The Lp Resolvents of Elliptic Operators with Uniformly Continuous Coefficients

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J. Differential Equations 188 (2003) 555–568

The Lp resolvents of elliptic operators with uniformly continuous coefﬁcients

Yoichi Miyazaki*,1 School of Dentistry, Nihon University, 8-13, Kanda-Surugadai 1-chome, Chiyoda-ku, Tokyo 101-8310, Japan

Received September 19, 2001; revised June 13, 2002

Abstract

We consider the strongly elliptic operator A of order 2m in the divergence form with bounded measurable coefﬁcients and assume that the coefﬁcients of top order are uniformly continuous. For 1opoN; A is a bounded linear operator from the Lp Sobolev space Hm;p into Hm;p: We will prove that ðA lÞ1 exists in Hm;p for some l and estimate its operator norm. r 2002 Elsevier Science (USA). All rights reserved.

Keywords: Lp theory; Elliptic operator; Non-smooth coefﬁcient; Divergence form; Resolvent; Uniformly continuous

1. Introduction

Let us consider the elliptic operator deﬁned in the n-dimensional Euclidean space Rn in the divergence form X a b AuðxÞ¼ D ðaabðxÞD uðxÞÞ; ð1:1Þ jaj;jbjpm

n where x ¼ðx1; y; xnÞ is a generic point in R and we use the notations pffiffiffiffiffiffiffi a a1 ? an D ¼ D1 Dn ; Dj ¼ 1@=@xj

*Corresponding author. E-mail address: [email protected]. 1 Supported partly by Sato Fund, Nihon University School of Dentistry.

0022-0396/03/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 0396(02)00109-2 556 Y. Miyazaki / J. Differential Equations 188 (2003) 555–568 for a multi-index a ¼ða1; y; anÞ of length jaj¼a1 þ ? þ an: Throughout this paper we assume the following. Smoothness: All the coefficients aab are bounded and measurable, and for jaj¼ jbj¼m; the coefficients aab are uniformly continuous. Strong ellipticity: There exists a constant d40 such that X aþb 2m aðx; xÞ¼ aabðxÞx Xdjxj jaj¼jbj¼m for all xARn and all xARn: We allowthe coefficients to be complex valued, although the principal symbol aðx; xÞ must be real valued. The notations will be given in the next section. We want to consider A in the Lp framework. In view of

Db aab Da Hm;p+Hjbj;p !Lp !Lp !Hjaj;p+Hm;p for jajpm and jbjpm one can regard A as a bounded operator from Hm;p to Hm;p for a ﬁxed p: By regarding aab as a multiplication operator we can also write X a b A ¼ D aabD : jaj;jbjpm

When p ¼ 2; we usually associate A with a sesquilinear form Z X b a B½u; v¼ aabðxÞ D uðxÞ D vðxÞ dx: n R jaj;jbjpm

In the L2 theory of the elliptic operator with non-smooth coefﬁcients the Garding( inequality plays a fundamental role. It leads to the existence of the resolvent ðA lÞ1 for some lAC and the estimate

1 1þðiþjÞ=2m jjðA lÞ jjHi;p-Hj;p pCjlj ð1:2Þ for 0pipm and 0pjpm when p ¼ 2 (see [[18, Lemma 3.3; 19, Lemma 7.6]]). The purpose of this paper is to obtain the same result for the resolvent in the Lp case as in 1 the L2 case. In other words, for 1opoN we will show the existence of ðA lÞ : Hm;p-Hm;p for some lAC and get estimates (1.2). In the paper [20] we proved the same result without assuming the uniform continuity when the operator is a small perturbation of an operator with constant coefﬁcients. We mean by small perturbation that

max sup jaabðxÞaabð0Þj jaj¼jbj¼m xARn is sufficiently small. For the small perturbation case we constructed a parametrix for A l by the Fourier multiplier to prove the result. In the case of this paper the proof Y. Miyazaki / J. Differential Equations 188 (2003) 555–568 557 is more complicated. We combine the partition of unity and the Fourier multiplier to construct a parametrix which turns out to be a pseudo-differential operator with double symbol (cf. [17,26]). If the coefficients are sufficiently smooth, we can write X a AuðxÞ¼ baðxÞD uðxÞ: ð1:3Þ jajp2m

