(\014nite.)456 3543 y(Pr)l(o)l(of.)43 b Fr(Supp)r(ose)27 b Fp(L;)14 b(L)1192 3512 y Fl(0)1241 3543 y Fr(are)26 b(p)r(ositiv)n(e)h(ev)n(en)g (in)n(tegers)f(with)h Fp(L)2427 3512 y Fl(0)2473 3543 y Fp(>)c Fr(2)p Fp(L;)40 b(L)23 b(>)f Fr(3)17 b(+)3031 3472 y Fn(p)p 3101 3472 49 4 v 3101 3543 a Fp(\025=m)p Fr(.)36 b(Let)456 3642 y Fp(@)5 b(Q)571 3654 y Fo(L)644 3642 y Fr(b)r(e)25 b(the)f(lattice)h(p)r(oin)n(ts)f(of)g Fu(Z)1539 3612 y Fo(d)1602 3642 y Fr(whic)n(h)h(form)f(the)g(b)r (oundary)g(of)g(the)h(cub)r(e)f Fp(Q)3029 3654 y Fo(L)3102 3642 y Fn(\032)e Fu(Z)3247 3612 y Fo(d)3311 3642 y Fr(and)456 3742 y(In)n(t\()p Fp(Q)660 3754 y Fo(L)709 3742 y Fr(\))i(=)e Fp(Q)918 3754 y Fo(L)968 3742 y Fn(n)p Fp(@)5 b(Q)1125 3754 y Fo(L)1173 3742 y Fr(.)37 b(W)-7 b(e)28 b(de\014ne)g(the)g(set)g Fp(U)1946 3754 y Fo(L)2023 3742 y Fr(b)n(y)456 3879 y Fp(U)513 3891 y Fo(L)585 3879 y Fr(=)23 b Fn(f)p Fp(x)g Fn(2)g Fp(@)5 b(Q)978 3891 y Fo(L)1050 3879 y Fr(:)23 b Fp(x)c Fr(+)f Fu(e)1289 3891 y Fo(i)1340 3879 y Fn(62)23 b Fp(Q)1484 3891 y Fo(L)1589 3879 y Fr(for)k(some)55 b Fp(i;)41 b Fr(1)22 b Fn(\024)h Fp(i)g Fn(\024)f Fp(d;)14 b Fr(and)55 b Fp(x)19 b Fn(\000)f Fu(e)2798 3891 y Fo(j)2856 3879 y Fn(2)24 b Fp(Q)3001 3891 y Fo(L)3105 3879 y Fr(for)j(all)55 b Fp(j;)42 b Fr(1)23 b Fn(\024)f Fp(j)28 b Fn(\024)23 b Fp(d)p Fn(g)p Fp(:)456 4020 y Fr(Observ)n(e)32 b(that)i(In)n(t\()p Fp(Q)1171 4032 y Fo(L)1221 4020 y Fr(\))22 b Fn([)h Fp(U)1410 4032 y Fo(L)1494 4020 y Fr(is)33 b(a)h(cub)r(e)g(con)n(taining)f Fp(L)2328 3990 y Fo(d)2400 4020 y Fr(lattice)h(p)r(oin)n(ts.)55 b(F)-7 b(or)33 b(eac)n(h)g Fp(x)h Fn(2)456 4119 y Fp(@)5 b(Q)571 4131 y Fo(L)620 4119 y Fp(;)41 b Fr(0)23 b Fn(\024)f Fp(t)h Fn(\024)g Fr(1,)k(w)n(e)h(de\014ne)f(a)h(function)g Fp(V)1874 4131 y Fo(x;t)1984 4119 y Fr(:)23 b(\012)2090 4131 y Fo(L)2136 4115 y Ff(0)2185 4119 y Fn(!)g Fu(R)k Fr(as)g(follo)n(ws:)562 4238 y Fu(\(a\):)41 b Fr(If)28 b Fp(x)c Fn(2)f Fp(U)1039 4250 y Fo(L)1116 4238 y Fr(then)961 4400 y Fp(V)1009 4412 y Fo(x;t)1096 4400 y Fr(\()p Fp(!)s Fr(\))g(=)f Fp(V)1392 4308 y Fj(\020)1442 4400 y Fp(!)s Fr(\()p Fp(x)d Fr(+)f Fu(e)1722 4412 y Fm(1)1759 4400 y Fr(\))h Fn(\000)f Fp(!)s Fr(\()p Fp(x)p Fr(\))p Fp(;)c(:::::;)g(t)p Fr([)p Fp(!)s Fr(\()p Fp(x)19 b Fr(+)f Fu(e)2581 4412 y Fo(j)2616 4400 y Fr(\))h Fn(\000)f Fp(!)s Fr(\()p Fp(x)p Fr(\)])896 4634 y(+\(1)g Fn(\000)g Fp(t)p Fr(\)[)p Fp(!)s Fr(\()p Fp(x)h Fn(\000)f Fr(\()p Fp(L)g Fn(\000)g Fr(1\))p Fu(e)1765 4646 y Fo(j)1800 4634 y Fr(\))h Fn(\000)f Fp(!)s Fr(\()p Fp(x)p Fr(\)])p Fp(;)c(:::::;)g(!)s Fr(\()p Fp(x)19 b Fr(+)f Fu(e)2592 4646 y Fo(d)2631 4634 y Fr(\))h Fn(\000)f Fp(!)s Fr(\()p Fp(x)p Fr(\))2931 4541 y Fj(\021)2981 4634 y Fp(;)661 4791 y Fr(where)24 b(w)n(e)g(in)n(tro)r(duce)g(the)h (in)n(terp)r(olation)f(parameter)f Fp(t)h Fr(for)g(an)n(y)g Fp(j)29 b Fr(with)c Fp(x)12 b Fr(+)g Fu(e)3169 4803 y Fo(j)3228 4791 y Fn(62)23 b Fp(Q)3372 4803 y Fo(L)3421 4791 y Fr(.)562 4891 y Fu(\(b\):)41 b Fr(If)28 b Fp(x)c Fn(2)f Fp(@)5 b(Q)1104 4903 y Fo(L)1153 4891 y Fn(n)p Fp(U)1252 4903 y Fo(L)1329 4891 y Fr(then)518 5053 y Fp(V)566 5065 y Fo(x;t)653 5053 y Fr(\()p Fp(!)s Fr(\))23 b(=)g Fp(V)950 4961 y Fj(\020)999 5053 y Fp(!)s Fr(\()p Fp(x)c Fr(+)f Fu(e)1279 5065 y Fm(1)1316 5053 y Fr(\))h Fn(\000)f Fp(!)s Fr(\()p Fp(x)p Fr(\))p Fp(;)c(:::::;)g(t)p Fr([)p Fp(!)s Fr(\()p Fp(x)20 b Fr(+)e Fu(e)2139 5065 y Fo(j)2173 5053 y Fr(\))h Fn(\000)f Fp(!)s Fr(\()p Fp(x)p Fr(\)])p Fp(;)c(::::;)g(!)s Fr(\()p Fp(x)20 b Fr(+)e Fu(e)2943 5065 y Fo(d)2981 5053 y Fr(\))h Fn(\000)f Fp(!)s Fr(\()p Fp(x)p Fr(\))3281 4961 y Fj(\021)3359 5053 y Fp(;)661 5216 y Fr(where)27 b(w)n(e)h(in)n(tro)r(duce)f(the)h (parameter)e Fp(t)i Fr(for)f(an)n(y)f Fp(j)33 b Fr(with)28 b Fp(x)19 b Fr(+)f Fu(e)2723 5228 y Fo(j)2781 5216 y Fn(2)23 b Fp(I)7 b(nt)p Fr(\()p Fp(Q)3080 5228 y Fo(L)3130 5216 y Fr(\))19 b Fn([)f Fp(U)3311 5228 y Fo(L)3361 5216 y Fr(.)p eop %%Page: 8 8 8 7 bop 456 226 a Fm(8)1108 b(JOSEPH)28 b(G.)h(CONLON)456 425 y Fr(F)-7 b(or)27 b(0)22 b Fn(\024)h Fp(t)g Fn(\024)f Fr(1)28 b(w)n(e)f(in)n(tro)r(duce)g(a)g(corresp)r(onding)f(Lagrangian)f Fn(L)2553 437 y Fo(t)2610 425 y Fr(on)i(\012)2785 437 y Fo(L)2831 421 y Ff(0)2885 425 y Fr(b)n(y)756 619 y Fn(L)813 631 y Fo(t)843 619 y Fr(\()p Fp(!)s Fr(\))c(=)1207 541 y Fj(X)1073 719 y Fo(x)p Fl(2)p Fo(Q)1208 735 y Fh(L)1248 723 y Ff(0)1274 719 y Fl(\000)p Fo(@)t(Q)1417 727 y Fh(L)1475 619 y Fp(V)18 b Fr(\()p Fn(r)p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))i(+)1992 541 y Fj(X)1943 719 y Fo(x)p Fl(2)p Fo(@)t(Q)2117 727 y Fh(L)2175 619 y Fp(V)2223 631 y Fo(x;t)2310 619 y Fr(\()p Fp(!)s Fr(\))e(+)2571 541 y Fj(X)2530 719 y Fo(x)p Fl(2)p Fo(Q)2665 734 y Fh(L)2705 722 y Ff(0)2755 563 y Fr(1)p 