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pp. 31–38 : Zengjian LouDiv-curl type theorems on Lipschitz domains.

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@article {Lou2005, author="Zengjian Lou", title={Div-curl type theorems on Lipschitz domains}, journal="Bull. Austral. Math. Soc.", fjournal={Bulletin of the Australian Mathematical Society}, volume="72", year="2005", number="1", pages="31--38", issn="0004-9727", coden="ALNBAB", language="English", date="24th January, 2005", classmath=" 42B30, 42B35", publisher={AMPAI, Australian Mathematical Society}, MRnumber="MR2162291", ZBLnumber="02212183", url="http://www.austms.org.au/Publ/Bulletin/V72P1/721-5024-Lou/index.shtml", acknowledgement={ This work is supported by NNSF of China (Grant No.10371069), NSF of Guangdong Province (Grant No.032038) and SRF for ROCS, State Education Ministry. This paper was done when the author visited the Centre for Mathematics and its Applications (CMA) of Mathematical Sciences Institute at the Australian National University. The author would like to thank Professor Alan \Mc {}Intosh for helpful discussions and for supporting his visit to CMA in September of 2003. He also likes to thank CMA for hospitality during the visit.}, abstract={ For Lipschitz domains of $\rn $ we prove div-curl type theorems, which are extensions to domains of the Div-Curl Theorem on $\rn $ by Coifman, Lions, Meyer and Semmes. Applying the div-curl type theorems we give decompositions of Hardy spaces on domains. }}

Bull. Austral. Math. Soc., Vol 72, No 1Metadata in BibTeX format

Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 1 10.wxyz/CV72P1 http://www.austms.org.au/Publ/Bulletin/V72P1/ Div-curl type theorems on Lipschitz domains Zengjian Lou 14 February 2006 2006 2 14 31 38 721-5024-Lou-2005 10.wxyz/C2005V72P1p31 http://www.austms.org.au/Publ/Bulletin/V72P1/721-5024-Lou/ For Lipschitz domains of $\rn $ we prove div-curl type theorems, which are extensions to domains of the Div-Curl Theorem on $\rn $ by Coifman, Lions, Meyer and Semmes. Applying the div-curl type theorems we give decompositions of Hardy spaces on domains. 42B30, 42B35 MR2162291 02212183 This work is supported by NNSF of China (Grant No.10371069), NSF of Guangdong Province (Grant No.032038) and SRF for ROCS, State Education Ministry. This paper was done when the author visited the Centre for Mathematics and its Applications (CMA) of Mathematical Sciences Institute at the Australian National University. The author would like to thank Professor Alan M cIntosh for helpful discussions and for supporting his visit to CMA in September of 2003. He also likes to thank CMA for hospitality during the visit. R.A. Adams Sobolev spaces Academic Press MR450957 R.A. Adams; \textit{Sobolev spaces} (Academic Press, New York, 1975). D.C. Chang The dual of Hardy spaces on a domain in \mathbb R n Forum Math. MR1253178 D.C. Chang; The dual of Hardy spaces on a domain in $\mathbb R^n$, \textit{Forum Math.} \textbf{6} (1994), pp.~65--81. D.C. Chang, G. Dafni and C. Sadosky A div-curl lemma in \BMO on a domain D.C. Chang, G. Dafni and C. Sadosky; A div-curl lemma in $\BMO $ on a domain (to appear). D.C. Chang, S.G. Krantz and E.M. Stein H p theory on a smooth domain in \mathbb R n and elliptic boundary value problems J. Funct. Anal. MR1223705 D.C. Chang, S.G. Krantz and E.M. Stein; $H^p$ theory on a smooth domain in $\mathbb R^n$ and elliptic boundary value problems, \textit{J. Funct. Anal.} \textbf{114} (1993), pp.~286--347. R. Coifman, P.L. Lions, Y. Meyer and S. Semmes Compensated compactness and Hardy spaces J. Math. Pures Appl. MR1225511 R. Coifman, P.L. Lions, Y. Meyer and S. Semmes; Compensated compactness and Hardy spaces, \textit{J. Math. Pures Appl.} \textbf{72} (1993), pp.~247--286. P.W. Jones Extension theorems for \BMO Indiana Univ. Math. J. MR554817 P.W. Jones; Extension theorems for $\BMO $, \textit{Indiana Univ. Math. J.} \textbf{29} (1980), pp.~41--66. Z. Lou Jacobian on Lipschitz domains of \mathbb R 2 Proc. Centre Math. Appl. Austral. Nat. Univ. MR1994518 Z. Lou; Jacobian on Lipschitz domains of $\mathbb R^2$, \textit{Proc. Centre Math. Appl. Austral. Nat. Univ.} \textbf{41} (2003), pp.~96--109. Z. Lou and A. M cIntosh Hardy spaces of exact forms on \mathbb R n Trans. Amer. Math. Soc. MR2115373 Z. Lou and A. M cIntosh; Hardy spaces of exact forms on $\mathbb R^n$, \textit{Trans. Amer. Math. Soc.} \textbf{357} (2005), pp.~1469--1496. Z. Lou and A. M cIntosh Hardy spaces of exact forms on Lipschitz domains in \mathbb R n Indiana Univ. Math. J. MR2060046 Z. Lou and A. M cIntosh; Hardy spaces of exact forms on Lipschitz domains in $\mathbb R^n$, \textit{Indiana Univ. Math. J.} \textbf{54} (2004), pp.~581--609. J. Necas Les méthodes directes en théorie des équations elliptiques Masson et Cie Paris Academia, Editeurs MR227584 J. Ne\ucas; \textit{Les m\'ethodes directes en th\'eorie des \'equations elliptiques}, (Masson et Cie Paris, Editors) (Academia, Editeurs, Prague, 1967).

