HARDY SPACE OF EXACT FORMS ON R N

Transcription

1 HARDY SPACE OF EXACT FORMS ON R N Zengjian Lou Alan M c Intosh Abstract Applying the atomic decomposition of the divergence-free Hardy space we characterize its dual as a variant of BMO. R Using the duality result we prove a div-curl type theorem: For b in L 2 loc (R3, ), sup b du dv is equivalent to the BMO-type norm of b, where the supremum is taken over all u, v H (R 3 ) with du L 2, dv L 2. This theorem can be used to get some coercivity results for polyconvex quadratic forms which come from the linearization of polyconvex variational integrals studied in nonlinear elasticity in R 3. In addition, we introduce Hardy spaces of exact forms onr N, study their atomic decompositions and dual spaces, and establish a div-curl type theorem on R N Mathematics Subject Classification(Amer. Math. Soc.): Primary 42B30; Secondary 58A0. Keywords and phrases: divergence-free Hardy space, Hardy space of exact forms, atomic decomposition, BM O, div-curl, coercivity CONTENTS. Statement of the main theorem 2. Divergence-free atomic decomposition 3. The dual space 4. Proof of the main theorem 5. Applications 5. Coercivity 5.2 Gårding s inequality 6. Hardy spaces of exact forms on R N 6. Definitions 6.2 Atomic decompositions and dual spaces 6.3 The div-curl type theorem on R N 6.4 A decomposition theorem Appendix A. Surjectivity of the curl operator Appendix B. Review of differential forms The authors are supported by Australian Government through the Australian Research Council. This paper was finished when the first author pursued his PhD in the Centre for Mathematics and its Applications of Mathematical Sciences Institute at the Australian National University. Typeset by AMS-TEX

2 2 ZENGJIAN LOU ALAN M c INTOSH. STATEMENT OF THE MAIN THEOREM In this paper, we consider the divergence-free Hardy space on R 3, give its divergencefree atomic decomposition and use this to characterize its dual space. Applying the duality relationship between the Hardy space and the BM O-type space we prove our main result of the paper, the following div-curl type theorem concerning an estimate of quadratic forms on R 3. Theorem 4.. Let b L 2 loc (R3, ). Then sup u,v W b du dv b BMOd (R 3, ), R 3 (.) where W = {w H (R 3 ) : dw L 2 (R 3, ) } and ( /2 b BMOd (R 3, ) := sup inf b g dx) 2, (.2) B g B B the supremum in (.2) being taken over all balls B in R 3, the infimum being taken over all g BMO(B, ) with dg = 0 in B. The implicit constants in (.) are absolute constants. Theorem 4. holds for N-dimensions and any form b. We are especially interested in the three-dimensional case, because, as shown in Section 5, Theorem 4. can be used to give some coercivity results of polyconvex quadratic forms which come from the linearization of polyconvex variational integrals studied in nonlinear elasticity by Ball [B]. These results answer questions of Kewei Zhang, who previously obtained analogous 2-dimensional results in [Z]. Extensions of these results to the case of Lipschitz domains in R N is contained in the sequels [LM] and [LM2] to this paper. The paper is organized as following. Section 2 provides the definition of the divergencefree Hardy space on R 3 and the proof of its divergence-free atomic decomposition. Using the atomic decomposition we characterize its dual in Section 3. The proof of the main theorem is in Section 4. Section 5 is devoted to applications of the main theorem to coercivity properties and Gårding s inequality of certain polyconvex quadratic forms. In Section 6, we introduce Hardy spaces of exact forms on R N, give their atomic decompositions, characterize their dual spaces and establish a div-curl theorem on R N. In addition, we give a decomposition theorem of these Hardy spaces into du dv quantities which is a generalization of a similar decomposition theorem by Coifman, Lions, Meyer and Semmes [CLMS]. In this paper, unless otherwise specified, C denotes a constant independent of functions and domains related to the inequalities. Such C may differ at different occurrences. 2. DIVERGENCE-FREE ATOMIC DECOMPOSITION In this section we introduce the divergence-free Hardy space and prove its divergencefree atomic decomposition. A similar decomposition was obtained by Gilbert, Hogan and Lakey in [GHL] by using a result of divergence-free wavelet decomposition of L 2 (R 3, R 3 )