2755 600 42 4 v 2755 676 a(2)2807 619 y Fp(m)2880 585 y Fm(2)2917 619 y Fp(!)s Fr(\()p Fp(x)p Fr(\))3083 585 y Fm(2)3121 619 y Fp(;)456 874 y Fr(where)33 b(as)g(in)g(the)h(de\014nition)g(of)g(the)g(measure)e(\(1.10\))h(w)n(e) g(are)g(iden)n(tifying)g(p)r(oin)n(ts)h(on)f(the)456 973 y(b)r(oundary)26 b(of)i Fp(Q)990 985 y Fo(L)1036 969 y Ff(0)1062 973 y Fr(.)37 b(W)-7 b(e)28 b(asso)r(ciate)e(with)i Fn(L)1860 985 y Fo(t)1917 973 y Fr(a)f(measure)1228 1145 y(exp)o([)p Fn(\000L)1499 1157 y Fo(t)1529 1145 y Fr(\()p Fp(!)s Fr(\)])1732 1066 y Fj(Y)1685 1244 y Fo(x)p Fl(2)p Fo(Q)1820 1260 y Fh(L)1860 1248 y Ff(0)1900 1145 y Fp(d!)s Fr(\()p Fp(x)p Fr(\))p Fp(=)p Fr(normalization)n Fp(:)-2216 b Fr(\(2.11\))456 1399 y(Eviden)n(tly)30 b(if)g Fp(t)e Fr(=)g(1)i(the)h(measure)e(\(2.11\))h(is)g(iden)n(tical)g(to)g(the)h (measure)f(\(1.10\))f(with)i Fp(Q)d Fr(=)456 1498 y Fp(Q)522 1510 y Fo(L)568 1494 y Ff(0)593 1498 y Fr(.)37 b(If)26 b Fp(t)d Fr(=)g(0)i(then)i(in)f(the)g(measure)f(\(2.11\))h(the)g(v)-5 b(ariables)25 b Fp(!)s Fr(\()p Fp(x)p Fr(\))p Fp(;)40 b(x)24 b Fn(2)f Fp(I)7 b(nt)p Fr(\()p Fp(Q)3010 1510 y Fo(L)3060 1498 y Fr(\))15 b Fn([)h Fp(U)3235 1510 y Fo(L)3284 1498 y Fr(,)26 b(are)456 1598 y(indep)r(enden)n(t)h(from)f (the)g(v)-5 b(ariables)25 b Fp(!)s Fr(\()p Fp(x)p Fr(\))p Fp(;)41 b(x)23 b Fn(2)h Fp(Q)2049 1610 y Fo(L)2095 1594 y Ff(0)2121 1598 y Fn(n)p Fr([)p Fp(I)7 b(nt)p Fr(\()p Fp(Q)2407 1610 y Fo(L)2456 1598 y Fr(\))16 b Fn([)g Fp(U)2632 1610 y Fo(L)2681 1598 y Fr(].)37 b(Hence)26 b(if)h(w)n(e)f(denote)456 1697 y(exp)r(ectation)h(w.r.)37 b(to)27 b(the)h(measure)f(\(2.11\))f(b) n(y)i Fp(<)22 b Fn(\001)h Fp(>)2200 1709 y Fo(Q)2252 1725 y Fh(L)2292 1713 y Ff(0)2314 1725 y Fh(;m;t)2463 1697 y Fr(w)n(e)k(ha)n(v)n(e)f(that)1210 1805 y Fj(D)1260 1897 y Fp(f)9 b Fr(\()p Fp(!)s Fr(\(0\)\))1535 1805 y Fj(E)1586 1955 y Fo(Q)1638 1971 y Fh(L)1678 1959 y Ff(0)1700 1971 y Fh(;m;)p Fe(1)1909 1897 y Fr(=)2057 1805 y Fj(D)2107 1897 y Fp(f)g Fr(\()p Fp(!)s Fr(\(0\)\))2382 1805 y Fj(E)2433 1955 y Fo(Q)2485 1971 y Fh(L)2525 1959 y Ff(0)2547 1971 y Fh(;m)2653 1897 y Fp(;)1210 2015 y Fj(D)1260 2107 y Fp(f)g Fr(\()p Fp(!)s Fr(\(0\)\))1535 2015 y Fj(E)1586 2165 y Fo(Q)1638 2181 y Fh(L)1678 2169 y Ff(0)1700 2181 y Fh(;m;)p Fe(0)1909 2107 y Fr(=)2057 2015 y Fj(D)2107 2107 y Fp(f)g Fr(\()p Fp(!)s Fr(\(0\)\))2382 2015 y Fj(E)2433 2165 y Fo(Q)2485 2173 y Fh(L;m)2631 2107 y Fp(:)456 2353 y Fr(W)-7 b(e)35 b(can)f(therefore)g(compare)1460 2261 y Fj(D)1511 2353 y Fp(f)9 b Fr(\()p Fp(!)s Fr(\(0\)\))1786 2261 y Fj(E)1837 2411 y Fo(Q)1889 2426 y Fh(L)1929 2414 y Ff(0)1951 2426 y Fh(;m)2064 2353 y Fr(and)2233 2261 y Fj(D)2284 2353 y Fp(f)g Fr(\()p Fp(!)s Fr(\(0\)\))2559 2261 y Fj(E)2609 2411 y Fo(Q)2661 2419 y Fh(L;m)2814 2353 y Fr(b)n(y)35 b(di\013eren)n(tiating)456 2505 y(\(2.11\))26 b(w.r.)37 b(to)27 b Fp(t)h Fr(and)g(using)f(the)h(fundamen)n(tal)f (theorem)h(of)f(calculus.)36 b(W)-7 b(e)28 b(ha)n(v)n(e)f(then,)539 2636 y Fj(D)589 2729 y Fp(f)9 b Fr(\()p Fp(!)s Fr(\(0\)\))864 2636 y Fj(E)915 2787 y Fo(Q)967 2802 y Fh(L)1007 2790 y Ff(0)1029 2802 y Fh(;m)1108 2729 y Fn(\000)1173 2636 y Fj(D)1223 2729 y Fp(f)g Fr(\()p Fp(!)s Fr(\(0\)\))1498 2636 y Fj(E)1549 2787 y Fo(Q)1601 2795 y Fh(L;m)1742 2729 y Fr(=)1866 2672 y(1)p 1840 2710 96 4 v 1840 2786 a Fp(L)1897 2762 y Fo(d)2131 2650 y Fj(X)1959 2832 y Fo(x)p Fl(2)p Fo(I)c(nt)p Fm(\()p Fo(Q)2220 2840 y Fh(L)2263 2832 y Fm(\))p Fl([)p Fo(U)2379 2840 y Fh(L)2437 2636 y Fj(n)2492 2729 y Fn(h)p Fp(f)k Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))p Fn(i)2837 2754 y Fo(Q)2889 2769 y Fh(L)2929 2757 y Ff(0)2952 2769 y Fh(;m;)p Fe(1)3096 2729 y Fn(\000)18 b(h)p Fp(f)9 b Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))p Fn(i)3524 2754 y Fo(Q)3576 2769 y Fh(L)3616 2757 y Ff(0)3639 2769 y Fh(;m;)p Fe(0)3764 2636 y Fj(o)3843 2729 y Fr(=)466 3002 y Fn(\000)p Fr(1)p 466 3039 107 4 v 471 3115 a Fp(L)528 3091 y Fo(d)595 2945 y Fj(Z)678 2965 y Fm(1)642 3134 y(0)730 3058 y Fp(dt)803 2941 y Fj(\034)879 2916 y(")937 3002 y Fp(d)p Fn(L)1037 3014 y Fo(t)1067 3002 y Fr(\()p Fp(!)s Fr(\))p 937 3039 249 4 v 1025 3115 a Fp(dt)1215 3058 y Fn(\000)1298 2941 y Fj(\034)1370 3002 y Fp(d)p Fn(L)1470 3014 y Fo(t)1500 3002 y Fr(\()p Fp(!)s Fr(\))p 1370 3039 V 1458 3115 a Fp(dt)1629 2941 y Fj(\035)1691 3141 y Fo(Q)1743 3156 y Fh(L)1783 3144 y Ff(0)1805 3156 y Fh(;m;t)1926 2916 y Fj(#)1988 2891 y(2)1988 3041 y(4)2216 2979 y(X)2044 3161 y Fo(x)p Fl(2)p Fo(I)c(nt)p Fm(\()p Fo(Q)2305 3169 y Fh(L)2348 3161 y Fm(\))p Fl([)p Fo(U)2464 3169 y Fh(L)2522 2966 y Fj(n)2577 3058 y Fp(f)k Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))19 b Fn(\000)f(h)q Fp(f)9 b Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))p Fn(i)3305 