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Bull. Austral. Math. Soc. 42b30, 42b35

Vol. 72 (2005) [31–38]

DIV-CURL TYPE THEOREMS ON LIPSCHITZ DOMAINS

Zengjian Lou

For Lipschitz domains of Rn we prove div-curl type theorems, which are extensionsto domains of the Div-Curl Theorem on Rn by Coifman, Lions, Meyer and Semmes.Applying the div-curl type theorems we give decompositions of Hardy spaces ondomains.

1. Introduction

In [4] two Hardy spaces are defined on domains Ω of Rn, one which is reasonablyspeaking the largest, and the other which in a sense is the smallest. The largest,

\mathcal{H} 1r(Ω), arises by restricting to Ω arbitrary elements of \mathcal{H} 1(Rn). The other, \mathcal{H} 1z(Ω),arises by restricting to Ω elements of \mathcal{H} 1(Rn) which are zero outside Ω. Norms onthese spaces are defined as following

\| f\| \mathcal{H} 1r(\Omega ) = inf \| F\| \mathcal{H} 1(Rn),

the infimum being taken over all functions F \in \mathcal{H} 1(Rn) such that F | \Omega = f ,

\| f\| \mathcal{H} 1z(\Omega ) = \| F\| \mathcal{H} 1(Rn),

where F is the zero extension of f to Rn.From [2], the dual of \mathcal{H} 1z(Ω) is BMOr(Ω), a space of locally integrable functions

with

\| f\| BMOr(\Omega ) = supQ\subset \Omega

\biggl( 1

| Q|

\int Q

\bigm| \bigm| f(x) - fQ\bigm| \bigm| 2 dx\biggr) 1/2

32 Z. Lou [2]

Let Ω denote a Lipschitz domain — an assumption which is enough to ensure

the existence of a bounded extension map from BMOr(Ω) to BMO(Rn) ([6]). We useH(Ω)n := H(Ω,Rn) to denote a space of functions f : Ω \rightarrow Rn (when n = 1, writeH(Ω)1 as H(Ω)). For simplicity we introduce the following spaces