3 HARDY SPACE OF EXACT FORMS ON R N 3 due to Lemarié-Rieusset [Le]. Our proof is different from that in [GHL] and is valid for forms of all degrees as is shown in Section 6. We first recall briefly some definitions and results of Hardy spaces and tent spaces which are used in this paper. The Hardy space H (R 3 ) is the space of locally integrable functions f for which M(f)(x) = sup ϕ t f(x) t>0 belongs to L (R 3 ), where ϕ C 0 (R 3 ), ϕ t (x) = t 3 ϕ( x t ), t > 0, R 3 ϕ(x) dx =, supp ϕ B(0, ), a ball centered at the origin with radius. The norm of H (R 3 ) is defined by f H (R 3 ) = M(f) L (R 3 ), where M is the maximal function. Among many characterizations of Hardy spaces, the atomic decomposition is an important one. An L 2 (R 3 ) function a is an H (R 3 )-atom if there is a cube or a ball B = B a in R N satisfying: ) supp a B; 2) a L 2 (R 3,R 3 ) B /2 ; 3) a(x) dx = 0. B It is obvious that any H (R 3 )-atom a is in H (R 3 ). The basic result about atoms is the following atomic decomposition theorem ([CW], [La]): A function f on R 3 belongs to H (R 3 ) if and only if f has a decomposition f = λ k a k, where the a k s are H (R 3 )-atoms and λ k <. Furthermore, ( ) f H (R 3 ) inf λ k, where the infimum is taken over all such decompositions, the constants of the proportionality are absolute constants. Here A B means that there are constants C and C 2 such that C A B C 2 A. Define the tent space N p (R 4 +) ( p < ) to consist of all measurable functions F on R 4 + for which S(F ) L p (R 3 ), where S(F ) is the square function defined by S(F )(x) = ( Γ(x) ) /2 F (y, t) 2 dydt t 4, Γ(x) = {(y, t) R 4 + : y x < t}, F N p (R 4 + ) = S(F ) L p (R 3 ). An N p (R 4 +)-atom is a function α supported in a tent T (B) = {(x, t) : x x 0 r t} of a ball B = B(x 0, r) in R 3, for which T (B) α(x, t) 2 dxdt t B 2/p.

4 4 ZENGJIAN LOU ALAN M c INTOSH When p <, Coifman, Meyer and Stein proved the following atomic decomposition theorem [CMS]: any F N p (R 4 +) can be written as F = λ k α k, where the α k are N p (R 4 +)-atoms and λ k C F N p (R 4 + ). In the proof of Theorem 2.3, we use the following two facts: one is that the operator defined by ( ) /2 g S ψ (g) := g ψ t (y) 2 dydt t 4 Γ(x) is bounded from H (R 3 ) to L (R 3 ) for ψ S(R 3 ) (the space of test functions) with ψ dx = 0, and S ψ (g) L (R 3 ) C g H (R 3 ) (2.) (see Theorems 3 and 4 in Chapter III of [St2]). The other is that the operator π ψ (a) = is bounded from N 2 (R 4 +) to L 2 (R 3 ) and 0 a(, t) ψ t dt t π ψ (a) L 2 (R 3 ) C a N 2 (R 4 + ) (2.2) ([CMS, Theorem 6]). Now we define the divergence-free Hardy space. Let H (R 3, R 3 ) denote the space of vector-valued functions with each component in H (R 3 ). Definition 2.. The divergence-free Hardy space on R 3 is defined as with norm H div(r 3, R 3 ) = {f H (R 3, R 3 ) : div f = 0 in R 3 } f H div (R 3,R 3 ) = f H (R 3,R 3 ), where div f is defined in the sense of distributions. Definition 2.2. A function a L 2 (R 3, R 3 ) is said to be an Hdiv (R3, R 3 )-atom if there is a cube or ball B = B a in R 3 satisfying (i) supp a B; (ii) a L 2 (R 3,R 3 ) B /2 ; (iii) a(x) dx = 0; B (iv) div a = 0 in R 3. The properties of tent spaces can be used to clarify various points in the theory of Hardy spaces, for example, atomic decompositions. Combining this with an idea of Auscher we prove a divergence-free atomic decomposition of the divergence-free Hardy space. For simplicity, we prove the result by using the language of forms. Let us interpret vector fields as two-forms. Then H (R 3, R 3 ) and H div (R3, R 3 ) become H (R 3, 2 ) and H d (R3, 2 ) respectively. Similarly we can define tent spaces and their atoms for forms. See Appendix B for definitions of exterior operator d and its formal adjoint δ and information on forms.

5 HARDY SPACE OF EXACT FORMS ON R N 5 Theorem 2.3. A function f on R 3 is in H d (R3, 2 ) if and only if it has a decomposition f = λ k a k, where the a k s are H d (R3, 2 )-atoms and λ k <. Furthermore, ( ) f H d (R 3, 2 ) inf λ k, where the infimum is taken over all such decompositions. The constants of the proportionality are absolute constants. Proof. The easy part is the if part, that is, assuming f has such a decomposition in D (R 3, 2 ) (the space of distributions). For then, if the sum is finite, f H d (R 3, 2 ) k λ k a k H (R 3, 2 ) C k λ k, (2.3) where we used the fact that a H (R 3, 2 ) C when a is an H (R 3, 2 )-atom. We now prove the only if part. Suppose f = i<j 3 f ij e i e j H d (R3, 2 ). Choose a function ϕ C 0 (R 3 ) with support in the unit ball, which satisfies 0 t ξ 2 ˆϕ(tξ) 2 dt =. (2.4) Define ( ) F (x, t) = tδ f ϕ t (x), x R 3, t > 0 (see Appendix B for the definition of δ). Then F (x, t) can be written as F (x, t) = 3 i<j 3 l= = i<j 3 l= t x l ( f ij ϕ t )(x) µ l (e i e j ) 3 f ij ( l ϕ) t (x)(δ li e j δ lj e i ). (2.5) Since f ij ( l ϕ) t N (R 4 +) by (2.), then F N (R 4 +, ) with F N (R 4 +, ) C f H (R 3, 2 ). By the atomic decomposition theorem for tent spaces, F has a decomposition F = λ k α k (2.6)