Tanabe [25] (or [23,24]) treated the Lp boundary value problem for the elliptic operator with bounded and measurable coefficients of the form (1.3) in a bounded or unbounded domain in Rn: Under the condition that the top order coefficients ba are uniformly continuous and that the coefficients of the formally adjoint operator of A have the same smoothness property, he showed the existence 1 of the resolvent ðA lÞ in the Lp space (1opoN) for some lAC and derived the estimate

1 1þj=2m jjðA lÞ jjLp-Hj;p pCjlj ð1:4Þ for 0pjp2m (see also [1,2,11]). Based on (1.4) he estimated the heat kernel and the resolvent kernel for A: His result cannot be applied to the operator of the form (1.1) if the coefficients aab are not so smooth, since his condition requires at least that aab are in Cm-class for jaj¼jbj¼m: One of the motivations of this paper is to expand the Lp theory discussed in [25] to the divergence form elliptic operators. Along the same line as in [25] we can derive from (1.2) the theorem on the heat kernel for the operator defined by (1.1), which Auscher and Qafsaoui [5] have already obtained by using the Morrey–Campanato spaces and showing the equivalence between elliptic regularity and heat kernel estimates. This theorem asserts that the heat kernel is in Cm1þe-class for any eAð0; 1Þ and gives the estimates for its derivatives. From this theorem one can derive the estimates on the resolvent kernel and its derivatives. Estimate (1.2) gives another approach to this theorem. We can also apply (1.2) to elliptic regularity and the asymptotic behavior of the spectral function. These topics will be discussed in subsequent papers. In recent years the theory of elliptic operators in the divergence form has been developed. We refer to Davies [16], Auscher and Tchamitchian [7], Auscher and Qafsaoui [5] and references therein (see also [3,6,8,10,13–15]). The result of this paper concerning the Lp resolvent has been partly obtained. Auscher, McIntosh and Tchamitchian [4] proved that A1 : H1;p-H1;p is a bounded operator 1 when np2 for the second order operator A: Barbatis [9] proved that ðA þ 1Þ : Lp-Hm;p for p sufficiently close to 2 when A is the operator defined on a bounded domain in Rn: Our approach is different from theirs and most part of our result is new. 558 Y. Miyazaki / J. Differential Equations 188 (2003) 555–568

2. Main result

p p n Let 1pppN: We denote by jjujjp the norm in the space L ¼ L ðR Þ: For an integer kX0 the Lp Sobolev space Hk;p ¼ Hk;pðRnÞ is deﬁned by

Hk;p ¼ff ALp : Daf ALp for jajpkg; and the norm is given by 0 1 1=p X @ a pA jjf jjk;p ¼ jjD f jjp : jajpk

For an integer ko0 the Lp Sobolev space Hk;p ¼ Hk;pðRnÞ is deﬁned to be the set of all functions f which can be writtenX as a p f ¼ D fa; faAL ; ð2:1Þ jajpk and the norm is given by 0 1 1=p X @ pA jjf jjk;p ¼ inf jjfajjp ; jajpk where the inﬁmum is taken over all ffagjajpk satisfying (2.1) (cf. [21,22]). We denote by BðX; YÞ the set of bounded linear operators from a Banach space X to a Banach space Y endowed with the norm jj Á jjX-Y and set BðXÞ¼BðX; XÞ: We set

M ¼ max jjaabjjN; jaj;jbjpm

n oðeÞ¼ max supfjaabðxÞaabðyÞj : x; yAR ; jx yjpeg; jaj;jbjpm

Lðr; yÞ¼flAC : jljXr; yparglp2p yg for r40 and yAð0; p=2Þ:

Main theorem. Let 1opoN: For any yAð0; p=2Þ there exists a constant C140 depending only on n; m; p; d; M; y and the function o such that when lALðC1; yÞ;