3083 y Fo(Q)3357 3098 y Fh(L)3397 3086 y Ff(0)3419 3098 y Fh(;m;t)3540 2966 y Fj(o)3595 2891 y(3)3595 3041 y(5)3665 2941 y(\035)3727 3141 y Fo(Q)3779 3156 y Fh(L)3819 3144 y Ff(0)3841 3156 y Fh(;m;t)3980 3058 y Fp(:)456 3313 y Fr(Hence)27 b(if)h(w)n(e)g(use)f(the)h(Sc)n(h)n(w)n(arz)e(inequalit)n (y)h(w)n(e)g(ha)n(v)n(e)g(that)456 3518 y(\(2.12\))751 3422 y Fj(\014)751 3472 y(\014)751 3522 y(\014)778 3426 y(D)829 3518 y Fp(f)9 b Fr(\()p Fp(!)s Fr(\(0\)\))1104 3426 y Fj(E)1155 3576 y Fo(Q)1207 3591 y Fh(L)1247 3579 y Ff(0)1269 3591 y Fh(;m)1366 3518 y Fn(\000)1449 3426 y Fj(D)1500 3518 y Fp(f)g Fr(\()p Fp(!)s Fr(\(0\)\))1775 3426 y Fj(E)1825 3576 y Fo(Q)1877 3584 y Fh(L;m)1996 3422 y Fj(\014)1996 3472 y(\014)1996 3522 y(\014)2046 3518 y Fn(\024)1060 3762 y Fr(1)p 1033 3799 96 4 v 1033 3875 a Fp(L)1090 3851 y Fo(d)1187 3818 y Fr(sup)1152 3888 y Fm(0)p Fl(\024)p Fo(t)p Fl(\024)p Fm(1)1361 3676 y Fj(*")1486 3762 y Fp(d)p Fn(L)1586 3774 y Fo(t)1616 3762 y Fr(\()p Fp(!)s Fr(\))p 1486 3799 249 4 v 1574 3875 a Fp(dt)1764 3818 y Fn(\000)1847 3701 y Fj(\034)1919 3762 y Fp(d)p Fn(L)2019 3774 y Fo(t)2049 3762 y Fr(\()p Fp(!)s Fr(\))p 1919 3799 V 2007 3875 a Fp(dt)2178 3701 y Fj(\035)2240 3901 y Fo(Q)2292 3916 y Fh(L)2332 3904 y Ff(0)2355 3916 y Fh(;m;t)2475 3676 y Fj(#)2524 3693 y Fm(2)2561 3676 y Fj(+)p Fm(1)p Fo(=)p Fm(2)2628 3928 y Fo(Q)2680 3943 y Fh(L)2720 3931 y Ff(0)2742 3943 y Fh(;m;t)1193 4153 y Fr(sup)1158 4223 y Fm(0)p Fl(\024)p Fo(t)p Fl(\024)p Fm(1)1367 4011 y Fj(*)1434 4058 y(\014)1434 4107 y(\014)1434 4157 y(\014)1647 4074 y(X)1475 4256 y Fo(x)p Fl(2)p Fo(I)c(nt)p Fm(\()p Fo(Q)1736 4264 y Fh(L)1780 4256 y Fm(\))p Fl([)p Fo(U)1896 4264 y Fh(L)1953 4153 y Fp(f)k Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))19 b Fn(\000)f(h)q Fp(f)9 b Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))p Fn(i)2681 4178 y Fo(Q)2733 4193 y Fh(L)2773 4181 y Ff(0)2795 4193 y Fh(;m;t)2930 4058 y Fj(\014)2930 4107 y(\014)2930 4157 y(\014)2958 4078 y Fm(2)2995 4011 y Fj(+)3062 4028 y Fm(1)p Fo(=)p Fm(2)3062 4283 y Fo(Q)3114 4298 y Fh(L)3154 4286 y Ff(0)3176 4298 y Fh(;m;t)3338 4153 y Fp(:)456 4437 y Fr(W)-7 b(e)23 b(can)f(estimate)g(the)h(RHS)g (of)g(\(2.12\))f(b)n(y)g(using)g(the)h(Brascamp-Lieb)d(inequalit)n(y)j (pro)n(vided)456 4537 y(w)n(e)c(can)h(obtain)g(a)g(suitable)g(lo)n(w)n (er)e(b)r(ound)j(on)e(the)i(Hessian)e Fn(L)2383 4507 y Fl(0)q(0)2383 4557 y Fo(t)2426 4537 y Fr(\()p Fp(!)s Fr(\))i(of)f(the)g(Lagrangian)e Fn(L)3273 4549 y Fo(t)3302 4537 y Fr(\()p Fp(!)s Fr(\).)456 4637 y(W)-7 b(e)28 b(ha)n(v)n(e)e (already)g(observ)n(ed)g(that)1527 4796 y Fn(L)1584 4762 y Fl(00)1584 4817 y Fm(1)1627 4796 y Fr(\()p Fp(!)s Fr(\))d Fn(\025)f(\000)p Fp(\025)p Fr(\001)2038 4808 y Fo(L)2084 4792 y Ff(0)2129 4796 y Fr(+)c Fp(m)2285 4762 y Fm(2)2350 4796 y Fp(;)456 4956 y Fr(where)25 b(\001)763 4968 y Fo(L)809 4952 y Ff(0)862 4956 y Fr(is)h(the)h(lattice)f(Laplacian)f(on) h Fp(Q)1896 4968 y Fo(L)1942 4952 y Ff(0)1994 4956 y Fr(with)h(p)r(erio)r(dic)f(b)r(oundary)f(conditions.)36 b(It)27 b(is)456 5056 y(easy)f(to)i(see)f(also)f(that)1210 5216 y Fn(L)1267 5181 y Fl(00)1267 5236 y Fm(0)1309 5216 y Fr(\()p Fp(!)s Fr(\))e Fn(\025)e(\000)p Fp(\025)14 b Fr([\001)1758 5228 y Fo(L)1826 5216 y Fn(\012)k Fp(I)26 b Fr(+)18 b Fp(I)25 b Fn(\012)18 b Fr(\001)2267 5228 y Fo(L)2313 5211 y Ff(0)2336 5228 y Fo(;L)2405 5216 y Fr(])h(+)f Fp(m)2603 5181 y Fm(2)2667 5216 y Fp(;)p eop %%Page: 9 9 9 8 bop 1520 226 a Fm(PDE)28 b(AND)h(FIELD)g(THEOR)-5 b(Y)1030 b(9)456 425 y Fr(where)23 b(\001)761 437 y Fo(L)834 425 y Fr(is)g(the)h(p)r(erio)r(dic)g(Laplacian)e(acting)h(on)g Fp(I)7 b(nt)p Fr(\()p Fp(Q)2319 437 y Fo(L)2369 425 y Fr(\))j Fn([)g Fp(U)2533 437 y Fo(L)2607 425 y Fr(and)23 b(\001)2833 437 y Fo(L)2879 421 y Ff(0)2901 437 y Fo(;L)2994 425 y Fr(is)h(the)g(Lapla-)456 525 y(cian)j(on)h Fp(Q)813 537 y Fo(L)859 521 y Ff(0)885 525 y Fn(n)p Fr([)p Fp(I)7 b(nt)p Fr(\()p Fp(Q)1171 537 y Fo(L)1220 525 y Fr(\))19 b Fn([)g Fp(U)1402 537 y Fo(L)1451 525 y Fr(])28 b(with)g(Neumann)h(b)r (oundary)e(conditions)g(on)h(the)g(b)r(oundary)456 624 y(of)f Fp(I)7 b(nt)p Fr(\()p Fp(Q)771 636 y Fo(L)821 624 y Fr(\))18 b Fn([)h Fp(U)1002 636 y Fo(L)1052 624 y Fr(.)36 b(More)27 b(generally)-7 b(,)26 b(for)h(0)c Fn(\024)g Fp(t)g Fn(\024)f Fr(1)28 b(there)f(is)g(the)h(inequalit)n(y) -7 b(,)977 766 y Fn(L)1034 732 y Fl(0)q(0)1034 787 y Fo(t)1077 766 y Fr(\()p Fp(!)s Fr(\))23 b Fn(\025)g(\000)p Fp(\025)1434 699 y Fj(\002)1468 766 y Fp(t)1498 732 y Fm(2)1535 766 y Fr(\001)1604 778 y Fo(L)1650 762 y Ff(0)1695 766 y Fr(+)18 b(\(1)g Fn(\000)g Fp(t)p Fr(\))2015 732 y Fm(2)2067 766 y Fn(f)o Fr(\001)2177 778 y Fo(L)2246 766 y Fn(\012)g Fp(I)25 b Fr(+)18 b Fp(I)26 b Fn(\012)18 b Fr(\001)2687 