6 6 ZENGJIAN LOU ALAN M c INTOSH with λ k C F N (R 4 + ),, where the α k s are N (R 4 +, )-atoms, i.e. there exist balls B k such that supp α k T (B k ) and Define T (B k ) α k (y, t) 2 dydt t ( ) dt πf = td F (, t) ϕ t 0 t. B k. (2.7) Then (2.6) gives πf = λ ka k, where a k = πα k. Let α k = 3 i= αi k ei, a i,l k = π l ϕ(α i k ), where the αi k s are N (R 4 +)-atoms, then a k = 3 i,l= a i,l k µ l(e i ). It is easy to check that the function a k satisfies the following conditions: ) supp a k 2B k, since l ϕ is supported in the unit ball and supp α k T (B k ); 2) a k dx = 0, since l ϕ S(R 3 ) and l ϕ dx = 0; 3) da k = 0 in R 3. We now prove that a k also satisfies the size condition: 4) 2B k a k 2 dx C 2B k, where C is independent of a k and B k. Since αk i are N (R 4 +)-atoms, then αk i N 2 (R 4 +). The boundedness of π ψ in (2.2) and (2.7) imply that a i,l k L2 (R 3 ) and a i,l k 2 L 2 (2B k ) C αi k 2 N 2 (R 4 + ) = C αk(y, i t) 2 χ(x y/t) dydt C R 3 R 4 + T (B k ) C 2B k, αk(y, i t) 2 dydt t t 4 dx where χ denotes the characteristic function in the unit ball. We proved that a k are Hd (R3, 2 )-atoms. Moreover we next show that f = λ k a k is an atomic decomposition of f, where the λ k s are the same as those in (2.6). To see this we only need to prove that f = πf. Applying the Fourier transform to f = (dδ + δd)f and the following two facts df(ξ) = iξ ˆf(ξ), δf(ξ) = iξ ˆf(ξ),

7 HARDY SPACE OF EXACT FORMS ON R N 7 we obtain f(ξ) = iξ δf(ξ) = iξ ( iξ ˆf(ξ) ) = ξ ( ξ ˆf(ξ) ), where we used the fact that df = 0. In addition f(ξ) = ξ 2 ˆf(ξ), so The assumption (2.4) on ϕ and (2.8) give πf (ξ) = = ξ ( ξ ˆf(ξ) ) = ξ 2 ˆf(ξ). (2.8) 0 = = ( ) itξ tδ(f ϕt )(ξ) ˆϕ(tξ) dt itξ t 2 ξ ( itξ ˆf(ξ) ˆϕ(tξ) 2) dt t ( ξ ˆf(ξ) ) ˆϕ(tξ) 2 dt t t ξ 2 ˆϕ(tξ) 2 ˆf(ξ) dt = ˆf(ξ). The desired result follows. The proof of Theorem 2.3 is completed 3. THE DUAL SPACE We now characterize the dual space of H d (R3, 2 ). Recall that under the duality (f, g) = f(x)g(x) dx, R 3 when suitably defined, the dual of H (R 3 ) is the real-valued BMO(R 3 ) space of functions f of bounded mean oscillation, i.e. f BMO(R 3 ) = sup B inf c R ( /2 f c dx) 2 <, B B the supremum being taken over all balls B in R 3. So we have H (R 3, 2 ) = BMO(R 3, ) under the pairing (f, g) = f g, R 3 when suitably defined [St2].

8 8 ZENGJIAN LOU ALAN M c INTOSH Definition 3.. Let BMO d (R 3, ) be the space of measurable functions G for which ( /2 G BMOd (R 3, ) = sup inf G g B dx) 2 <, B g B B B where the supremum is taken over all balls B in R 3, the infimum is taken over all functions g B L 2 (B, ) with dg B = 0 in B. Consider the Banach space BMO d (R 3, )/X 0 with the norm G + X 0 BMOd (R 3, )/X 0 = G BMOd (R 3, ), where X 0 = {G BMO d (R 3, ) : G BMOd (R 3, ) = 0}. We show that it is the dual of H d (R3, 2 ). To prove this we first prove a lemma. Let D(R 3, 2 ) denote the vector space finitely generated by H d (R3, 2 )-atoms. By Theorem 2.3, one has that D(R 3, 2 ) is dense in H d (R3, 2 ). Lemma 3.2. For g BMO(R 3, ), R 3 g h = 0 for all h D(R 3, 2 ) if and only if dg = 0 in R 3. Proof. Note that dϕ is an H d (R3, 2 )-atom if ϕ C 0 (R 3, ). Then R 3 g dϕ = 0 if R 3 g h = 0 for all h D(R 3, 2 ). Hence dg = 0 in R 3. Suppose h D(R 3, 2 ) and h = k λ k a k, where the a k s are Hd (R3, 2 )-atoms, i.e. a k L 2 (R 3, 2 ) supported in balls B k and da k = 0 in B k. By Proposition A. in Appendix A, there exist functions ϕ k H0 (B k, ) such that a k = dϕ k, where H0 (B k, ) denotes the Sobolev space H (B k, ) with zero boundary values. From Green s formula, for g BMO(R 3, ) with dg = 0 in R 3, we have g h = λ k g a k R 3 k B k = k λ k B k g dϕ k = k λ k B k dg ϕ k = 0 for all h D(R 3, 2 ). Now we are ready to prove our main result of this section.