ðA lÞ1 : Hm;p-Hm;p exists and

1 1þðiþjÞ=2m jjðA lÞ jjHi;p-Hj;p pCn;m;p;d;M;yjlj for 0pipmand0pjpm: Y. Miyazaki / J. Differential Equations 188 (2003) 555–568 559

The following is an immediate consequence of the main theorem.

p Corollary. Let 1opoN: Deﬁne the operator ALp in L by

m;p p DðALp Þ¼fuAH : AuAL g; ALp u ¼ Au:

1 Then the resolvent ðALp lÞ exists and

1 1þj=2m jjðALp lÞ jjLp-Hj;p pCn;m;p;d;M;yjlj for 0pjpm:

3. Preliminary

In this section we prepare some lemmas to prove the main theorem.

Mihlin’s multiplier theorem. Let 1opoN: Assume that for a function P : Rn-R there exists a constant K40 such that

a jxjj jjDaPðxÞjpK for jajp½n=2þ1: Then the Fourier multiplier PðDÞ deﬁned by Z PðDÞuðxÞ¼ð2pÞn eixxPðxÞ uˆðxÞ dx is a bounded operator in Lp and

jjPðDÞjjLp-Lp pCn;pK:

Proof. See [12, Theorem 6.1.6]. &

n 1 Lemma 3.1. Let 1opoN and ﬁx x0AR : Then we have ðaðx0; DÞlÞ A BðHi;p; Hj;pÞ and

1 1þðiþjÞ=2m jjðaðx0; DÞlÞ jjHi;p-Hj;p pCn;m;p;d;M;yjlj ð3:1Þ for 0pipmand0pjpm when lALð1; yÞ: More precisely, we have

a 1 1þjaj=2m jjD ðaðx0; DÞlÞ jjLp-Lp pCn;m;p;d;M;yjlj ð3:2Þ for 0pjajp2m: 560 Y. Miyazaki / J. Differential Equations 188 (2003) 555–568

1 Proof. For simplicity of notation, we set aðxÞ¼aðx0;PxÞ and Pl ¼ðaðx0; DÞlÞ : k;p b A p By the deﬁnition of the H -norm we have, for f ¼ jbjpi D fb; fb L ;

p X X X p a p aþb jjPlf jj ¼ jjD Plf jj ¼ D Plfb : j;p p jajpj jajpj jbjpi p

Hence (3.1) is reduced to (3.2). a a 1 It remains to prove (3.2). Since the symbol of D Pl is x ðaðxÞlÞ ; we have only to estimate xa X jxjjbjDb ¼jxjjbj C DbgðxaÞDgðaðxÞlÞ1 aðxÞl b;g gpb

1 for jbjp½n=2þ1 to apply Mihlin’s multiplier theorem. Noting that DgðaðxÞlÞ is a linear combination of the terms

Dg1 aðxÞ?Dgk aðxÞ ðaðxÞlÞkþ1 with g1 þ ? þ gk ¼ g; jg1jX1; y; jgkjX1; 1pkpjgj; and using the inequality

s s s jlj sup p2 ; 0psp1; s40 s l dðlÞ where dðlÞ is the distance from l to the half line ½0; NÞ (see [19, Proof of Lemma 7.5]), we have a Xjbj jajþ2mk Xjbj kþjaj=2m jbj b x jxj aðxÞ jxj D p C pC aðxÞl kþ1 kþ1 k¼1 jaðxÞlj k¼1 jaðxÞlj Xjbj kþjaj=2m jlj C C l 1þjaj=2m: p kþ1 p j j k¼1 dðlÞ

Then Mihlin’s multiplier theorem gives the lemma. &

Let Z be the set of all integers.