778 y Fo(L)2733 762 y Ff(0)2755 778 y Fo(;L)2824 766 y Fn(g)2866 699 y Fj(\003)2919 766 y Fr(+)g Fp(m)3075 732 y Fm(2)3112 766 y Fp(:)-2679 b Fr(\(2.13\))456 911 y(It)28 b(follo)n(ws)e(from)h(\(2.13\))g(and)h(Brascamp-Lieb)d (that)j(if)g(0)23 b Fn(\024)f Fp(t)i Fn(\024)e Fr(1)p Fp(=)p Fr(2,)k(then)456 1175 y(\(2.14\))765 1033 y Fj(*)832 1079 y(\014)832 1129 y(\014)832 1179 y(\014)1045 1096 y(X)873 1278 y Fo(x)p Fl(2)p Fo(I)5 b(nt)p Fm(\()p Fo(Q)1134 1286 y Fh(L)1178 1278 y Fm(\))p Fl([)p Fo(U)1294 1286 y Fh(L)1351 1175 y Fp(f)k Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))19 b Fn(\000)f(h)q Fp(f)9 b Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))p Fn(i)2079 1199 y Fo(Q)2131 1215 y Fh(L)2171 1203 y Ff(0)2193 1215 y Fh(;m;t)2328 1079 y Fj(\014)2328 1129 y(\014)2328 1179 y(\014)2355 1100 y Fm(2)2393 1033 y Fj(+)2460 1050 y Fm(1)p Fo(=)p Fm(2)2460 1305 y Fo(Q)2512 1320 y Fh(L)2552 1308 y Ff(0)2574 1320 y Fh(;m;t)2168 1488 y Fn(\024)2265 1432 y Fr(4)p Fp(L)2364 1402 y Fo(d)p 2265 1469 137 4 v 2310 1545 a Fp(\025)2626 1409 y Fj(X)2454 1591 y Fo(x)p Fl(2)p Fo(I)c(nt)p Fm(\()p Fo(Q)2715 1599 y Fh(L)2758 1591 y Fm(\))p Fl([)p Fo(U)2874 1599 y Fh(L)2959 1488 y Fp(H)i Fr(\()p Fp(x)p Fr(\))p Fp(G)p Fr(\()p Fp(x)p Fr(\))p Fp(;)456 1720 y Fr(where)1506 1865 y Fp(H)g Fr(\()p Fp(x)p Fr(\))84 b(=)1924 1773 y Fj(D)1975 1865 y Fp(f)2025 1831 y Fl(0)2048 1865 y Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))p 2278 1793 299 4 v Fp(f)2328 1841 y Fl(0)2352 1865 y Fr(\()p Fp(!)s Fr(\(0\)\))2577 1773 y Fj(E)2628 1923 y Fo(Q)2680 1938 y Fh(L)2720 1926 y Ff(0)2742 1938 y Fh(;m;t)2905 1865 y Fp(;)958 1986 y Fj(\024)1002 2103 y Fn(\000)p Fr(\001)1136 2115 y Fo(L)1204 2103 y Fr(+)1297 2047 y(4)p Fp(m)1412 2017 y Fm(2)p 1297 2084 152 4 v 1349 2160 a Fp(\025)1459 1986 y Fj(\025)1516 2103 y Fp(G)p Fr(\()p Fp(x)p Fr(\))h(=)d Fp(\016)s Fr(\()p Fp(x)p Fr(\))p Fp(;)71 b(x)23 b Fn(2)h Fp(I)7 b(nt)p Fr(\()p Fp(Q)2539 2115 y Fo(L)2588 2103 y Fr(\))19 b Fn([)f Fp(U)2769 2115 y Fo(L)2847 2103 y Fp(:)456 2294 y Fr(W)-7 b(e)28 b(conclude)f(that)h (the)g(LHS)g(of)f(\(2.14\))g(is)h(b)r(ounded)g(b)n(y)1206 2434 y(4)p Fp(L)1305 2404 y Fo(d)p 1206 2471 137 4 v 1250 2547 a Fp(\025)1352 2490 y(H)7 b Fr(\(0\))1720 2411 y Fj(X)1548 2593 y Fo(x)p Fl(2)p Fo(I)e(nt)p Fm(\()p Fo(Q)1809 2601 y Fh(L)1852 2593 y Fm(\))p Fl([)p Fo(U)1968 2601 y Fh(L)2054 2490 y Fp(G)p Fr(\()p Fp(x)p Fr(\))24 b(=)2359 2434 y Fp(L)2416 2404 y Fo(d)p 2352 2471 111 4 v 2352 2547 a Fp(m)2425 2523 y Fm(2)2499 2490 y Fp(H)7 b Fr(\(0\))p Fp(:)456 2727 y Fr(In)33 b(view)g(of)g(\(2.13\))g(one)f (has)h(an)g(inequalit)n(y)g(lik)n(e)f(\(2.3\))h(for)g(the)h(measure)e Fp(<)65 b Fn(\001)h Fp(>)3187 2739 y Fo(Q)3239 2755 y Fh(L)3279 2743 y Ff(0)3301 2755 y Fh(;m;t)3421 2727 y Fr(.)456 2834 y(Since)23 b Fp(f)32 b Fr(satis\014es)23 b(\(2.4\))g(it)h(follo)n(ws)e(that)i(there)f(is)h(a)f(constan)n(t)f Fp(C)2502 2846 y Fo(d)2565 2834 y Fr(dep)r(ending)i(only)f(on)g Fp(d)g Fn(\025)g Fr(3)456 2933 y(suc)n(h)k(that)1482 3074 y Fp(H)7 b Fr(\(0\))23 b Fn(\024)g Fp(A)1837 3040 y Fm(2)1888 3074 y Fr(exp[2)p Fp(B)2147 3040 y Fm(2)2184 3074 y Fp(C)2243 3086 y Fo(d)2282 3074 y Fp(=\025)p Fr(])p Fp(:)456 3214 y Fr(W)-7 b(e)28 b(conclude)f(then)h(that)g(if)g(0)23 b Fn(\024)f Fp(t)h Fn(\024)g Fr(1)p Fp(=)p Fr(2,)j(there)i(is)f(the)h (inequalit)n(y)-7 b(,)456 3454 y(\(2.15\))765 3312 y Fj(*)832 3334 y(\014)832 3384 y(\014)832 3434 y(\014)832 3483 y(\014)1045 3375 y(X)873 3557 y Fo(x)p Fl(2)p Fo(I)5 b(nt)p Fm(\()p Fo(Q)1134 3565 y Fh(L)1178 3557 y Fm(\))p Fl([)p Fo(U)1294 3565 y Fh(L)1351 3454 y Fp(f)k Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))19 b Fn(\000)f(h)q Fp(f)9 b Fr(\()p Fp(!)s Fr(\()p Fp(x)p Fr(\)\))p Fn(i)2079 3479 y Fo(Q)2131 3495 y Fh(L)2171 3483 y Ff(0)2193 3495 y Fh(;m;t)2328 3359 y Fj(\014)2328 3409 y(\014)2328 3458 y(\014)2355 3379 y Fm(2)2393 3312 y Fj(+)2460 3584 y Fo(Q)2512 3600 y Fh(L)2552 3588 y Ff(0)2574 3600 y Fh(;m;t)2473 3768 y Fn(\024)2578 3711 y Fp(L)2635 3681 y Fo(d)p 2570 3749 V 2570 3825 a Fp(m)2643 3801 y Fm(2)2718 3768 y Fp(A)2780 3733 y Fm(2)2831 3768 y Fr(exp[2)p Fp(B)3090 3733 y Fm(2)3127 3768 y Fp(C)3186 3780 y Fo(d)3225 3768 y Fp(=\025)p Fr(])p Fp(:)456 3936 y Fr(W)-7 b(e)39 b(can)f(mak)n(e)g(a) h(similar)f(argumen)n(t)f(for)i(1)p Fp(=)p Fr(2)h Fn(\024)h Fp(t)h Fn(\024)f Fr(1)e(b)n(y)f(replacing)g(\001)2980 3948 y Fo(L)3068 3936 y Fr(b)n(y)h(\001)3264 3948 y Fo(L)3310 3932 y Ff(0)3375 3936 y Fr(in)456 4036 y(the)30 b(pro)r(of)f(of)h (\(2.15\).)43 b(W)-7 b(e)30 b(can)f(estimate)h(the)g(exp)r(ectation)g (of)g Fp(d)p Fn(L)2622 4048 y Fo(t)2651 4036 y Fr(\()p Fp(!)s Fr(\))p Fp(=dt)g Fr(on)g(the)g(RHS)h(of)456 4135 y(\(2.12\))23 b(similarly)f(to)i(the)g(w)n(a)n(y)e(w)n(e)i(obtained)f (\(2.15\).)35 b(Th)n(us)23 b(b)n(y)g(Brascamp-Lieb)f(there)i(is)f(the) 456 4235 y(inequalit)n(y)-7 b(,)456 4487 y(\(2.16\))765 4345 y Fj(*")890 4431 y Fp(d)p Fn(L)990 4443 y Fo(t)1020 4431 y Fr(\()p Fp(!)s Fr(\))p 890 4468 249 4 v 978 4544 a Fp(dt)1167 4487 y Fn(\000)1250 4370 y Fj(\034)1322 4431 y Fp(d)p Fn(L)1422 4443 y Fo(t)1452 4431 y Fr(\()p Fp(!)s Fr(\))p 1322 4468 V 1410 4544 a Fp(dt)1581 4370 y Fj(\035)1644 4570 y Fo(Q)1696 4585 y Fh(L)1736 4573 y Ff(0)1758 4585 y Fh(;m;t)1879 4345 y Fj(#)1927 4362 y Fm(2)1964 4345 y Fj(+)2031 4598 y Fo(Q)2083 4613 y Fh(L)2123 4601 y Ff(0)2146 4613 y Fh(;m;t)1687 4741 y Fn(\024)1870 4663 y Fj(X)1802 4841 y Fo(x;y)r Fl(2)p Fo(Q)1993 4856 y Fh(L)2033 4844 y Ff(0)2100 4741 y Fn(h)p Fp(g)2172 4753 y Fo(t)2201 4741 y Fr(\()p Fp(!)s(;)14 b(x)p Fr(\))28 b Fp(G)2497 4753 y Fo(t)2527 4741 y Fr(\()p Fp(x;)14 b(y)s Fr(\))28 b Fp(g)2787 4753 y Fo(t)2816 4741 y Fr(\()p Fp(!)s(;)14 b(y)s Fr(\))p Fn(i)3048 4766 y Fo(Q)3100 4782 y Fh(L)3140 4770 y Ff(0)3162 4782 y Fh(;m;t)3324 4741 y Fp(;)456 4976 y Fr(where)33 b Fp(G)767 4988 y Fo(t)796 4976 y Fr(\()p Fp(x;)14 b(y)s Fr(\))35 b(is)e(the)i(k)n(ernel)d(of)i(the)g(in)n(v)n(erse)f(of)g(the)h(op)r (erator)e(on)i(the)g(RHS)g(of)g(\(2.13\).)456 5075 y(The)27 b(function)h Fp(g)991 5087 y Fo(t)1048 5075 y Fr(has)f(the)h(prop)r (ert)n(y)e(that)984 5216 y Fp(g)1024 5228 y Fo(t)1053 5216 y Fr(\()p Fp(!)s(;)14 b(x)p Fr(\))83 b(=)g(0)p Fp(;)96 b(x)24 b Fn(62)f Fp(N)9 b Fr(\()p Fp(@)c(Q)2020 5228 y Fo(L)2069 5216 y Fr(\))p Fp(;)p eop %%Page: 10 10 10 9 bop 456 226 a Fm(10)1075 b(JOSEPH)28 b(G.)h(CONLON)938 429 y Fn(j)p Fp(g)1001 441 y Fo(t)1030 429 y Fr(\()p Fp(!)s(;)14 b(x)p Fr(\))p Fn(j)83 b(\024)g Fp(C)6 b Fr(\003)1823 350 y Fj(X)1624 532 y Fo(y)r Fl(2)p Fo(N)g Fm(\()p Fo(x)p Fm(\))p Fl(\\)p Fo(N)g Fm(\()p Fo(@)t(Q)2075 540 y Fh(L)2117 532 y Fm(\))2157 429 y Fn(j)p Fp(!)s Fr(\()p Fp(y)s Fr(\))p Fn(j)p Fp(;)97 b(x)23 b Fn(2)h Fp(N)9 b Fr(\()p Fp(@)c(Q)2858 441 y Fo(L)2907 429 y Fr(\))p Fp(;)456 665 y Fr(for)35 b(some)h(univ)n(ersal)e(constan)n(t)i Fp(C)6 b Fr(.)62 b(Here)36 b Fp(N)9 b Fr(\()p Fp(@)c(Q)2089 677 y Fo(L)2138 665 y Fr(\))36 b(is)g(the)h(neigh)n(b)r(orho)r(o)r(d)e(of)h Fp(@)5 b(Q)3198 677 y Fo(L)3283 665 y Fr(with)456 765 y(radius)34 b(1.)58 b(By)35 b Fp(N)9 b Fr(\()p Fp(x)p Fr(\))35 b(w)n(e)g(mean)f(the)i(union)f(of)f(the)i(neigh)n(b)r(orho)r (o)r(ds)d(of)i Fp(x)g Fr(and)g Fp(x)24 b Fn(\000)f Fr(\()p Fp(L)g Fn(\000)456 864 y Fr(1\))p Fu(e)574 876 y Fo(j)608 864 y Fp(;)69 b Fr(1)23 b Fn(\024)g Fp(j)28 b Fn(\024)22 b Fp(d)p Fr(,)28 b(with)g(radius)f(1.)37 b(Observ)n(e)25 b(that)j Fp(G)2200 876 y Fo(t)2258 864 y Fr(satis\014es)1086 1045 y Fp(G)1151 1057 y Fo(t)1181 1045 y Fr(\()p Fp(x;)14 b(y)s Fr(\))24 b Fn(\025)e Fr(0)p Fp(;)69 b(x;)14 b(y)26 b Fn(2)d Fp(Q)1913 1057 y Fo(L)1959 1041 y Ff(0)1985 1045 y Fp(;)2089 967 y Fj(X)2050 1145 y Fo(y)r Fl(2)p Fo(Q)2183 1160 y Fh(L)2223 1148 y Ff(0)2263 1045 y Fp(G)2328 1057 y Fo(t)2357 1045 y Fr(\()p Fp(x;)14 b(y)s Fr(\))24 b(=)2705 989 y(1)p 2670 1026 111 4 v 2670 1102 a Fp(m)2743 1078 y Fm(2)2791 1045 y Fp(:)456 1278 y Fr(W)-7 b(e)28 b(conclude)f(that)h(the)g(LHS)g(of)f(\(2.16\))g(is)h(b)r(ounded)g(ab)r (o)n(v)n(e)e(b)n(y)569 1414 y Fp(C)628 1426 y Fo(d)667 1414 y Fr(\003)725 1384 y Fm(2)p 569 1451 193 4 v 611 1527 a Fp(m)684 1503 y Fm(2)890 1391 y Fj(X)786 1573 y Fo(x)p Fl(2)p Fo(N)6 b Fm(\()p Fo(@)t(Q)1045 1581 y Fh(L)1088 1573 y Fm(\))1156 1403 y Fj(\012)1195 1470 y Fp(!)s Fr(\()p Fp(x)p Fr(\))1361 1436 y Fm(2)1399 1403 y Fj(\013)1438 1503 y Fo(Q)1490 1519 y Fh(L)1530 1507 y Ff(0)1553 1519 y Fh(;m;t)1696 1470 y Fn(\024)1794 1414 y Fp(C)1853 1426 y Fo(d)1892 1414 y Fr(\003)1950 1384 y Fm(2)p 1794 1451 V 1835 1527 a Fp(m)1908 1503 y Fm(2)2115 1391 y Fj(X)2011 1573 y Fo(x)p Fl(2)p Fo(N)g Fm(\()p Fo(@)t(Q)2270 1581 y Fh(L)2313 1573 y Fm(\))2353 1470 y Fp(G)2418 1482 y Fo(t)2447 1470 y Fr(\()p Fp(x;)14 b(x)p Fr(\))25 b Fn(\024)2764 1414 y Fp(C)2823 1426 y Fo(d)2862 1414 y Fr(\003)2920 1384 y Fm(2)p 2764 1451 V 2806 1527 a Fp(m)2879 1503 y Fm(4)2967 1470 y Fn(j)p Fp(N)9 b Fr(\()p Fp(@)c(Q)3213 1482 y Fo(L)3262 1470 y Fr(\))p Fn(j)p Fp(;)456 1701 y Fr(for)27 b(some)g(constan)n(t)g Fp(C)1185 1713 y Fo(d)1252 1701 y Fr(dep)r(ending)h(only)f(on)g Fp(d)p Fr(.)38 b(Eviden)n(tly)27 b(the)h(n)n(um)n(b)r(er)f(of)h (lattice)g(p)r(oin)n(ts)456 1801 y Fn(j)p Fp(N)9 b Fr(\()p Fp(@)c(Q)702 1813 y Fo(L)751 1801 y Fr(\))p Fn(j)27 b Fr(in)g Fp(N)9 b Fr(\()p Fp(@)c(Q)1152 1813 y Fo(L)1201 1801 y Fr(\))27 b(is)f(b)r(ounded)h(b)n(y)g Fp(C)1854 1813 y Fo(d)1893 1801 y Fp(L)1950 1771 y Fo(d)p Fl(\000)p Fm(1)2100 1801 y Fr(for)f(some)g(constan)n(t)g Fp(C)2826 1813 y Fo(d)2892 1801 y