9 HARDY SPACE OF EXACT FORMS ON R N 9 Theorem 3.3. If G + X 0 BMO d (R 3, )/X 0, then the linear functional L defined by L(h) = G h, (3.) R 3 initially defined on D(R 3, 2 ), has a unique bounded extension to H d (R3, 2 ). Conversely, if L is in H d (R3, 2 ), then there exists a unique G + X 0 BMO d (R 3, )/X 0 such that (3.) holds. The map G + X 0 L given by (3.) is a Banach isomorphism between BMO d (R 3, )/X 0 and H d (R3, 2 ). Proof. Let G BMO d (R 3, ). Define L(h) = G h, h D(R 3, 2 ). R 3 If G BMOd (R 3, ) = 0 then for any ball B, inf G g 2 dx = 0. (3.2) g L 2 (B, ),dg=0 B Since {g L 2 (B, ) : dg = 0 in B} is a closed subspace of L 2 (B, ) (see, for example, [ISS, Corollary 5.2]), (3.2) implies that G L 2 (B, ) with dg = 0 in B for all B R 3. Hence dg = 0 in R 3. By Lemma 3.2 we have L(h) = G h = 0 for all h D(R 3, 2 ). R 3 Therefore we can define ρ : BMO d (R 3, )/X 0 D(R 3, 2 ) by ρ (G + X 0 )(h) = G h, h D(R 3, 2 ). R 3 The proof that, for every G BMO d (R 3, ), the linear functional (3.) is defined and bounded on Hd (R3, 2 ) depends on the inequality G h C G BMOd (R 3, ) h H R 3 d (R 3, 2 ) (3.3) for G BMO d (R 3, ) and h in the dense subspace D(R 3, 2 ) H d (R3, 2 ). Similar to the proof of Lemma 3.2, write h D(R 3, 2 ) as a finite sum of atoms a k supported in balls B k. For all g k L 2 (B k, ) with dg k = 0 in B k, we have G h R 3 k k k λ k (G g k ) a k B k ( ) /2 λ k G g k 2 dx a k L B k ( ) /2 λ k G g k 2 dx, B k B k 2(Bk, 2)

10 0 ZENGJIAN LOU ALAN M c INTOSH where we used the size condition of a k. This gives G h ( ) /2 λ k inf G g k 2 dx R 3 g k B k k B k C h H d (R 3, ) G 2 BMOd (R 3, ). Then (3.3) is proved. Therefore each G BMO d (R 3, ) gives a bounded linear functional on the dense subspace D(R 3, 2 ), and thus on H d (R3, 2 ). Let Y 0 := {g BMO(R 3, ) : dg = 0 in R 3 }. Note that Hd (R3, 2 ) is a closed subspace of H (R 3, 2 ). Applying the Hahn-Banach Theorem and Lemma 3.2, one finds that Y 0 is a closed subspace of BMO(R 3, ) and the map ρ 2 : L G + Y 0 is a Banach isomorphism between H d (R3, 2 ) and Y := BMO(R 3, )/Y 0, and L op G + Y 0 Y, where L H d (R3, 2 ) is defined as in (3.), L op is the operator norm of L. Define ρ 3 : G + Y 0 G + X 0 from Y to BMO d (R 3, )/X 0. We next show that ρ 3 is well-defined and bounded. For G BMO(R 3, ), g Y 0 and a ball B R 3, we have ( /2 ( /2 inf G g B dx) 2 inf G g c dx) 2, g B B B c B B where the infimum in the left-hand side is taken over all g B L 2 (B, ) with dg B = 0 in B and that in the right-hand side is taken over all constant forms c. Hence G BMOd (R 3, ) inf g Y 0 G g BMO(R 3, ) = G + Y 0 Y. It is straightforward to check that ρ 3 ρ 2 ρ = I, ρ ρ 3 ρ 2 = I, where I denotes the identity map. Hence the theorem is proved. 4. PROOF OF THE MAIN THEOREM. Using the duality result in the previous section we next prove our main theorem of this paper: the following div-curl type theorem on R 3. The N dimensional case is studied in Section 6.