L2div(Ω)n =

\bigl\{ f \in L2(Ω)n : div f = 0, \nu \cdot f | \partial \Omega = 0, \| f\| L2(\Omega )n 6 1

\bigr\} ;

L2curl(Ω)n =

\bigl\{ f \in L2(Ω)n : curl f = 0, \nu \times f | \partial \Omega = 0, \| f\| L2(\Omega )n 6 1

\bigr\} ,

where \nu denotes the outward unit normal vector. When Ω = Rn

L2div(Rn)n =\bigl\{ f \in L2(Rn)n : div f = 0, \| f\| L2(Rn)n 6 1

\bigr\} ;

L2curl(Rn)n =\bigl\{ f \in L2(Rn)n : curl f = 0, \| f\| L2(Rn)n 6 1

\bigr\} .

In [5, Theorems II.1 and III.2], among other results, Coifman, Lions, Meyer and

Semmes established the following theorems.

Theorem CLMS1. Let 1 < p, q

[3] Div-curl type theorems 33

(2) If b \in BMOz(Ω), then

(1.4) \| b\| BMOz(\Omega ) \approx supe,f

\int \Omega

b e \cdot f dx,

the supremum being taken over all e = E| \Omega , f = F | \Omega , E \in L2div(Rn)n, F \in L2curl(Rn)n.The implicit constants in (1.3) and (1.4) depend only on the domain Ω and on the

dimension n.

Remark. Results for other BMO-type spaces, such as dual of divergence-free Hardy

spaces, can be found in [8] and [9].

Corollary 1.2.

(1) A function b \in BMOr(Ω) if and only if there exists a constant C such that\int \Omega

b e \cdot f dx 6 C for all e \in L2div(Ω)n and f \in L2curl(Ω)n.

(2) A function b \in BMOz(Ω) if and only if there exists a constant C such

that

\int \Omega

b e \cdot f dx 6 C for all e = E| \Omega and f = F | \Omega with E \in L2div(Rn)n,

F \in L2curl(Rn)n.

Here and afterwards, unless otherwise specified, C denotes a constant depending

only on the domain Ω and the dimension n. Such C may differ at different occurrences.

Applying Theorem 1 we have the following theorem which gives decompositions

of \mathcal{H} 1z(Ω) and \mathcal{H} 1r(Ω) into quantities of forms “e \cdot f”.

Theorem 1.3.

(1) Any function u \in \mathcal{H} 1z(Ω) can be written as

u =\infty \sum

k=1

\lambda k ek \cdot fk,

where ek \in L2div(Ω)n, fk \in L2curl(Ω)n and\infty \sum

k=1

| \lambda k|

34 Z. Lou [4]

2. Proof of Theorem 1.1

To prove Theorem 1.1, we need the following lemmas.

Lemma 2.1. ([6, Theorem 1]) Let b \in BMOr(Ω). Then there exists B \in BMO(Rn) such that

b = B| \Omega

and

\| B\| BMO(Rn) 6 C\| b\| BMOr(\Omega ).(2.1)

Lemma 2.2. ([7, Theorem 3.1]) Let b be a locally integrable function on Ω.Then

\| b\| BMOr(\Omega ) \approx \| b\| BMOH(\Omega ),(2.2)

where

\| b\| BMOH(\Omega ) = supQ

\Bigl( 1| Q|

\int Q

| b - bQ| 2 dx\Bigr) 1/2

,

the supremum being taken over all cubes Q with 2Q \subset Ω, the implicit constants in(2.2) depend only on Ω and n.

Lemma 2.3. For b \in L2loc(Ω)

(2.3) \| b\| BMOH(\Omega ) 6 C supe,f

\int \Omega

b e \cdot f dx,

the supremum being taken over all e \in L2div(Ω)n and f \in L2curl(Ω)n.The proof of Lemma 2.3 is given in the last section.