Fr(dep)r(ending)h(only)456 1901 y(on)g Fp(d)p Fr(.)37 b(Hence)28 b(there)f(is)h(a)f(constan)n(t)g Fp(C)1680 1913 y Fo(d)1746 1901 y Fr(dep)r(ending)h(only)g(on)f Fp(d)h Fr(suc)n(h)f(that)936 1985 y Fj(*")1061 2071 y Fp(d)p Fn(L)1161 2083 y Fo(t)1191 2071 y Fr(\()p Fp(!)s Fr(\))p 1061 2108 249 4 v 1149 2184 a Fp(dt)1338 2127 y Fn(\000)1421 2010 y Fj(\034)1494 2071 y Fp(d)p Fn(L)1594 2083 y Fo(t)1623 2071 y Fr(\()p Fp(!)s Fr(\))p 1494 2108 V 1581 2184 a Fp(dt)1752 2010 y Fj(\035)1815 2210 y Fo(Q)1867 2225 y Fh(L)1907 2213 y Ff(0)1929 2225 y Fh(;m;t)2050 1985 y Fj(#)2098 2002 y Fm(2)2135 1985 y Fj(+)2202 2237 y Fo(Q)2254 2253 y Fh(L)2294 2241 y Ff(0)2317 2253 y Fh(;m;t)2461 2127 y Fn(\024)2558 2071 y Fp(C)2617 2083 y Fo(d)2656 2071 y Fr(\003)2714 2040 y Fm(2)2751 2071 y Fp(L)2808 2040 y Fo(d)p Fl(\000)p Fm(1)p 2558 2108 374 4 v 2690 2184 a Fp(m)2763 2160 y Fm(4)2941 2127 y Fp(:)-2508 b Fr(\(2.17\))456 2370 y(The)27 b(result)h(follo)n(ws)e(no)n (w)h(from)g(\(2.12\),)g(\(2.15\),)g(\(2.17\).)p 3384 2370 4 57 v 3388 2318 50 4 v 3388 2370 V 3437 2370 4 57 v 456 2526 a Fu(Prop)s(osition)j(2.1.)40 b Fg(L)l(et)h Fr(\012)g Fg(b)l(e)h(the)g(sp)l(ac)l(e)g(of)g(functions)g Fp(!)47 b Fr(:)d Fu(Z)2616 2496 y Fo(d)2700 2526 y Fn(!)g Fu(R)d Fg(and)h Fn(F)50 b Fg(b)l(e)41 b(the)456 2626 y(c)l(orr)l(esp)l(onding)h(Bor)l(el)h(algebr)l(a)f(gener)l(ate)l(d)g (by)g(\014nite)f(dimensional)i(r)l(e)l(ctangles.)73 b(Then)42 b(if)456 2726 y Fp(d)23 b Fn(\025)g Fr(3)29 b Fg(ther)l(e)h(is)g(a)g (unique)f(pr)l(ob)l(ability)j(me)l(asur)l(e)e Fp(P)41 b Fg(on)30 b Fr(\(\012)p Fp(;)14 b Fn(F)8 b Fr(\))30 b Fg(with)g(the)g(pr)l(op)l(erties:)562 2845 y Fu(\(a\):)41 b Fg(If)31 b Fp(N)g Fn(\025)23 b Fr(1)29 b Fg(and)h Fp(f)i Fr(:)23 b Fu(R)1447 2815 y Fo(N)1533 2845 y Fn(!)g Fu(C)30 b Fg(is)g(a)g Fp(C)1964 2815 y Fm(2)2031 2845 y Fg(function)g (satisfying)h(the)f(ine)l(quality)1349 2987 y Fn(j)p Fp(f)1422 2953 y Fl(00)1464 2987 y Fr(\()p Fp(z)t Fr(\))p Fn(j)23 b(\024)g Fp(A)14 b Fr(exp[)p Fp(B)t Fn(j)p Fp(z)t Fn(j)p Fr(])p Fp(;)103 b(z)26 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Fg(iter)l(ate)h Fp(!)1617 4797 y Fo(t;N)1752 4785 y Fg(of)g(\(2.23\))h(and)f(the)g(solutions)g(of)g(\(2.21\))i (satisfy)e(the)456 4884 y(ine)l(quality,)552 4988 y Fj(*)619 5012 y(\024)697 5130 y Fr(sup)663 5199 y Fm(0)p Fl(\024)p Fo(s)p Fl(\024)p Fo(t)870 5130 y Fn(j)p Fp(!)945 5142 y Fo(s)981 5130 y Fr(\(0\))18 b Fn(\000)g Fp(!)1240 5142 y Fo(s;N)1354 5130 y Fr(\(0\))p Fn(j)1483 5012 y Fj(\025)1527 5030 y Fm(2)1564 4988 y Fj(+)1631 5237 y Fo(Q;m;W)1880 5130 y Fn(\024)p eop %%Page: 13 13 13 12 bop 1520 226 a Fm(PDE)28 b(AND)h(FIELD)g(THEOR)-5 b(Y)997 b(13)1514 412 y Fr([)p Fp(c)1573 424 y Fo(d)1612 412 y Fr(\(\003)18 b(+)g Fp(m)1876 382 y Fm(2)1913 412 y Fr(\))p Fp(t)p Fr(])1998 382 y Fm(2)p Fo(N)p 1472 449 665 4 v 1472 525 a Fr([1)g Fn(\000)g Fp(c)1674 537 y Fo(d)1713 525 y Fr(\(\003)g(+)g Fp(m)1977 501 y Fm(2)2014 525 y Fr(\))p Fp(t)p Fr(])2099 501 y Fm(2)2176 468 y Fp(C)2235 480 y Fo(d)2274 468 y Fr(\(\003)h(+)f Fp(m)2539 434 y Fm(2)2576 468 y Fr(\))2608 434 y Fm(2)2645 468 y 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Fh(n)1927 803 y Fl(\000)p Fo(t)2004 811 y Fe(1)p Fh(;N)2109 791 y Fr(\()p Fp(x)2188 803 y Fo(n)2234 791 y Fr(\))2266 724 y Fj(\001)2318 720 y(\014)2318 770 y(\014)2346 791 y Fp(!)2398 803 y Fm(0)2435 724 y Fj(\013)2474 824 y Fo(Q;m;W)2227 931 y Fr(=)22 b Fp(g)17 b Fr(\()p Fp(!)2455 943 y Fm(0)2492 931 y Fr(\()p Fp(y)2565 943 y Fm(1)2602 931 y Fr(\))p Fp(;)d(!)2723 943 y Fm(0)2760 931 y Fr(\()p Fp(y)2833 943 y Fm(2)2871 931 y Fr(\))p Fp(;)g(:::;)g(!)3098 943 y Fm(0)3135 931 y Fr(\()p Fp(y)3208 943 y Fo(p)3246 931 y Fr(\)\))g Fp(;)456 1073 y Fr(where)24 b(the)h(v)-5 b(ariables)24 b Fp(y)1216 1085 y Fm(1)1253 1073 y Fp(;)14 b(y)1331 1085 y Fm(2)1368 1073 y Fp(;)g(:::;)g(y)1552 1085 y Fo(p)1615 1073 y Fr(include)25 b(the)h(v)-5 b(ariables)23 b Fp(x)2427 1085 y Fm(1)2465 1073 y Fp(;)14 b(x)2549 1085 y Fm(2)2586 1073 y Fp(;)g(:::;)g(x)2776 1085 y Fo(n)2847 1073 y Fr(and)25 b(neigh)n(b)r(ors)f(of)456 1175 y(them)i(in)f Fu(Z)818 1145 y Fo(d)883 1175 y Fr(within)h(a)f(distance)h Fp(N)9 b 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Fp(;)e(:::;)g(!)2019 1811 y Fm(0)2056 1799 y Fr(\()p Fp(y)2129 1811 y Fo(p)2167 1799 y Fr(\)\))q Fn(i)2264 1827 y Fo(Q)2316 1835 y Fh(L)2360 1827 y Fo(;m)2568 1799 y Fr(exists)p Fp(:)456 1973 y Fr(Next)23 b(w)n(e)g(use)g(Lemma)f(2.3)h (to)g(sho)n(w)f(that)h(the)g(limit)h(\(2.26\))f(exists)f(pro)n(vided)g Fp(c)2970 1985 y Fo(d)3009 1973 y Fr(\003\()p Fp(t)3129 1985 y Fo(n)3183 1973 y Fn(\000)9 b Fp(t)3287 1985 y Fm(1)3324 1973 y Fr(\))24 b Fp(<)456 2073 y Fr(1.)36 b(In)28 b(fact)g(on)f(restricting)f Fp(m)i Fr(to)g(satisfy)f