jjj m aj(x)

15 Pages·1998·0.2 MB·English

CONSTRUCTIONS WITH ANALYTIC SEMIGROUPS AND ABSTRACT
EXPONENTIAL DECAY RESULTS FOR EIGENFUNCTIONS
SIGURDANGENENT
PresentedontheoccasionofH.Amann’s60thbirthday
Contents
1. Introduction
2. Resolutions
3. Localization
4. AnExample
5. Exponentialdecayoftheresolvent
6. Exponentialdecayofeigenfunctions
7. Exponentialdecayintheexample
8. Applicationto“Fredholmsolutions”ofNonlinearEllipticSystems
1. INTRODUCTION
Inthisnotewepresentsomeconstructionswithgeneratorsofanalyticsemigroupswhich
areanabstractversionofthefamiliarmethodof“freezingthecoefficients”toproveellip-
tic estimates for differentialoperatorswith continuouscoefficientsor Ho¨lder-continuous
coefficients. Asasideresultweobtainanabstractexponentialdecayresultfor,sayeigen-
functionscorrespondingtoisolatedeigenvalues.
Afterrecallingsomebasicdefinitionsinsection1.1wefirstintroducearesolutionofan
operator inaBanachcouple (section2)andthendiscusslocalized
resolutionAs,:wEh1ic!h aEll0ow us to formulate a(nEa1b;sEtr0a)ct version of the method of “freezing
the coefficients” (section 3). In section 4 we show how all this can be applied to show
thatsystems of parabolicoperators generateanalytic semigroups
j
in spaces. In this example the Ares=ult isjjcje(cid:20)rmtaianjly(xn)Dot new, but it illustrates the the-
p
oryLand suggests generalizations. In section 5 we show that the matrix elements of the
P
resolventof an with respectto a localizedresolutionare exponentiallyde-
caying,providedAce2rtaHinolc(oEm)mutatorsaresufficientlysmall(Intheexampleofsystemsof
differential operators on this would imply that the integral kernel of the resolvent is
Rd
exponentiallydecaying.) Thisresultimmediatelyimpliesexponentiallocalizationofgen-
eralizedeigenvaluescorrespondingtoisolatedeigenvalues offinitemultiplicityagain,
assumingcertaincommutatorsaresmallenough(seesectio(cid:21)n06). Fortheexampleitturns
outonecanalwayschoosea localizedresolutionforwhichtherelevantcommutatorsare
adequatelysmall. Thiswaywefindthatgeneralizedeigenfunctions(correspondingtoiso-
lated eigenvalues with finite multiplicity) of parabolic operators
j
alwaysdecayexponentiallyat . ThisfacthasanimplicatioAnfo=rclasjjsji(cid:20)camlsaojl(uxt)ioDns
of nonlinear elliptic systems, nxam=e1ly, if an entire solution of an elliptic system of
P
u(x)
1
2 SIGURDANGENENT
PDEs is such that the linearized equation at has as
m
isolateFde(uig;eDnvua;l:u:e:o;fDfiniute)m=ult0iplicity,then exists(sectioun8). (cid:21) = 0
limjxj!1u(x)
I havefoundthe theoryof analyticsemigroupsparticularlyusefulto proveshorttime
existence and regularity results for an ever increasing class of nonlinear diffusion equa-
tions. Inthemiddle80iesIfoundtheworks[1][2][3]particlarlyinspiring. Thereadercan
find much information in [4], (see also [13][5][11][14][12]). There is some overlap be-
tweenthetheorypresentedhereandtheworkinvolumesIandIIof[4]. Ihopethereader
willfindthedifferentperspectiveuseful.
1.1. Some definitions and facts. A Banach couple is a pair of Banach
spaces with denselyembeddedin . AnEop=era(tEor1; E0) generates
ananalEyt1ic(cid:26)seEm0igroupEif1forsome all E0 wAith2L(E1; Ea0re)invertible
! >0 (cid:21)(cid:0)A:E1 !E0 <(cid:21)>!
while .
(cid:0)1
sup<(cid:21)>! ((cid:21)(cid:0)A) L(E0;E1) <1
Wedenotethe(cid:13)setof (cid:13) whichgenerateananalyticsemigroupby .