11 HARDY SPACE OF EXACT FORMS ON R N Theorem 4.. Let b L 2 loc (R3, ). Then sup b du dv b BMOd (R 3, ), (4.) u,v W R 3 where W = {w H (R 3 ) : dw L 2 (R 3, ) }. The implicit constants in (4.) are absolute constants. Proof. Suppose b BMO d (R 3, ). From Theorem II. in [CLMS] (see also Lemma 6.9 in Section 6), du dv H (R 3, 2 ) for u, v H (R 3 ) and there exists an absolute constant C such that du dv H (R 3, 2 ) C du L 2 (R 3, ) dv L 2 (R 3, ). (4.2) Further, du dv Hd (R3, 2 ). From (3.3), (4.2) and the assumption on u and v, we have b du dv b BMOd (R 3, ) du dv H d (R 3, 2 ) R 3 C b BMOd (R 3, ). On the other hand, we need to prove that there exists an absolute constant C such that for all balls B R 3 ( /2 inf b g B dx) 2 C sup g B B B u,v W b du dv, (4.3) R 3 where the infimum is taken over all g B L 2 (B, ) with dg B = 0 in B. Since the left-hand side of (4.3) is invariant by scaling, to prove (4.3) we need only to show that for the unit ball B 0, there exist u 0 H0 (B 0 ), v 0 H0 (2B 0 ) with du 0 L 2 (R 3, ), dv 0 L 2 (R 3, ) such that inf g 0 ( ) /2 b g 0 2 dx C b du 0 dv 0, (4.4) B 0 B 0 where the infimum is taken over all g 0 L 2 (B 0, ) with dg 0 = 0 in B 0. We now prove (4.4). Let H = {h L 2 (B 0, ) : δh = 0 in B 0, n h B0 = 0}. Since H is a closed subspace of L 2 (B 0, ), we have the decomposition: L 2 (B 0, ) = H H, (4.5) where H denotes the orthogonal complement of H in L 2 (B 0, ) and H = {dq : q H (B 0 )} (ref. [GR, Theorem 2.7]). Let b L 2 (B 0, ), (4.5) gives that b = h + dq, where h H, q H (B 0 ). Then we have b du 0 dv 0 = h du 0 dv 0 B 0 B 0

12 2 ZENGJIAN LOU ALAN M c INTOSH and inf g 0 ( ) /2 b g 0 2 dx h L B 0 2(B0, ). Therefore to prove (4.4) it is sufficient to prove that there exists an absolute constant C such that h L 2 (B 0, ) C h du 0 dv 0 (4.6) B 0 for all h H. Applying Proposition A. in Appendix A, for h H, there exists ϕ H 0 (B 0, ) and an absolute constant C 0 such that h = dϕ and ( is the Hodge star operator). Thus we have Dϕ L 2 (B 0, ) C 0 h L 2 (B 0, ) (4.7) h 2 L 2 (B 0, ) B = h dϕ 0 = h d(ϕ dx + ϕ 2 dx 2 + ϕ 3 dx 3 ) B 0 3 max h dϕ i dx i i 3 B 0 := 3 h dϕ i0 dx i0 (4.8) B 0 for some choice of i 0 ( i 0 3). Define u 0 = ϕ i0. C 0 h L 2 (B 0, ) It is obvious that u 0 H0 (B 0 ) and du 0 L 2 (R 3, ) by (4.7). We now construct v 0. Let ψ 0 C0 (R 3 ) such that { in B0 ; ψ 0 = 0 outside 2B 0. Define v 0 = γx i0 ψ 0, i 0 3, where γ > 0 is a constant so that dv 0 L2 (R 3, ). It is easy to check that v 0 C0 (2B 0 ) and dv 0 = γdx i0 in B 0, So (4.8) and the construction of u 0 and v 0 give dϕ i0 h L2 (B 0, ) 3C 0 γ h B 0 C 0 h L2 (B 0, ) = 3C 0 γ h du 0 dv 0. B 0 γdx i0 This proves (4.6). The proof of Theorem 4. is completed.

13 HARDY SPACE OF EXACT FORMS ON R N 3 Remarks. () It is easy to check that the following estimate of the quadratic form det Du can be derived from Theorems II. and III.2 in [CLMS] b BMO(R 2 ) sup b det Du dx, (4.9) u R 2 where the supremum is taken over all u H (R 2, ) with du i L 2 (R 2, ), i =, 2. Theorem 4. is an extension of (4.9) to three-dimensions. (2) We are especially interested in the three-dimensional case, because, as shown in the following section, Theorem 4. can be used to give coercivity properties and Gårding s inequality of some polyconvex quadratic forms. 5. APPLICATIONS In the study of homogenization of linearized elasticity, Geymonat, Müller and Triantafyllidis [GMT] considered the following system div α A i,j α,β (x ε ) βu j = f in Ω (5.) u Ω = 0, where A i,j α,β (x) is a periodic measurable function, i, j, α, β N. A quantity Λ is introduced which gives a criterion of whether an elliptic system satisfying the Legendre- Hadamard condition can be homogenized, namely Λ = inf { R N A i,j α,β (x) αu i β u j dx R N Du 2 dx : u C 0 (R N, ) It was proved in [GMT] that if Λ > 0 some homogenization results can be obtained for the system (5.). If Λ < 0, the system cannot be homogenized. Zhang asked the following question: what conditions on the coefficient A i,j α,β of the system imply that Λ 0? For N = 2, this question was answered by Zhang in [Z]. Motivated by [Z] we answer the question for N = 3. Suppose that A i,j α,β (x) αu i β u j can be written in the form A i,j α,β (x) αu i β u j = B i,j α,β (x) αu i β u j + b ij (x)(adj Du) i,j, (5.2) where A i,j α,β, Bi,j α,β L (R 3 ), B i,j α,β αu i β u j C Du 2, adj Du denotes the adjoint matrix of Du for u H (R 3, ) (the summation convention is understood). We are interested in forms of this type in three dimensions, because they arrive naturally from the linearization of polyconvex variational integrals studied in nonlinear elasticity by Ball in [B]. When i =, the last term in (5.2) becomes 3 b j (x)(adj Du),j j= = b (x)(adj Du), + b 2 (x)(adj Du),2 + b 3 (x)(adj Du),3 = det b b 2 b 3 u 2 2 u 2 3 u 2 u 3 2 u 3 3 u 3 := b du 2 du 3. }.