n Lemma 3.2. Let fcsgsAZ be a family of real valued continuous functions on R satisfying

jcsðxÞjpK for some K40 independent of s and x. Assume that there exists an integer T such n a that for any xAR the number of integers s satisfying csðxÞ 0 is less than T. Then Y. Miyazaki / J. Differential Equations 188 (2003) 555–568 561 we have X p p p jjcsf jjppTK jjf jjp: sAZ P p p Proof. The lemma immediately follows from s jcsðxÞj pTK : & P p p Lemma 3.3. Let ffsgsAZ be a family of L -functions. Assume that sAZ jjfsjjpoN and n that there exists an integer T such that forP any xAR the number of integers s satisfying a p fsðxÞ 0 is less than T. Then the series sAZ fs converges strongly in L ; and

p X X p1 p fs pT jjfsjjp: sAZ p s

Proof. Let q satisfy 1=p þ 1=q ¼ 1 and set IðxÞ¼fsAZ: fsðxÞa0g: Then Ho¨ lder’s inequality gives 0 1 0 1 1=q 1=p X X X X @ qA @ pA fsðxÞ p jfsðxÞjp 1 jfsðxÞj sAZ sAIðxÞ sAIðxÞ sAIðxÞ ! X 1=p 1=q p p T jfsðxÞj ; sAZ from which the lemma easily follows. &

4. Proof of the main theorem

n Step 1: For x ¼ðx1; y; xnÞAR we define the norm jxjN ¼ maxfjxij : i ¼ 1; y; ng: We take a family of functions fjsgsAZn as follows. Choose and fix a A N n non-negative valued function j C0 ðR Þ such that jðxÞ¼1 for jxjNp1=2 and jðxÞ¼0 for jxjNX2=3: We fix e with 0oeo1; which will be determined later, and set

jðx=e sÞ j x P sð Þ¼ 2 1=2 f tAZn jðx=e tÞ g for sAZn: Then it follows that X 2 a jaj jsðxÞ ¼ 1; jD jsðxÞjpCn;me for jajp2m: ð4:1Þ sAZn

Step 2: We construct a parametrix for A l by X 1 Pl ¼ jsPsljs; Psl ¼ðaðes; DÞlÞ : sAZn 562 Y. Miyazaki / J. Differential Equations 188 (2003) 555–568

Pl is an inﬁnite sum of the compositions of three operators:

j P j Hm;p !s Hm;p !sl Hm;p !s Hm;p:

Noting that fjsgsAZn is locally ﬁnite, we have X X a b ðA lÞPl ¼ D ðaab aabðesÞÞD jsPsljs s jaj¼jbj¼m X X a b þ D aabD jsPsljs s jajþjbjo2m X þ ðaðes; DÞlÞjsPsljs s

¼ I þ Rl; where I is the identity and

Xm Xm ð1Þ ð2Þ ð1Þ ð1Þ Rl ¼ Rl þ Rl ; Rl ¼ Rkll; k¼0 l¼0 X X X ð1Þ a b Rkll ¼ D asabD jsPsljs; s jaj¼k jbj¼l ( aabðxÞaabðesÞðjaj¼jbj¼mÞ; asabðxÞ¼ aabðxÞðjajþjbjo2mÞ; X Rð2Þ ¼ ðaðes; DÞlÞj P j I: l s s sl s Using the Leibniz formula and (4.1), we have ! X X X pffiffiffiffiffiffiffi ð2Þ a þ b jgj ðgÞ aþbg R ¼ ð 1Þ aabðesÞj D Pslj : l s jaj¼jbj¼m 0ogpaþb g s s

Applying again the Leibniz formula to the terms with ja þ b gj4m; we can write

Xm Xm ð2Þ ð2Þ Rl ¼ Rkll; k¼0 l¼0 X X X X ð2Þ a ðgÞ b Rkll ¼ bsabgD js D Psljs; s jaj¼k jbj¼l 0ojgj¼2mjajjbj Y. Miyazaki / J. Differential Equations 188 (2003) 555–568 563 where bsabg is a constant satisfying

jbsabgjpCm;nM:

ð1Þ ð2Þ Step 3: In order to evaluate the operator norms of Rkll and Rkll; we set ( pffiffiffi oð neÞðk ¼ l ¼ mÞ; Mkl ¼ Mekþl2m ðk þ lo2mÞ:

2m Let 0pipm; 0pkpmP; 0plpm; lALð1; yÞ and jljXe : Applying Lemmas 3.1– m i;p 3.3, we have, for f ¼ D fmAH ; j m jpi p X X X X Rð1Þ f p a Dbj P j Dmf jj kll jjk;pp sab s sl s m a s b m p

p X X X X ðbb0Þ b0þm0 ðmm0Þ ¼ C 0 Cm;m0 asabj D Pslj fm b;b s s a s;b;m b0 b m0pm p p X jbb0j 1þjb0þm0j=2m p ðmm0Þ p p C ðMkle jlj Þ jjjs fmjjp s;b;m;b0;m0 X jbb0jjmm0j 1þjb0þm0j=2m p p p C ðMkle jlj Þ jjfmjjp b;m;b0;m0 X 1þðlþiÞ=2m p p p CðMkljlj Þ jjfmjjp; jmjpi which gives ð1Þ 1þðiþlÞ=2m jjRklljjHi;p-Hk;p pCMkljlj :

In a similar manner we have, for k þ l 2m; o p X X X X X Rð2Þ f p b jðgÞDbP j Dmf jj kll jjk;pp sabg s sl s m a s b g m p

p X X X ðgÞ bþm0 ðmm0Þ ¼ Cm;m0 bsabgjs D Psljs fm 0 a s;b;g;m m pm p X jgj 1þjbþm0j=2m p ðmm0Þ p p C ðMe jlj Þ jjjs fmjjp s;b;g;m;m0 X jgjjmm0j 1þjbþm0j=2m p p p C ðMe jlj Þ jjfmjjp b;g;m;m0 564 Y. Miyazaki / J. Differential Equations 188 (2003) 555–568 X jgj 1þðlþjmjÞ=2m p p p C ðMe jlj Þ jjfmjjp g;m X kþl2m 1þðiþlÞ=2m p p p CðMe jlj Þ jjfmjjp; jmjpi which gives ð2Þ 1þðiþlÞ=2m jjRklljjHi;p-Hk;p pCMkljlj

ð2Þ ð1Þ ð2Þ for k þ lo2m: It is clear that Rmml ¼ 0: We set Rkll ¼ Rkll þ Rkll to get Xm Xm Rl ¼ Rkll; ð4:2Þ k¼0 l¼0

1þðiþlÞ=2m jjRklljjHi;p-Hk;p pC2Mkljlj : ð4:3Þ

1 m;p Step 4: We shall showthat ðI þ RlÞ exists in H and that XN 1 N ðI þ RlÞ ¼ ðRlÞ ð4:4Þ N¼0 holds where the sum is convergent in BðHm;pÞ for appropriate l: Taking into account X X X YN N ? Rl ¼ Rkhlhl ð4:5Þ k1;l1 k2;l2 kN ;lN h¼1 with k1; l1; y; kN ; lN taken over 0; 1; y; n; and

Rk l l Rk l l Hm;p ¼ HkNþ1;p N!N HkN ;p-?-Hk2;p !1 1 Hk1;p+Hm;p with kNþ1 ¼ m; and using (4.3) we have N jjRl jjHm;p-Hm;p

X X YN ? 1þðkhþ1þlhÞ=2m p C2Mkhlh jlj k1;l1 kN ;lN h¼1 X X YN ? ðmk1Þ=2m 1þðkhþlhÞ=2m ¼ jlj C2Mkhlh jlj k ;l k ;l h¼1 1 1 N N ! Xm Xm N 1=2 1þðkþlÞ=2m pjlj C2Mkljlj k¼0 l¼0ffiffiffi 1=2 p 1 1=2m N pjlj fC3ðoð n eÞþMe jlj Þg ; ð4:6Þ Y. Miyazaki / J. Differential Equations 188 (2003) 555–568 565 where we have used jljXe2m: Since the coefficients of top order are uniformly continuous, it follows that oðeÞ-0ase-0: So we take e so that ffiffiffi p 1 C3oð n eÞp4:

2m Then there exists C4X1 such that jljXC4 implies jljXe and ffiffiffi p 1 1=2m 1 C3ðoð n eÞþMe jlj Þp2: ð4:7Þ X PIn the following we assume jlj C4: From (4.6) and (4.7) the Neumann series N N m;p 1 m;p N¼0ðRlÞ converges in BðH Þ: Therefore ðI þ RlÞ exists in H and (4.4) holds for lALðC4; yÞ: 1 Step 5: We shall claim the existence of ðA lÞ and express it by Pl and Rl for appropriate l: From the result of step 4 we observe that, for lALðC4; yÞ;

1 m;p ðA lÞPlðI þ RlÞ ¼ I in H ð4:8Þ

1 m;p m;p and that PlðI þ RlÞ : H -H is a bounded operator. Let 1=p þ 1=q ¼ 1: We deﬁne A0; the dual operator of A by X 0 jajþjbj b a A ¼ ð1Þ D aabD ; jaj;jbjpm which is considered as a bounded operator from Hm;q to Hm;q: Since the principal 0 0 m;q- m;q symbol of A is the same as that of A; there exist C4 and Ql : H H such that A 0 for l LðC4; yÞ;

0 m;q ðA lÞQl ¼ I in H : ð4:9Þ

Let us denote by X 0 the dual space of a Banach space X: Then we have ðHm;qÞ0 ¼ Hm;p by the pairing Z X a /u; f S ¼ D uðxÞfaðxÞ dx n R jajpm P A m;q a A p m;q 0 m;p for u H and f ¼ jajpm D fa; fa L : We also have ðH Þ ¼ H : Taking the dual of (4.9), we obtain

0 m;p QlðA lÞ¼I in H ð4:10Þ

A 0 0 for l LðC4; yÞ; where Ql is the dual operator of Ql: Combining (4.8) and(4.10), we conclude that ðA lÞ1 exists and

XN 1 1 N ðA lÞ ¼ PlðI þ RlÞ ¼ PlðRlÞ : ð4:11Þ N¼0 566 Y. Miyazaki / J. Differential Equations 188 (2003) 555–568

Step 6: We shall evaluate the operator norm of ðA lÞ1: Taking into account

Rk l l Rk l l P Hi;p ¼ HkNþ1;p N!N HkN ;p-?-Hk2;p !1 1 Hk1;p !l Hj;p

with kNþ1 ¼ i and using (4.3) and (4.5), we have

N jjPlRl jjHi;p-Hj;p

X X YN ? 1þðk1þjÞ=2m 1þðkhþ1þlhÞ=2m pC5 jlj C2Mkhlh jlj k1;l1 kN ;lN h¼1 X X YN 1þðiþjÞ=2m ? 1þðkhþlhÞ=2m pC5jlj C2Mkhlh jlj k ;l k ;l h¼1 1 1 N N ! Xm Xm N 1þðiþjÞ=2m 1þðkþlÞ=2m pC5jlj C2Mkljlj k¼0 l¼0ffiffiffi 1þðiþjÞ=2m p 1 1=2m N pC5jlj fC3ðoð neÞþMe jlj Þg :

This combined with (4.7) and (4.11) gives

1 1þðiþjÞ=2m jjðA lÞ jjHi;p-Hj;p p2C5jlj :

Thus we complete the proof of the main theorem. &

Remark. The parametrix Pl we constructed in the proof of the main theorem is a pseudodifferential operator with the symbol

X j ðxÞj ðyÞ P ðx; x; yÞ¼ s s ; l a es; x l sAZn ð Þ

2m which is in the S1;0;0 class, that is, Plðx; x; yÞ satisﬁes

a b g 2mjbj jDxDxDyPlðx; x; yÞjpCa;b;gð1 þjxjÞ for any multi-indexes a; b and g: This can be easily seen from X pffiffiffiffiffiffiffi 1 DaDbDg P ¼ ð 1ÞjajþjgjjðaÞðxÞjðgÞðyÞDb : x x y l s s a es; x l sAZn ð Þ

m;p m;p By the same calculation as in (4.6) we observe that PlABðH ; H Þ: Y. Miyazaki / J. Differential Equations 188 (2003) 555–568 567

Acknowledgments

The author is grateful to the referee for careful reading of the manuscript, valuable suggestions and pointing out recent works especially by Auscher and Qafsaoui [5].

References

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