Thissetis open(cid:13)(cid:13)in A2(cid:13)(cid:13)L(E, m1;eEan0i)nga smallperturbation(in the Hopoelr(aEto)r
norm) of a generatoLr(Eof1;anE0a)nalytic semigroup also generates one. OEt1he!r pEer0turbations
which do not destroy the property of generatingan analytic semigroup are addition of a
compact operator , or an operator of zero relative bound
(meaningforall K : tEhe1re!isEan0 suchthaLt : E1 ! E0 for
any ) " > 0 C" <1 kLxkE0 (cid:20) "kxkE1 +C"kxkE0
x2E1:
2. RESOLUTIONS
ConsideraBanachcouple . Bydefinitionaresolution of consists
(cid:14) "
ofanotherBanachcouple aEndtwomapsofBanachcouplEes(cid:0)!F (cid:0)!Eand E
whichsatisfy F (cid:14) :E !F ":F !E
"(cid:14) =idE:
Aresolutionofanoperator is,bydefinition,anoperator
0
A 2 L(E1; E0) A 2 L(F1; F0)
where isaresolutionoftheBanachcouple .
(cid:14) "
E (cid:0)!F (cid:0)!E E
Foranyresolution of wedefinethecommutators
0
A A
0
[A; (cid:15)] = A(cid:15)(cid:0)(cid:15)A
0
[A; (cid:14)] = A(cid:14)(cid:0)(cid:14)A:
Theseareboundedoperators: and . Ingenerala
resolutionwillbequiteuseless[,Au;n(cid:15)le]s2stLhe(Fc1o;rrEes0p)ondin[gAc;o(cid:14)m]m2uLta(tEor1s;aFre0)insomesense
small.
We will call the resolution exact if the commutators and vanish. If
0
thesecommutatorsarecompactAoperators(from to , or[A; (cid:15)t]o )[tAhe;n(cid:14)]wewillcall
the resolution compact. If the commutatorsareFop1eratEor0switEh1relatFiv0e bound zero, then
we’llsaytheresolutionhasrelativeboundzero.
Theorem2.1. Let have a resolution , andassume that
0
this resolution is eiAthe2r cLo(mEp1a;cEt,0o)r of relative boundAzero2. LT(hFen1; F0) implies
0
. A 2 Hol(F)
A2Hol(E)
AnalyticSemigroupsandExponentialDecay 3
PROOF. Let bethegivenresolutionandassumethattheresolutionis
0
compact.ItfolloAws2froLm(F1; F0) that
(cid:15)(cid:14) =idE
(cid:25)F =(cid:14)(cid:15)
isaboundedprojectionin . Thisprojectionallowsustosplit as
F F
range kern
F = ((cid:14))(cid:8) ((cid:15)):
Thematrixofthelinearoperator withrespecttothissplittingcanbewrittenas
0
A
0 P Q
A =
R S
with , (cid:18) (cid:19) ,etc.
P 2L(range((cid:14))1; range((cid:14))0) R2L(range((cid:14))1; kern((cid:15))0)
Theoffdiagonalpartsof , and ,aregivenby
0
A R Q
0 0
R=(1(cid:0)(cid:25)F)A(cid:25)F =[A; (cid:25)F](cid:25)F
0 0
Q=(cid:25)FA(1(cid:0)(cid:25)F)=(cid:25)F[(cid:25)F; A]
andsince
0 0
[A; (cid:25)F]=[A; (cid:14)(cid:15)]=[A; (cid:14)](cid:15)+(cid:14)[A; (cid:15)]
the commutator is a compact operator. Therefore and are also compact.
0
Compactperturba[Atio;n(cid:25)sFof]an stillbelongto R ,soQthat
0
A 2Hol(F) Hol(F)
P 0
2Hol(F);
0 S
which implies that (cid:18) . (cid:19)But is an isomorphism onto its range, with
inverse ,so P 2 Hol(range((cid:14))) (cid:14)
(cid:15)
0
(cid:15)A(cid:14) =(cid:15)P(cid:14) 2Hol(E):
Finally,
0 0
(cid:15)A(cid:14)(cid:0)A=(cid:15)A(cid:14)(cid:0)(cid:15)(cid:14)A=(cid:15)[A; (cid:14)]
isagainacompactoperatorandwemayconcludethat .