14 4 ZENGJIAN LOU ALAN M c INTOSH For i = 2, 3, we have similarly and 3 b 2j (x)(adj Du) 2,j = b 2 du du 3 j= 3 b 3j (x)(adj Du) 3,j = b 3 du du 2, j= where b i = (b i, b i2, b i3 ). Let a(u) denote the following polyconvex quadratic form a(u) = Du 2 + b du 2 du 3 + b 2 du du 3 + b 3 du du 2, (5.3) where b, b 2 and b 3 are one-forms. So the question when Λ 0 becomes: find necessary conditions of b i such that R 3 a(u) dx 0 for all u H (R 3, ). In fact the conditions are that b i BMOd (R 3, ) cannot be too large. We prove this by using Theorem 4.. Another application of Theorem 4. is to prove the weak coercivity property - Gårding s inequality. 5. Coercivity For coercivity we give an almost necessary and sufficient condition on b i such that a(u) dx 0 for all u H (R 3, ). R 3 Proposition 5.. Let a(u) be the expression shown in (5.3). () There exists an absolute constant C such that max i 3 b i BMOd (R 3, ) C implies that a(u) dx R 2 Du 2 L 2 (R 3, ) 3 for all u H (R 3, ). (2) If a(u) dx 0 for all u H (R 3, ), then there exists an absolute constant C R 3 2 such that max b i BMOd (R i 3 3, ) C 2. (5.4) Proof. () Let b i BMO d (R 3, ) and u H (R 3, ). From (3.3) and (4.2), we have R 3 a(u) dx Du 2 L 2 (R 3, ) ( b BMOd (R 3, ) du 2 du 3 H d (R 3, 2 ) + b 2 BMOd (R 3, ) du du 3 H d (R 3, 2 ) + b 3 BMOd (R 3, ) du du 2 H d (R 3, 2 ) ( Du 2 L 2 (R 3, ) C max b i BMOd (R i 3 3, ) du 2 L 2 (R 3, ) du 3 L 2 (R 3, ) + du L2 (R 3, ) du 3 L2 (R 3, ) + du L2 (R 3, ) du 2 L2 (R 3, ) ( ) C max b i BMOd (R i 3 3, ) Du 2 L 2 (R 3, ). ) )

15 HARDY SPACE OF EXACT FORMS ON R N 5 So implies that max b i BMOd (R i 3 3, ) C = 2C a(u) dx R 2 Du 2 L 2 (R 3, ) 3 for all u H (R 3, ). (2) From Theorem 4., there exist absolute constants C and C such that C b BMOd (R 3, ) sup b du dv C b BMOd (R 3, ). u,v W R 3 For any ε > 0, there exist u ε, v ε H (R 3 ) with du ε L 2 (R 3, ), dv ε L 2 (R 3, ) such that C b BMOd (R 3, ) ε b du ε dv ε. R 3 (5.5) For b i and ε =, (5.5) gives b i d( u ) dv C b i BMOd (R 3, ) +. R 3 (5.6) Let w = (0, u, v ). Since a(u) dx 0 for all u H (R 3, ), in particular R 3 a(w ) dx 0. Combining this with (5.6) we get R 3 0 Dw 2 dx + b i d( u ) dv R 3 R 3 Hence Proposition 5. is proved. Dw 2 L 2 (R 3, ) C b i BMOd (R 3, ) + 3 C b i BMOd (R 3, ). max i 3 b i BMOd (R 3, ) C 2 = 3 C. 5.2 Gårding s Inequality The proof of Theorem 4. implies the following result Lemma 5.2. Let Ω R 3 be an open domain and b L 2 loc (Ω, ). Then there exists an absolute constant C 3 such that ( /2 sup inf b g dx) 2 C 3 sup b du dv, (5.7) B g B B u,v W Ω where the supremum in the left-hand side is taken over all balls B with 2B Ω, the infimum is taken over all g L 2 (B, ) with dg = 0 in B, and W = {w H 0 (Ω) : dw L 2 (Ω, ) }. Remark. The two sides of (5.7) are actually equivalent when Ω is a special Lipschitz domain or a bounded strongly Lipschitz domain. See Theorem 6. of [LM]. Let us denote the left-hand side of (5.7) by b BMO H d (Ω, ).