Fp(c)1835 2085 y Fo(d)1873 2073 y Fr(\(\003)19 b(+)f Fp(m)2138 2043 y Fm(2)2175 2073 y Fr(\)\()p Fp(t)2269 2085 y Fo(n)2333 2073 y Fn(\000)g Fp(t)2446 2085 y Fm(1)2483 2073 y Fr(\))24 b Fp(<)e Fr(1)28 b(w)n(e)f(ha)n(v)n(e)f(that)1101 2144 y Fj(\014)1101 2194 y(\014)1101 2244 y(\014)1101 2293 y(\014)1142 2264 y Fn(h)q Fp(f)c Fr(\()p Fp(!)1322 2276 y Fm(0)1359 2264 y Fr(\()p Fp(x)1438 2276 y Fm(1)1476 2264 y Fr(\))p Fp(;)14 b(!)1597 2276 y Fo(t)1622 2284 y Fe(2)1655 2276 y Fl(\000)p Fo(t)1732 2284 y Fe(1)1768 2264 y Fr(\()p Fp(x)1847 2276 y Fm(2)1885 2264 y Fr(\))p Fp(;)g(:::::;)g(!)2158 2276 y Fo(t)2183 2284 y Fh(n)2224 2276 y Fl(\000)p Fo(t)2301 2284 y Fe(1)2337 2264 y Fr(\()p Fp(x)2416 2276 y Fo(n)2462 2264 y Fr(\)\))q Fn(i)2559 2289 y Fo(Q;m;W)456 2264 y Fr(\(2.28\))988 2556 y Fn(\000)1067 2488 y Fj(\012)1105 2556 y Fp(f)1169 2488 y Fj(\000)1207 2556 y Fp(!)1259 2568 y Fm(0)1296 2556 y Fr(\()p Fp(x)1375 2568 y Fm(1)1413 2556 y Fr(\))p Fp(;)g(!)1534 2568 y Fo(t)1559 2576 y Fe(2)1591 2568 y Fl(\000)p Fo(t)1668 2576 y Fe(1)p Fh(;N)1773 2556 y Fr(\()p Fp(x)1852 2568 y Fm(2)1890 2556 y Fr(\))p Fp(;)g(:::::;)g(!)2163 2568 y Fo(t)2188 2576 y Fh(n)2229 2568 y Fl(\000)p Fo(t)2306 2576 y Fe(1)p Fh(;N)2410 2556 y Fr(\()p Fp(x)2489 2568 y Fo(n)2535 2556 y Fr(\))2567 2488 y Fj(\001)q(\013)2645 2589 y Fo(Q;m;W)2885 2435 y Fj(\014)2885 2485 y(\014)2885 2535 y(\014)2885 2585 y(\014)858 2871 y Fn(\024)995 2815 y Fr([)p Fp(c)1054 2827 y Fo(d)1093 2815 y 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Fn(\024)23 b Fr(2)41 b(exp[)p Fp(C)3073 4185 y Fo(d)3112 4173 y Fp(A)3174 4139 y Fm(2)3211 4173 y Fp(=\025)p Fr(])p Fp(;)456 4318 y Fr(pro)n(vided)26 b Fp(d)d Fn(\025)g Fr(3.)37 b(W)-7 b(e)28 b(also)e(ha)n(v)n(e)g(from)i(the)g(argumen)n(t)e (of)i(Lemma)f(2.3)g(that)552 4484 y Fn(h)q Fr(exp)13 b([)p Fp(A)p Fn(j)p Fp(!)885 4496 y Fo(t)915 4484 y Fr(\()p Fp(x)p Fr(\))19 b Fn(\000)f Fp(!)1180 4496 y Fo(t;N)1288 4484 y Fr(\()p Fp(x)p Fr(\))p Fn(j)p Fr(])q Fn(i)1478 4512 y Fo(Q;m;W)1727 4484 y Fr(=)23 b Fn(h)p Fr(exp)14 b([)p Fp(A)p Fn(j)p Fp(!)2148 4496 y Fo(t)2177 4484 y Fr(\(0\))19 b Fn(\000)f Fp(!)2437 4496 y Fo(t;N)2544 4484 y Fr(\(0\))p Fn(j)p Fr(])p Fn(i)2729 4512 y Fo(Q;m;W)1117 4702 y Fn(\024)1204 4560 y Fj(*)1271 4702 y Fr(exp)1412 4560 y Fj(")1460 4702 y Fp(A)48 b Fr(sup)1536 4771 y Fm(0)p Fl(\024)p Fo(s)p Fl(\024)p Fo(t)1743 4702 y Fn(k)p Fp(!)1837 4714 y Fo(s;)p Fm(1)1943 4702 y Fn(\000)18 b Fp(!)2078 4714 y Fo(s;)p Fm(0)2166 4702 y Fn(k)2208 4560 y Fj(,)2314 4609 y(n)2369 4702 y Fr(1)g Fn(\000)g Fp(c)2548 4714 y Fo(d)2587 4702 y Fr(\(\003)g(+)g Fp(m)2851 4667 y Fm(2)2888 4702 y Fr(\))p Fp(t)2950 4609 y Fj(o)3006 4560 y(#+)3121 4810 y Fo(Q;m;W)456 4939 y Fr(pro)n(vided)26 b Fp(c)833 4951 y Fo(d)872 4939 y Fr(\(\003)18 b(+)g Fp(m)1136 4909 y Fm(2)1173 4939 y Fr(\))p Fp(t)24 b(<)e Fr(1.)37 b(W)-7 b(e)28 b(also)e(ha)n(v)n(e)h(the)h(inequalit)n(y)-7 b(,)735 5130 y(sup)701 5200 y Fm(0)p Fl(\024)p Fo(s)p Fl(\024)p Fo(t)908 5130 y Fn(k)p Fp(!)1002 5142 y Fo(s;)p Fm(1)1108 5130 y Fn(\000)18 b Fp(!)1243 5142 y Fo(s;)p Fm(0)1331 5130 y Fn(k)k(\024)h Fp(C)1542 5142 y Fo(d)1581 5130 y Fr(\(\003)18 b(+)g Fp(m)1845 5096 y Fm(2)1882 5130 y Fr(\))1936 5051 y Fj(X)1928 5229 y Fo(x)p Fl(2)p Fo(Q)2077 5130 y Fp(e)2116 5096 y Fl(\000j)p Fo(x)p Fl(j)2263 5013 y Fj(\024)2306 5130 y Fn(j)p Fp(!)2381 5142 y Fm(0)2418 5130 y Fr(\()p Fp(x)p Fr(\))p Fn(j)i Fr(+)52 b(sup)2655 5200 y Fm(0)p Fl(\024)p Fo(s)p Fl(\024)p Fo(t)2862 5130 y Fn(j)p Fp(B)2948 5142 y Fo(s)2984 5130 y Fr(\()p Fp(x)p Fr(\))p Fn(j)3118 5013 y Fj(\025)3176 5130 y Fp(:)p eop %%Page: 15 15 15 14 bop 1520 226 a Fm(PDE)28 b(AND)h(FIELD)g(THEOR)-5 b(Y)997 b(15)456 425 y Fr(If)28 b(w)n(e)f(use)g(the)h(inequalit)n(y)-7 b(,)1183 617 y Fp(e)1222 583 y Fo(")p Fl(j)p Fo(z)r Fl(j)1354 617 y Fn(\024)22 b Fr(4)1483 541 y Fl(p)p 1537 541 32 3 v 1537 583 a Fo(")1587 617 y Fr(cosh)1758 553 y Fn(p)p 1827 553 39 4 v 64 x Fp(")h(z)t(;)68 b Fr(0)23 b Fp(<)g(")f Fn(\024)2335 561 y Fr(1)p 2335 598 42 4 v 2335 674 a(4)2386 617 y Fp(;)70 b(z)26 b Fn(2)d Fu(R)p Fp(;)456 803 y Fr(then)28 b(it)g(is)f(clear)g(from)g(\(2.3\))g(that)1008 918 y Fj(*)1075 1060 y Fr(exp)1216 893 y Fj(2)1216 1042 y(4)1271 1060 y Fp(A)1355 981 y Fj(X)1347 1159 y Fo(x)p Fl(2)p Fo(Q)1496 1060 y Fp(e)1535 1026 y Fl(\000j)p Fo(x)p Fl(j)1667 1060 y Fn(j)p Fp(!)1742 1072 y Fm(0)1779 1060 y Fr(\()p Fp(x)p Fr(\))p Fn(j)1913 893 y Fj(3)1913 1042 y(5)1970 918 y(+)2037 1193 y Fo(Q;m)2195 1060 y Fn(\024)c Fp(C)2342 1072 y Fo(d)2394 1060 y Fr(exp[)p 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1922 58 4 v 2812 1998 a Fr(\003)2880 1824 y Fj(\023)2941 1841 y Fo(n)3076 1885 y Fr(\003)p 3038 1922 135 4 v 3038 1998 a(2)p Fp(\025\021)3182 1941 