A2Hol(E)
Iftheresolutionisnotcompactbutofvanishingrelativeboundinstead,thenthesame
proofwillwork.Q.E.D.
Givenaresolutionof onecantrytoexpresstheresolventof interms
oftheresolventof . LeAt 2L(E1; E0) A
0
A
0 0 (cid:0)1
R((cid:21))=((cid:21)(cid:0)A)
betheresolventof ,forsome .
A0 (cid:21)2C
Asafirstapproximationtotheresolventof weintroduce
A
0
S((cid:21))=(cid:15)R((cid:21))(cid:14):
Tomeasurebyhowmuch failstobetheresolventof ,weconsider
S((cid:21)) A
0
T((cid:21))=1E(cid:0)((cid:21)(cid:0)A)S((cid:21))=[A; (cid:15)]R((cid:21))(cid:14):
4 SIGURDANGENENT
Formallyarightinversefor isgivenby
(cid:21)(cid:0)A
right-
(cid:0)1 (cid:0)1
((cid:21)(cid:0)A) =S((cid:21))[((cid:21)(cid:0)A)S((cid:21))]
(cid:0)1
= S((cid:21))[1E (cid:0)T((cid:21))]
2
= S((cid:21))[1E +T((cid:21))+T((cid:21)) +(cid:1)(cid:1)(cid:1)]
1
0 0 k
= (cid:15)(cid:14)R((cid:21))f(cid:14)[A; (cid:15)]R((cid:21))g (cid:14)
k=1
X
1
0 0 k
= (cid:15)R((cid:21)) f(cid:14)[A; (cid:15)]R((cid:21))g (cid:14):
"k=1 #
If the infinite series convergesin the opXeratornorm then the terms of the series decayat
leastexponentially,andthe series representsan actualrightinversefor . Thusthe
seriesconvergesifandonlyifforsome onehas (cid:21)(cid:0)A
k (cid:21)1
0 k
kf(cid:14)[A; (cid:15)]R((cid:21))g kL(F0) <1;
or,whatisequivalent,ifthespectralradiusof islessthanone.
0
(cid:14)[A; (cid:15)]R((cid:21)) 2L(E0)
Toobtainaleftinversefor oneconsiders
(cid:21)(cid:0)A
T^((cid:21))=1E (cid:0)S((cid:21))((cid:21)(cid:0)A)=(cid:0)(cid:15)R0((cid:21))[A; (cid:14)];
andonetriestosumtheseries
left-
(cid:0)1
((cid:21)(cid:0)A) =
= [1E +T^((cid:21))+T^((cid:21))2+(cid:1)(cid:1)(cid:1)]S((cid:21))
1
0 k 0
= (cid:15) f(cid:0)R((cid:21))[A; (cid:14)](cid:15)g R((cid:21))(cid:14):
"k=1 #
Again,anecessaryandsufficientXconditionforsummabilityoftheseriesisthatthespec-
trumof asoperatoron iscontainedintheopenunitdisk. Wecansumma-
0
rizethisRas(f(cid:21)o)l[lAow;s(cid:14).](cid:15) F0
Theorem2.2. If hasaresolution ,and liesintheresolventset
of ,then alsoAbe2lonLg(sEt1o;tEhe0)resolventsetof iAf0 (cid:21) 2 C
0
A (cid:21) A
and
0 0
rspec(R((cid:21))[A; (cid:14)](cid:15))<1 rspec((cid:14)[A; (cid:15)]R((cid:21))) <1:
Theresolventof at isthengivenby
A (cid:21)
(1) 1
(cid:0)1 0 0 k
((cid:21)(cid:0)A) = (cid:15)R((cid:21)) f(cid:14)[A; (cid:15)]R((cid:21))g (cid:14)
"k=0 #
X
(2) 1
0 k 0
= (cid:15) f(cid:0)R((cid:21))[A; (cid:14)](cid:15)g R((cid:21))(cid:14):
"k=0 #
X
3. LOCALIZATION
Let be a resolution of the Banach couple . We will say that the
(cid:14) (cid:15)
resolutiEon(cid:0)is!locFali(cid:0)ze!difEtheBanachcouple canbewrittenasaEdirectsum
F
((cid:11))
F =‘p(cid:0) F
(cid:11)2A
M
AnalyticSemigroupsandExponentialDecay 5
where isafiniteorcountableindexset.Moreprecisely,foreach wehaveaBanach
A (cid:11)2A
couple forwhich isaclosedsubspaceof ( ),andeach canbe
((cid:11)) ((cid:11))
writtenFinexactlyonewFajyasanormconvergentsumFj j =0; 1 x2Fj
((cid:11))
x= x
(cid:11)2A
andthe normof isequivalenttotheexXpression
Fj x
1=p
((cid:11)) p
kx kFj :
(cid:11)2A !