Proof of Theorem 1.1: (1) Let B \in BMO(Rn) be an extension ofb \in BMOr(Ω) such that b = B| \Omega and (2.1) holds. For e \in L2div(Ω)n, f \in L2curl(Ω)n,define

E =

\left\{ e in Ω;0 in Rn \setminus Ω,F =

\left\{ f in Ω;0 in Rn \setminus Ω.Since div e = 0 on Ω and e \cdot \nu | \partial \Omega = 0, it is easy to show that divE = 0 on Rn. SoE \in L2div(Rn)n. Similarly, curl f = 0 on Ω and f \times \nu | \partial \Omega = 0 imply that curlF = 0

[5] Div-curl type theorems 35

on Rn. Therefore F \in L2curl(Rn)n. By duality \mathcal{H} 1(Rn)\ast = BMO(Rn), Lemma 2.1 and(1.1), we have\int

\Omega

b e \cdot f dx =\int

RnB E \cdot F dx 6 \| B\| BMO(Rn)\| E \cdot F\| \mathcal{H} 1(Rn)

6 C\| b\| BMOr(\Omega )\| E\| L2(Rn)n\| F\| L2(Rn)n

= C\| b\| BMOr(\Omega )\| e\| L2(\Omega )n\| f\| L2(\Omega )n 6 C\| b\| BMOr(\Omega ).

The proof of the reversed inequality in (1.3) follows from (2.2) and (2.3).

(2) Let b \in BMOz(Ω) and B be its zero extension to Rn. Then B \in BMO(Rn)and \| B\| BMO(Rn) = \| b\| BMOz(\Omega ). Using (1.1) again,\int

\Omega

b e \cdot f dx =\int

RnB E \cdot F dx 6 \| B\| BMO(Rn)\| E \cdot F\| \mathcal{H} 1(Rn)

6 C\| b\| BMOz(\Omega )\| E\| L2(Rn)n\| F\| L2(Rn)n

6 C\| b\| BMOz(\Omega )

for all e = E| \Omega , f = F | \Omega , E \in L2div(Rn)n, F \in L2curl(Rn)n.

For the converse, let b \in BMOz(Ω) and define B as above. Applying (1.2) yields

\| b\| BMOz(\Omega ) = \| B\| BMO(Rn) 6 C supE\in L2div,F\in L

2curl

\int RnB E \cdot F dx

= C supe=E| \Omega ,f=F | \Omega ,E\in L2div,F\in L

2curl

\int \Omega

b e \cdot f dx.

Theorem 1.1 is proved.

3. Proof of Theorem 1.3

The proof of Theorem 1.3 relies on Theorem 1.1 and the following facts from

functional analysis which can be found in [5, Lemmas III.1, III.2].

Lemma 3.1. Let V be a bounded subset of a normed vector space X. Weassume that V (closure of V for the norm of X) contains the unit ball (centred at 0)

of X. Then, any x in that ball can be written as

x =\infty \sum

j=0

1

2jyj,

where yj \in V for all j > 0.

36 Z. Lou [6]

Lemma 3.2. Let V be a bounded symmetric (x \in V \Rightarrow - x \in V ) subset of anormed vector space X. Then, the closed convex hull \widetilde V of V (in X) contains a ballcentred at 0 if and only if, for any l \in X\ast ,

\| l\| X\ast \approx supx\in V

\langle l, x\rangle .

Proof of Theorem 1.3: (1) Let X = \mathcal{H} 1z(Ω) and

V =\bigl\{ e \cdot f : e \in L2div(Ω)n, f \in L2curl(Ω)n

\bigr\} .

It is easy to check that V is a bounded subset of X. In fact, for e \in L2div(Ω)n,f \in L2curl(Ω)n, let E and F be their zero extensions to Rn respectively. ThenE \in L2div(Rn)n, F \in L2curl(Rn)n. From Theorem CLMS1, E \cdot F \in \mathcal{H} 1(Rn) and

\| E \cdot F\| \mathcal{H} 1(Rn) 6 C\| E\| L2(Rn)n\| F\| L2(Rn)n 6 C.