16 6 ZENGJIAN LOU ALAN M c INTOSH Lemma 5.3. Let Ω R 3 be an open domain, a(u) be the expression shown in (5.3) and Ω a(u) dx 0 for all u H 0 (Ω, ). Then there exists an absolute constant C 4 such that max b i BMO H i 3 d (Ω, ) C 4. Proof. Using Lemma 5.2, similar to the proof of Proposition 5. (2), we can prove the proposition. The details are omitted. For b L 2 loc (R3, ), define b = lim sup l 0 B inf g ( /2 b g dx) 2, (5.8) B B where the supremum is taken over all balls B R 3 with radius less than l > 0, the infimum is taken over all g L 2 (B, ) with dg = 0 in B. Proposition 5.4. Assuming Gårding s inequality holds for a(u) dx, that is, there exist R 3 constants λ 0 > 0, λ 0 such that a(u) dx λ 0 Du 2 dx λ u 2 dx (5.9) R 3 R 3 R 3 for all u H (R 3, ). Then max b i C 4, i 3 where C 4 is the same constant in Lemma 5.3. Proof. From (5.8), there exists a sequence of balls B rk = B(x k, r k ) R 3 with r k 0 such that ( ) /2 inf b i g 2 dx b i. (5.0) g B rk B rk Suppose that v := v dx + v 2 dx 2 + v 3 dx 3 H (R 3, ) is supported in 2B rk. By (5.9), Dv(x) 2 dx + b (x) dv 2 (x) dv 3 (x) 2B rk 2B rk + b 2 (x) dv (x) dv 3 (x) + b 3 (x) dv (x) dv 2 (x) λ 0 Dv(x) 2B 2 dx λ v(x) 2 dx. (5.) rk 2B rk Set x = x k + 2r k y and let v k (y) = v(x k + 2r k y), b k i (y) = b i(x k + 2r k y) in (5.) we have Dv k (y) 2 dy + b k (y) dv2 k (y) dv3 k (y) B(0,) B(0,) + b k 2(y) dv k (y) dv3 k (y) + b k 3(y) dv k (y) dv2 k (y) λ 0 Dv k (y) 2 dy λ (2r k ) 2 v k (y) 2 dy B(0,) B(0,) (5.2)

17 HARDY SPACE OF EXACT FORMS ON R N 7 ( for all v k H ) 0 B(0, ),. For r k sufficiently small, by Poincaré s inequality the righthand side of (5.2) is non-negative, i.e. a(v k ) dy 0 B(0,) ( for all v k H ) 0 B(0, ),. This yields ( max inf i 3 g B(0, /2) B(0,/2) b k i (y) g(y) 2 dy) /2 C 4 (5.3) by Lemma 5.3, where the infimum is taken over all g L 2( B(0, /2), ) with dg = 0 in B(0, /2). Let y = x x k 2r k, (5.3) gives ( ) /2 max inf b i (x) g(x) 2 dx C 4, (5.4) i 3 g B rk B rk where g(x) L 2 (B rk, ) with d g = 0. Combining (5.2) with (5.4) we have The proof of Proposition 5.4 is finished. max b i C 4. i 3 6. HARDY SPACES OF EXACT FORMS ON R N In this section we introduce Hardy spaces of exact forms on R N and study their atomic decompositions and dual spaces. Using duality results we prove that Theorem 4. holds on R N when b, u, v are respectively k, m, l-forms with k + m + l + 2 = N. 6. Definitions Definition 6.. For 0 l N, the Hardy space of l-forms is defined as with the norm for f = I f Idx I. H (R N, l ) = {f : R N l : each component of f is in H (R N )} f H (R N, l ) = I f I H (R N ) Definition 6.2. Let l N. The Hardy space of exact l-forms is defined as with the norm H d(r N, l ) = {f H (R N, l ) : f = dg for some g D (R N, l )} f H d (R N, l ) = f H (R N, l ). Remark. When l = N, H d (RN, l ) is isomorphic to the usual Hardy space H (R N ). When l = N, H d (RN, l ) is isomorphic to the divergence-free Hardy space H div (RN, R N ) := {f H (R N, R N ) : div f = 0 in R N }.