ThroughoutthisarticlealldirectsumXs willbeunderstoodtobe sumsinthesense
justexplained. (cid:11)2A ‘p
Wewillsaythatan hasfiniLtesupportif forafinite
((cid:11)1) ((cid:11)m)
numberof x 2 F.iWedenotethespacexof2 Fwiithfinit:e:s:uppForitby fin. Since
every (cid:11)1i;s:a::n;o(cid:11)rmmc2onAvergentsumof ’sthespxace finisdensein . Fi
((cid:11)) L L
x2Fi x Fi Fi
Given such a direct sum, we define the canonical projections and
((cid:11))
inclusions by p(cid:11) : F ! F
((cid:11))
i(cid:11) :F !F
(3) p(cid:11)(x)=x((cid:11)) if x= (cid:11)2Ax((cid:11))
((cid:11))
( i(cid:11)(x)=xforallx2FjP (j =0; 1)
If isaboundedoperator,thenwecanconsideritsassoci-
T 2 L(Fi; Fj) (i; j = 0;1)
atedmatrix ,with ,definedby
((cid:12)) ((cid:11))
fT(cid:11)(cid:12)g T(cid:11)(cid:12) 2L(Fi ; Fj )
(4)
T(cid:11)(cid:12) =p(cid:11)Ti(cid:12):
Foroperators wewillalsodefinethe“matrixelements”
((cid:12)) ((cid:11))
by T :E !E T(cid:11)(cid:12) 2L(Fi ; Fj )
(5)
T(cid:11)(cid:12) =p(cid:11)(cid:14)T"i(cid:12):
The graph of a localized resolution. We can define the matrix of the projection
(cid:25)F =(cid:14)(cid:15)
(cid:25)(cid:11)(cid:12) =p(cid:11)(cid:25)F i(cid:12);
andwecandefineagraph ,whoseverticesaretheelementsof ,andinwhich
and areconnectedifeithGer(E; F) or . A (cid:11)
(cid:12) (cid:25)(cid:11)(cid:12) 6=0 (cid:25)(cid:12)(cid:11) 6=0
Wewillalwaysassumethatthegraph isuniformlylocallyfinite. Thismeans
bydefinitionthat G(E; F)
(6) numberofedgescontaining
def
nG = maxf (cid:11)g
(cid:11)2A
isfinite.
Let be given. Then an operator will be called a
0
localizeAdr2esoLlu(tEio1n;oEf0),if A 2 L(F1; F0)
A
1. Thedecomposition is invariant.
((cid:11)) 0
2. Theoperator isFdo=mina(cid:11)te2dAbFytheprAojection ,inthesensethat
whenever (cid:14)A(cid:15) . (cid:25)F ((cid:14)A(cid:15))(cid:11)(cid:12) =0
L
(cid:25)(cid:11)(cid:12) =0
6 SIGURDANGENENT
By invariancewemean:
0
A
(7)
0 ((cid:11)) ((cid:11))
A(F1 )(cid:26)F0 (8(cid:11)2A):
Lemma3.1. If ,thentherestriction of to belongsto .