Therefore e \cdot f \in \mathcal{H} 1z(Ω) with \| e \cdot f\| \mathcal{H} 1z(\Omega ) 6 C. Applying Theorem 1.1 (1) and Lemmas3.1 and 3.2, we have the decomposition of Theorem 1.3 (1).

(2) Let X = \mathcal{H} 1r(Ω) and

V =\bigl\{ e \cdot f : e = E| \Omega , f = F | \Omega , E \in L2div(Rn)n, F \in L2curl(Rn)n

\bigr\} .

Similar to the case (1), we have e \cdot f \in \mathcal{H} 1r(Ω) with

\| e \cdot f\| \mathcal{H} 1r(\Omega ) = infe\cdot f=G| \Omega ,G\in \mathcal{H} 1(Rn)

\| G\| \mathcal{H} 1(Rn) 6 \| E \cdot F\| \mathcal{H} 1(Rn) 6 C

for e \cdot f \in V . Using Theorem 1.1 (2) and those two lemmas again we finish the proofof Theorem 1.3.

4. Proof of Lemma 2.3

To prove Lemma 2.3 we need the following result due to Nec̆as (see [10, Lemma 7.1,

Chapter 3]). In Lemma 4.1, W 1,20 (Ω)n denotes the closure of C\infty 0 (Ω)

n in the Sobolev

spaceW 1,2(Ω)n and\nabla \varphi =\bigl( (\partial \varphi i)/(\partial xj)

\bigr) n\times n a n\times nmatrix (see [1] for Sobolev spaces).

Lemma 4.1. Let Ω be a Lipschitz domain in Rn. If f \in L2(Ω) has zero integral,then there exists \varphi \in W 1,20 (Ω)n such that

f = div \varphi

and

\| \nabla \varphi \| L2(\Omega )n\times n 6 C\| f\| L2(\Omega ).

Corollary 4.2. Let Q be a cube in Rn. If f \in L2(Q) has zero integral, thenthere exists \varphi \in W 1,20 (Q)n such that f = div \varphi and

\| \nabla \varphi \| L2(Q)n\times n 6 C0\| f\| L2(Q)

[7] Div-curl type theorems 37

for a constant C0 independent of Q.

Proof of Lemma 2.3: Suppose b \in L2loc(Ω). We shall show that for all cubesQ with 2Q \subset Ω there exists e \in L2div(Ω)n and f \in L2curl(Ω)n such that

(4.1)

\biggl( 1

| Q|

\int Q

| b - bQ| 2 dx\biggr) 1/2

6 C

\bigm| \bigm| \bigm| \bigm| \int \Omega

b e \cdot f dx\bigm| \bigm| \bigm| \bigm| .

Let h = b - bQ, then h \in L2(Q) with\int

Q

h dx = 0. From Corollary 4, there exists

\varphi := (\varphi 1, . . . , \varphi n) \in W 1,20 (Q)n such that h = div\varphi and

(4.2) \| \nabla \varphi \| L2(Q)n\times n 6 C0\| h\| L2(Q),

where C0 is independent of Q. So

\| h\| 2L2(Q) =\int

Q

hn\sum

i=1

\partial \varphi i\partial xi

dx 6 n max16i6n

\bigm| \bigm| \bigm| \bigm| \int Q

h\partial \varphi i\partial xi

dx

\bigm| \bigm| \bigm| \bigm| = n

\bigm| \bigm| \bigm| \bigm| \int Q

h\partial \varphi i0\partial xi0

dx

\bigm| \bigm| \bigm| \bigm| (4.3)for some choice of i0 (i0 = 1, . . . , n). Assuming without loss of generality that i0 = 1

in (4.3). To prove (4.1), it is sufficient to show that

(4.4)

\bigm| \bigm| \bigm| \bigm| \int Q

h\| h\| - 1L2(Q)\partial \varphi 1\partial x1

dx

\bigm| \bigm| \bigm| \bigm| 6 C| Q| 1/2\bigm| \bigm| \bigm| \bigm| \int Q

h e \cdot f dx\bigm| \bigm| \bigm| \bigm| .