y(:)-2749 b Fr(\(3.16\))456 2147 y(Eviden)n(tly)30 b(for)g(an)n(y)g Fp(\021)i(>)c Fr(0)i(w)n(e)h(can)f(mak)n(e)g(the)h(RHS)h(of)e(\(3.16\))g(as)h(small)f (as)g(w)n(e)h(please)f(b)n(y)456 2246 y(simply)d(taking)g Fp(n)h Fr(large)e(enough.)36 b(The)28 b(result)f(follo)n(ws)g(from)g (this)h(and)f(\(3.14\).)p 3384 2246 4 57 v 3388 2193 50 4 v 3388 2246 V 3437 2246 4 57 v 555 2346 a Fu(Ac)m(kno)m(wledgemen) m(t:)32 b Fr(This)21 b(researc)n(h)e(w)n(as)h(partially)f(supp)r(orted) i(b)n(y)g(NSF)g(under)g(gran)n(t)456 2445 y(DMS-0138519.)1708 2675 y Fq(References)456 2808 y Ft([1])34 b(H.)20 b(Brascamp)f(and)h (E.)g(Lieb,)g Fk(On)i(extensions)g(of)h(the)f(Brunn-Minkowski)g(and)h (Pr)n(\023)-35 b(ekop)l(a-L)l(eind)t(ler)24 b(the)l(o-)565 2891 y(r)l(ems,)g(including)e(ine)l(qualities)h(for)f(lo)l(g)h(c)l(onc) 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(t,)e(a)n(v)l(ailable)h(at)h(www.math.lsa.umic)n(h.edu/)p Fb(\030)p Ft(conlon.)456 3555 y([5])34 b(J.)18 b(Conlon)h(and)f(A.)g (Naddaf,)h Fk(Gr)l(e)l(ens)i(F)-5 b(unctions)21 b(for)f(El)t(liptic)h (and)g(Par)l(ab)l(olic)h(Equations)g(with)e(R)l(andom)565 3638 y(Co)l(e\016cients)p Ft(,)j(New)h(Y)-6 b(ork)24 b(J.)f(Math.)h Fd(6)f Ft(\(2000\),)i(153-225,)f(MR)f(2001j:35282.)456 3721 y([6])34 b(R.)28 b(Durrett,)h Fk(Pr)l(ob)l(ability:)41 b(the)l(ory)30 b(and)h(examples)p Ft(,)e(second)h(edition,)f(Duxbury)f (Press,)g(Belmon)n(t)g(CA.)565 3804 y(1996.)d(MR)e(98m:60001.)456 3887 y([7])34 b(T.)23 b(F)-6 b(unaki)24 b(and)g(H.)e(Sp)r(ohn,)i Fk(Motion)i(by)e(me)l(an)i(curvatur)l(e)g(fr)l(om)g(the)f(Ginzbur)l (g-L)l(andau)i Fb(r)p Fc(\036)e Fk(interfac)l(e)565 3970 y(mo)l(del)p Ft(,)g(Comm.)c(Math.)j(Ph)n(ys.)f Fd(185)g Ft(\(1997\),)i(1-36,)e(MR)g(98f:60206.)456 4053 y([8])34 b(G.)28 b(Giacomin,)g(S.)f(Olla)g(and)h(H.)f(Sp)r(ohn,)i Fk(Equilibrium)h(\015uctuations)g(for)g Fb(r)p Fc(\036)f Fk(interfac)l(e)g(mo)l(del)p Ft(,)h(Ann.)565 4136 y(Probab.)24 b Fd(29)f Ft(\(2001\),)i(1138-1172,)g(MR)e(1872740.)456 4219 y([9])34 b(B.)e(Hel\013er)f(and)h(J.)g(Sj\177)-35 b(ostrand,)34 b Fk(On)f(the)g(c)l(orr)l(elation)h(for)g(Kac-like)e(mo)l (dels)j(in)e(the)g(c)l(onvex)g(c)l(ase)p Ft(,)h(J.)565 4302 y(Statist.)25 b(Ph)n(ys.)e Fd(74)g Ft(\(1994\),)i(349-409,)g(MR)e (95q:82022.)456 4385 y([10])35 b(S.)21 b(Kozlo)n(v,)h Fk(A)n(ver)l(aging)i(of)g(r)l(andom)i(structur)l(es)p Ft(,)c(Dokl.)g(Ak)l(ad.)f(Nauk.)h(SSSR)h Fd(241)d Ft(\(1978\),)k (1016-1019,)565 4468 y(MR)f(80e:60078.)456 4551 y([11])35 b(C.)29 b(Landim,)h(S.)g(Olla)f(and)i(H.)e(T.)g(Y)-6 b(au,)32 b Fk(Conve)l(ction-di\013usion)g(e)l(quation)g(with)g(sp)l(ac) l(e-time)g(er)l(go)l(dic)565 4634 y(r)l(andom)c(\015ow)p Ft(,)c(Probab.)f(Theory)i(Relat.)e(Fields)g Fd(112)g Ft(\(1998\),)i(203-220,)g(MR)e(99j:35084.)456 4717 y([12])35 b(A.)23 b(Naddaf)g(and)h(T.)f(Sp)r(encer,)i Fk(On)g(homo)l(genization)i (and)g(sc)l(aling)f(limit)f(of)h(some)g(gr)l(adient)g(p)l(erturb)l(a-) 565 4800 y(tions)g(of)g(a)g(massless)h(fr)l(e)l(e)e(\014eld)p Ft(,)f(Comm.)d(Math.)j(Ph)n(ys.)f Fd(183)g Ft(\(1997\),)i(55-84,)f(MR)f (98m:81089.)456 4883 y([13])35 b(H.)c(Osada)i(and)g(H.)f(Sp)r(ohn,)j Fk(Gibbs)e(me)l(asur)l(es)j(r)l(elative)d(to)h(Br)l(ownian)i(motion)p Ft(,)f(Ann.)d(Probab.)g Fd(27)565 4967 y Ft(\(1999\),)26 b(1183-1207,)e(MR)g(2001f:82024.)456 5050 y([14])35 b(G.)21 b(P)n(apanicolaou)h(and)g(S.)f(V)-6 b(aradhan,)22 b Fk(Boundary)i (value)g(pr)l(oblems)g(with)g(r)l(apid)t(ly)g(oscil)t(lating)f(r)l (andom)565 5133 y(c)l(o)l(e\016cients)p Ft(,)c(V)-6 b(olume)17 b(2)h(of)37 b Fk(Col)t(l.)21 b(Math.)f(So)l(c.)h(Janos)g(Bolya)p Ft(,)f Fd(27)p Ft(,)d(Random)h(\014elds,)h(Amsterdam,)d(North)565 5216 y(Holland)24 b(Publ.)f(Co.)g(1981,)i(pp.)e(835-873,)i(MR)e (84k:58233.)p eop %%Page: 20 20 20 19 bop 456 226 a Fm(20)1075 b(JOSEPH)28 b(G.)h(CONLON)456 425 y Ft([15])35 b(T.)17 b(Sp)r(encer,)i Fk(Sc)l(aling,)j(the)d(fr)l(e) l(e)h(\014eld)h(and)g(statistic)l(al)f(me)l(chanics)p Ft(,)f(in)e(The)h(Legacy)h(of)f(Norb)r(ert)f(Wiener:)565 508 y(A)j(Cen)n(tennial)h(Symp)r(osium)d(\(Cam)n(bridge)h(MA,)g(1994\)) i(Pro)r(c.)f(Symp)r(os.)e(Pure)i(Math.)g Fd(60)p Ft(,)f(Amer.)f(Math.) 565 591 y(So)r(c.,)24 b(Pro)n(vidence)h(1997,)f(MR)f(98k:82009.)456 674 y([16])35 b(V.)22 b(Zhik)n(o)n(v,)i(S.)f(Kozlo)n(v)h(and)g(O.)f (Oleinik,)f Fk(Homo)l(genization)27 b(of)e(Di\013er)l(ential)g(Op)l(er) l(ators)i(and)f(Inte)l(gr)l(al)565 757 y(F)-5 b(unctionals)p Ft(,)25 b(Springer)e(V)-6 b(erlag,)23 b(Berlin,)f(1994,)j(MR)e (96h:35003b.)555 913 y Fs(University)i(of)g(Michigan,)f(Dep)l(ar)l (tment)h(of)f(Ma)l(thema)l(tics,)h(Ann)f(Arbor,)h(MI)g(48109-1109)555 996 y Fk(E-mail)h(addr)l(ess)5 b Ft(:)33 b Fa(conlon@umich.edu)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0302031559473--