0 ((cid:11)) 0 ((cid:11)) ((cid:11))
A 2Hol(F) A A F Hol(F )
Proof.Let belongtotheresolventsetof . Then(7)triviallyimplies
0
(cid:21) A
0 (cid:0)1 ((cid:11)) ((cid:11))
((cid:21)(cid:0)A) (F0 )(cid:26)F1 (8(cid:11)2A):
Indeed, if then we can write as , with
((cid:11)) 0 (cid:0)1
f(cid:11) 2 F0 g = ((cid:21) (cid:0) A) f(cid:11) g = (cid:12)2Ag(cid:12)
,anditfollowsfrom
((cid:12))
g(cid:12) 2F1 P
0 0 ((cid:11))
((cid:21)(cid:0)A)g = ((cid:21)(cid:0)A)g(cid:12) =f(cid:11) 2F0
(cid:12)2A
and the uniqueness of the decomposXition of into the ’s that for all .
g g(cid:12) g(cid:12) = 0 (cid:12) 6= (cid:11)
Hence .
((cid:11))
g 2F1
Sincethenormoftheresolventrestrictedto canneverbemorethanitsnormon
((cid:11))
,thisshowsthat . Q.E.D. F0
((cid:11)) ((cid:11))
F0 A 2Hol(F )
Asaconversetothislemmawehavethefollowingobservation.
Lemma3.2. Assume that all the ’s belong to , that an exists for
which any with Ab((cid:11)el)ongsto the reHsoollv(eFn(t(cid:11)s)e)t of all the! 2 R’s, and for
which (cid:21) 2 C <((cid:21)) (cid:21) ! A((cid:11))
((cid:11)) (cid:0)1
M =<(s(cid:21)u)p(cid:21)!(cid:11)su2pAk((cid:21)(cid:0)A ) kL(F0((cid:11));F1((cid:11))) <1:
Then .
0
A 2Hol(F)
Indeed,if then isintheresolventsetof . Theresolventisgivenby
0
<((cid:21))(cid:21)! (cid:21) A
0 (cid:0)1 ((cid:11)) (cid:0)1
((cid:21)(cid:0)A) ff(cid:11)g=f((cid:21)(cid:0)A ) f(cid:11)g
andits normdoesnotexceed .
L(F0; F1) M
4. AN EXAMPLE
Let with ,andconsiderthe -order
(matrixE)d=ifferWenpmtia(Rlodp;eRranto);rLp(Rd; Rn) 1<p<1 mth
(cid:0) (cid:1)
j
A= aj(x)D 2L(E1; E0):
jjj(cid:20)m
Herethecoefficients areconXtinuous matrixvaluedfunctions,andthesubscript
isamultiindex. Weawj(ixll)showthat d(cid:2)d ,ifthe areuniformlycontinuous,
ajndiftheprincipalsymbol A 2 Hol(E) aj(x)
(8)
def j
Am(x;(cid:24)) = aj(x)((cid:0)i(cid:24))
jjj=m
satisfies X
(9)
(cid:0)1 c
((cid:21)I (cid:0)Am(x;(cid:24))) (cid:20) m
j(cid:21)j+j(cid:24)j
(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
AnalyticSemigroupsandExponentialDecay 7
forall ,all ,all with andsomeconstant . Thisisof
coursexw2ellRkdnown(cid:24),b2utCtdhepro(cid:21)of2weCgivebe<lo(cid:21)w(cid:21)gecn;eralizestomanyothcer>si0tuationsand
shouldalsomotivatetheconstructionswhichfollowinthissection.