We next construct e and f . Define

f =\Bigl( - \partial \varphi 1\partial xi

, 0, . . . , 0,\partial \varphi 1\partial x1

, 0, . . . , 0\Bigr) C - 10 \| h\| - 1L2(Q),

where (\partial \varphi 1)/(\partial x1) is the i-th component of f . Then f \in L2(Q)n with div f = 0 and\| f\| L2(Q)n 6 1 by (4.2).

Let \psi 0 \in C\infty 0 (Rn) such that

\psi 0 =

\left\{ 1 on [ - 1, 1]n;0 outside [ - 2, 2]n.Define

e = \gamma C0| Q| - 1/2\nabla \bigl( (xi - x0i )\psi Q(x)

\bigr) , 1 6 i 6 n,

38 Z. Lou [8]

where \psi Q(x) = \psi 0

\Bigl( (x - x0)

\big/ \bigl( l(Q)/2

\bigr) \Bigr) , x0 = (x01, . . . , x

0n) and l(Q) denote the centre

and the side-length of the cube Q, \gamma > 0 is a normalisation constant (independent of x0

and l(Q)) so that \| e\| L2(\Omega )n 6 1. It is obvious that e \in C\infty 0 (2Q) and e = \gamma C0| Q| - 1/2\varepsilon ion Q, where \varepsilon i = (0, . . . , 0, 1, 0, . . . , 0), 1 is the i-th component of \varepsilon i. From the

construction of e and f , we get

e \cdot f = \gamma | Q| - 1/2\| h\| - 1L2(Q)\partial \varphi 1\partial x1

on Q

and (4.4) is proved.

Note. It should be added that at the time the paper was finished, the author was

unfortunately unaware of a similar but unpublished work [3] (with different proof).

Thanks go to Galia Dafni (Department of Mathematics & Statistics, Concordia Uni-

versity, Canada) for informing us her paper with Chang and Sadosky.

References

ˆM [1] R.A. Adams, Sobolev spaces (Academic Press, New York, 1975).ˆM [2] D.C. Chang, ‘The dual of Hardy spaces on a domain in Rn', Forum Math. 6 (1994),

65–81.ˆ [3] D.C. Chang, G. Dafni and C. Sadosky, ‘A div-curl lemma in BMO on a domain' (to

appear).ˆM [4] D.C. Chang, S.G. Krantz and E.M. Stein, ‘\mathcal{H} p theory on a smooth domain in Rn and

elliptic boundary value problems', J. Funct. Anal. 114 (1993), 286–347.ˆˆM [5] R. Coifman, P.L. Lions, Y. Meyer and S. Semmes, ‘Compensated compactness and Hardy

spaces', J. Math. Pures Appl. 72 (1993), 247–286.ˆˆM [6] P.W. Jones, ‘Extension theorems for BMO', Indiana Univ. Math. J. 29 (1980), 41–66.ˆM [7] Z. Lou, ‘Jacobian on Lipschitz domains of R2', Proc. Centre Math. Appl. Austral. Nat.

Univ. 41 (2003), 96–109.ˆM [8] Z. Lou and A. McIntosh, ‘Hardy spaces of exact forms on Rn', Trans. Amer. Math. Soc.

357 (2005), 1469–1496.ˆM [9] Z. Lou and A. McIntosh, ‘Hardy spaces of exact forms on Lipschitz domains in Rn',

Indiana Univ. Math. J. 54 (2004), 581–609.ˆM [10] J. Nec̆as, Les méthodes directes en théorie des équations elliptiques, (Masson et Cie Paris,

Editors) (Academia, Editeurs, Prague, 1967).

Department of MathematicsShantou UniversityShantouGuangdong 515063Peoples Repbulic of Chinae-mail: zjlou@stu.edu.cn

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Index of Volume 72 No. 1Div-curl type theorems on Lipschitz domainsIntroduction Proof of Theorem 1.1 Proof of Theorem 1.3 Proof of Lemma 2.3 References