Chooseaboundedopensymmetricneighbourhood oftheorigin,whosetrans-
lates cover (e.g. asufficientlylaOrge(cid:26)spRhdere). Wethendefineforany
“scale”O+(cid:11)ja(cid:11)nd2Zd thesRedt
r >0 (cid:11)2Zd
(10) (cid:8) (cid:9)
def
O(cid:11);r = fr((cid:11)+x)jx2Og
For any the cover . This coveringis uniformlylocally finite: since is
boundedrit>con0tainsOon(cid:11)l;yrafiniteRnudmber oflatticepoints,andhenceany intersOects
exactly other s. N O(cid:11);r
N O(cid:12);r
Next we constructa suitable partitionof unity. Choose so that the
1
translatesby of alsocover . Define 0 (cid:20) 2 Cc (O)
Zd fxj (x)>0g Rd
def (x(cid:0)(cid:11))
’(cid:11)(x) =
2
(cid:12)2Zd (x(cid:0)(cid:12))
and
q
P
def x
’(cid:11);r(x) = ’(cid:11)( ):
r
Then
(11)
’(cid:11);r(x(cid:0)(cid:11))2 =1 (8x2Rd):
(cid:11)2Zd
X
We nowdescribetheresolution. Theindexsetwillbe ,andforeach
weput A = Zd (cid:11) 2 Zd
F((cid:11)) =(Wpm(Rd; Rn); Lp(Rd; Rn))
sothattheBanachcouple isdefinedby
F
((cid:11))
F =‘p(cid:0) F :
(cid:11)2A
M
Thus is the space of all sequences of functions for
((cid:11))
whichFtjhe(sju=m 0; 1) ff(cid:11)g(cid:11)2Zd f(cid:11) 2 Fj
p p
kff(cid:11)g(cid:11)2AkFj = kf(cid:11)kFj((cid:11))
(cid:11)2A
isfinite. X
Weconstructthemorphisms usingthepartitionofunity . Forany
(cid:14) (cid:15)
wedefinetheE !coFm!ponEentof tobe f’(cid:11)g
th
f 2Ej (j =0; 1) (cid:11) (cid:14)f
(12)
((cid:14)f)(cid:11) =’(cid:11);r(x)f(x);
andforany weput
ff(cid:11)g(cid:11)2A 2Fj (j =0; 1)
(13)
((cid:15)ff(cid:11)g)(x)= ’(cid:12);r(x)f(cid:12)(x):
(cid:12)2A
X
8 SIGURDANGENENT
Observethatthislastsumislocallyfiniteandthatboth and
((cid:11)) ((cid:11))
(cid:15):Fj !Ej (cid:14) :Ej !Fj
areboundedoperators. Asanexample,letusverifythat isbounded. Let
((cid:11))
begiven.Then (cid:14) : F1 ! E1
f 2Wpm(Rd; Rn)
p p
k(cid:14)fkF1 = k((cid:14)f)(cid:11)kF1((cid:11))
(cid:11)2A
X
p
= k’(cid:11)fkWpm( Rd;Rn)
(cid:11)2A
X
p
(cid:20) c kfkWpm(O(cid:11);r)
(cid:11)2A
X
(cid:20) NckfkE1:
where, as above, is the number of lattice points in . So is indeed
bounded. N O (cid:14) : E1 ! F1
Forany onehas
f 2Fj
2
((cid:15)(cid:14)f)(x)= ’(cid:12)(x) f(x)=f(x)
(cid:12)2A
X
sothat ,andtherefore isindeedalocalizedresolutionof .
(cid:14) (cid:15)
(cid:15)(cid:14) =1E E (cid:0)!F (cid:0)!E E
Theprojection actsasfollows:
(cid:25)F =(cid:14)(cid:15)
((cid:25)Fff(cid:11)g)(cid:11)(x)= ’(cid:11)(x)’(cid:12)(x)f(cid:12)(x);
(cid:12)2A
from which one can see what the graph X is: the vertices are the lattice points
,and areconnectedifandGo(Enl;yFif) . Inparticularthisgraphis
u(cid:11)n2ifoArmlylo(cid:11)ca;ll(cid:12)y2finAite,andwehave . (cid:11)(cid:0)(cid:12) 2 O
nG =N
Interpretationofthematrixcoefficients ofanoperator
. Itisevidentthatforany onehas T(cid:11)(cid:12) = p(cid:11)(cid:14)T"i(cid:12) T : E1 !
E0 f 2E1
(14)
T(cid:11)(cid:12)(f)=’(cid:11);rT(’(cid:12);rf)
sothat moreorlessrepresentsthecomponentin oftheimageunder ofthe
componTe(cid:11)n(cid:12)t(ifn) of . O(cid:11);r T
O(cid:12);r f
Next,weintroduceanoperator bysayinghowitactsoneach :
0 ((cid:11))
A 2L(F1; F0) F1
A((cid:11))f(cid:11) (x)= a(j(cid:11))(x)Djf(cid:11)(x) (8x2R)
jjj(cid:20)m
(cid:16) (cid:17) X
for any . The coefficients coincide with the coefficients of the
((cid:11)) ((cid:11))
f(cid:11) 2 F1 aj (x) aj(x)
originaloperator onthesupportof Outsideofthesesupportsthe arecontin-
((cid:11))
uousextensionsoAfthe forwhic’h(cid:11): aj (x)
aj(x)
(15)
((cid:11)) ((cid:11))
xo2sRcdaj (x)=x2ospstc’(cid:11)aj (x)
holds.
1
Forvectorvaluedfunctions oneshoulddefined tobethesmallest suchthattherangeof is
cont1ainedinaballofradius .f oscf r>0 f
r=2
AnalyticSemigroupsandExponentialDecay 9
The operator defined by is a resolution of . Observe that
0 0 ((cid:11))
the componeAntof ,i.e. A f,fc(cid:11)ogin=cidesAwithf(cid:11) onthesupportof A,butischosen
th 0 ((cid:11))
arbi(cid:11)trarilyoutsideofthAisset. A A ’(cid:11)
(cid:8) (cid:9)
Inordertoseehowgoodtheresolution is,wewillcomputethecommutators
0
and .ItturnsoutthatthesecommutatoArsdonotdependonwhatthe ’sdoou[Ats;id(cid:15)e]
((cid:11))
of th[eA;su(cid:14)p]portof . This givesus the freedomto choose any waAy we like on the
((cid:11))
complementofthe’s(cid:11)upportof . TheparticularchoicewhicAhwehavemadeensuresthat
thecoefficients of ’(cid:11)donotoscillatemorethanthe doon .
((cid:11)) ((cid:11))
aj (x) A aj(x) O(cid:11);r
In general the commutator of a differential operator with a multiplication operator is
anotherdifferentialoperatoroflowerorder. Thepointofthefollowingcomputationsisto
showthat and aremadeupofdifferentialoperatorsoforder .
[A; (cid:15)] [A; (cid:14)] m(cid:0)1
If ,then
ff(cid:11)g2F1
((cid:11))
[A; (cid:15)](cid:1)ff(cid:11)g=A ’(cid:11);rf(cid:11) (cid:0) ’(cid:11);rA ff(cid:11)g
((cid:11)2A ) (cid:11)2A
X X
j
= aj(x)[D ; ’(cid:11);r]f(cid:11)
jjj(cid:20)m(cid:11)2A
X X
k j(cid:0)k
= aj(x) (cid:13)j;kD (’(cid:11);r)D (f(cid:11))
(cid:11)2A jjj(cid:20)m0<k(cid:20)j
X X X
(16) j(cid:0)k
((cid:11)) D (f(cid:11))
= bj;k(x) jkj
r
jjj(cid:20)m0<k(cid:20)j(cid:11)2A
X X X
wherethe arefixedintegerswhichappearascoefficientswhenoneexpandsthecom-
mutator (cid:13)j;k ,andwhere
j
[D ; ’(cid:11);r]
(17)
((cid:11)) k x(cid:0)(cid:11)
bj;k(x)=aj(x)(cid:13)j;k D ’ :
r
(cid:18) (cid:19)
(cid:0) (cid:1)
Thecoefficients areuniformlybounded,i.e. thereisaconstant suchthat
((cid:11))
bj;k(x) M <1
((cid:11))
bj;k(x) <M
(cid:12) (cid:12)
for all and all multiindices (cid:12) (cid:12) , and the supportof is contained
in x.2CRondsequently the operator0(cid:12) < k (cid:20)i(cid:12)sjbounded from b(j(cid:11);k)(x) to
O(cid:11);r ,andtherefore ,byth[Aei;n(cid:15)t]erpolationinequality (cid:11)2AWpm(cid:0)1(Rd;Rn)
Lp(Rd;Rn) [A; (cid:15)]
L
(18)
1(cid:0)1=m 1=m
kukWpm(cid:0)1 (cid:20)CkukWpm kukLp
isanoperatorofzerorelativeboundfrom to .
F1 E0
10 SIGURDANGENENT
Asfor ,wecomputethatfor onehas
[A; (cid:14)] f 2E1
((cid:11))
([A; (cid:14)](cid:1)f)(cid:11) = A (’(cid:11)(x)f(x))(cid:0)’(cid:11)(x)Af(x)
j
= aj(x) D ; ’(cid:11);r f
jjj=m
X (cid:2) (cid:3)
k j(cid:0)k
= (cid:13)j;kD (’(cid:11);r)D (f)
jjj=m0<k(cid:20)j
X X
j(cid:0)k
((cid:11)) D (f)
= bj;k(x) k
r
jjj=m0<k(cid:20)j
X X
Againthecoefficients areuniformlybounded,fromwhichonededucesthat
((cid:11))
isaboundedoperatorbfrjo;km(x) to . Theinterpolati[oAn;in(cid:14)-]
equality(18)impliesthat Wpm(cid:0)1(Rd;Rn)haszer(cid:11)o2rAelLatpiv(Redb;oRunnd).
[A; (cid:14)]:E1 !F0
So isaresolutionof withvanishingrelaLtivebound,andifwecanshowthat
0 0
A,thenwehavealsoAshownthat . A 2
Hol(F) A2Hol(E)
Inordertoverifythat weconsidereach separately,withtheintention
0 ((cid:11))
A 2Hol(F) A
of usinglemma (3.2). By assumptionthe coefficients are almostconstantif one
((cid:11))
choosesthescale smallenough. Thus,givenanyaj (x) onecanchoose so
smallthatconstanrt > 0 matrices existforwhich (cid:22)0 > 0 r > 0
((cid:11))
d(cid:2)d a(cid:22)j
(19)
a(j(cid:11))(x)(cid:0)a(cid:22)(j(cid:11)) (cid:20)(cid:22)0; 8x2Rd
holds for all . So the ’s are uniformly small perturbations of the constant
(cid:12) ((cid:11)) (cid:12)
coefficientope(cid:11)ra2torsA (cid:12) A (cid:12)
(cid:12) (cid:12)
((cid:11)) ((cid:11)) j
A0 = a(cid:22)j D ;
jjj(cid:20)m
X
allofwhichbelongto . Theresolventof isgivenbyconvolutionwith
((cid:11)) ((cid:11))
Hol(F ) A0
(cid:0)1
(cid:0)1 ((cid:11)) j
F 8 (cid:21)(cid:0) a(cid:22)j ((cid:0)i(cid:24)) 9
0 1
>< jjXj(cid:20)m >=
where denotesthe Fouriertransform. An appropriatemultipliertheoremthenprovides
@ A
uswithFaconstant suc>:hthat >;
M <1
((cid:11)) (cid:0)1
k((cid:21)(cid:0)A0 ) kL(F0((cid:11));F1((cid:11))) (cid:20)M
whenever for some fixed . Thereforeif the size of the perturbation
<((cid:21)is)s(cid:21)ma!llenough,the ! w>il0lsatisfythesameestima(cid:22)te0with insteadof
((cid:11)) ((cid:11)) ((cid:11))
A . B(cid:0)ylAem0ma(3.2)thisimpliesthAat ,andthus .2M
0
M A 2Hol(F) A2Hol(E)
5. EXPONENTIALDECAY OF THERESOLVENT
Consideralocalizedresolution , , of and
(cid:14) " 0
assumethatforsome intheresolvEen(cid:0)t!setFof(cid:0)!onEehAas2 Hol(F) A 2 Hol(E)
0
(cid:21) A
(20) and
0 0
k(cid:14)[A; (cid:15)]R((cid:21))kL(F0;F1) (cid:20)# kR((cid:21))[(cid:14); A](cid:15)kL(F0;F1) (cid:20)#
forsome . ByTheorem2.2 belongstotheresolventsetof ,andtheresolventis
# < 1 (cid:21) A