18 8 ZENGJIAN LOU ALAN M c INTOSH Definition 6.3. We say that a is an H d (RN, l )-atom if (i) there exists b L 2 (R N, l ) supported in a cube Q in R N such that a = db; (ii) a and b satisfy size conditions: a L 2 (Q, l ) Q /2, b L 2 (Q, l ) l(q) Q /2, where l(q) denotes the side-length of Q. 6.2 Atomic Decompositions and Dual Spaces The main result of this section is the following atomic decomposition theorem for H d (RN, l ). We also characterize its dual by using this decomposition. Theorem 6.4. Let l N. An l form f on R N is in Hd (RN, l ) if and only if it has a decomposition f = λ k a k, (6.) where the a k s are H d (RN, l )-atoms and λ k <. Furthermore, ( ) f H d (R N, l ) inf λ k, where the infimum is taken over all such decompositions. The constants of the proportionality depend only on the dimension N. Proof. We first prove the if part. Suppose that f can be written as (6.) with λ k <, (6.2) where the a k s are Hd (RN, l )-atoms, i.e. there exist b k L 2 (R N, l ) supported in cubes Q k such that a k = db k and b k L 2 (R N, l ) l(q k ) Q k /2. We need to show that g := λ k b k exists in D (R N, l ), for then f = dg Hd (RN, l ). To prove g D (R N, l ), it is sufficient to show that the sum λ kb k is convergent in the sense of distributions. From (6.2), n λ k 0 as m, n. k=m Combining this with the size condition of b k, for any ϕ C 0 (R N, N l+ ) with compact

19 HARDY SPACE OF EXACT FORMS ON R N 9 support Ω, R N ( n ) n λ k b k ϕ λ k k=m C C C k=m Q k Ω b k ϕ n λ k b k L 2 (Q k Ω, ) Q l k Ω /2 k=m n λ k l(q k ) Q k /2 Q k Ω /2 k=m n λ k 0 as m, n, k=m where the constant C depends only on ϕ. The convergence of λ kb k is proved. The proof of the only if part is similar to that of Theorem 2.3 in Section 2, so we only give an outline of the proof. From the proof of Theorem 2.3, we know that any f Hd (RN, l ) can be written as ( ) dt f = td tδ(f ϕ t ) ϕ t 0 t, (6.3) where ϕ C 0 (R N ) with support in the unit ball and 0 t ξ 2 ˆϕ(tξ) 2 dt =. For y R N, t > 0, define F (y, t) = tδ(f ϕ t )(y). Let f = I f Ie I l, then F (y, t) = i,i t i (f I ϕ t )(y)µ i (e I ) = i,i f I ( i ϕ) t (y)µ i (e I ). Applying (2.), we get F N (R N+ +, l ) and From atomic decompositions for tent spaces, with F N (R N+ +, l ) C f H (R N, l ). F = λ k α k (6.4) λ k C F N (R N+ +, l ), where the α k s are N (R N+ +, l )-atoms. Define ( ) dt a k = td α k (, t) ϕ t 0 t.

20 20 ZENGJIAN LOU ALAN M c INTOSH From (6.3) and (6.4), we have f = λ k a k, where a k is supported in a ball 2B k and satisfies the size condition: a k L 2 (2B k, l ) C 2B k /2 for a constant C independent of k. Applying Lemma 6.7 () in Section 6 to a k, there exists b k L 2 (R N, l ) supported in 2B k such that a k = db k and b k L 2 (2B k, l ) C r(2b k ) 2B k /2, where r(2b k ) denotes the radius of the ball 2B k and C is independent of k. Let Q k be a smallest cube containing 2B k. Then and a k L 2 (Q k, l ) C Q k /2 b k L2 (Q k, l ) C l(q k ) Q k /2 for some constants C independent of k. We have proved that a k is an Hd (RN, l )-atom. The proof of Theorem 6.4 is finished. Now we consider dual spaces of Hd (RN, l ). Let BMO d (R N, k ) (0 k N) be the space of all locally integrable functions G : R N k with ( /2 G BMOd (R N, k ) = sup inf G g B dx) 2 <, B g B B where the supremum is taken over all balls B in R N and the infimum is taken over all g B L 2 (B, k ) with dg B = 0 in B. Consider BMO d (R N, k )/X 0 with the norm G + X 0 BMOd (R N, k )/X 0 = G BMOd (R N, k ), where X 0 = {G BMO d (R N, k ) : G BMOd (R N, k ) = 0}. We see that when k = 0, BMO d (R N, 0 )/X 0 reduces to the usual BMO-space on R N. The following theorem is an analogue of Theorem 3.3, which reveals that the dual of H d (RN, l ) is the space BMO d (R N, N l )/X 0. Its proof is similar to that of Theorem 3.3, so we skip the details. Let D(R N, l ) denote the vector space finitely generated by H d (RN, l )-atoms. Theorem 6.4 implies that D(R N, l ) is dense in H d (RN, l ). Theorem 6.5. Let l N. functional L defined by B If G + X 0 BMO d (R N, N l )/X 0, then the linear L(h) = G h, R N (6.5) initially defined in D(R N, l ), has a unique bounded extension to H d (RN, l ). Conversely, if L H d (RN, l ), then there exists a unique G + X 0 BMO d (R N, N l )/X 0 such that (6.5) holds. The map G + X 0 L given by (6.5) is a Banach isomorphism between BMO d (R N, N l )/X 0 and H d (RN, l ). 6.3 The Div-Curl Type Theorem on R N We now prove the div-curl type theorem on R N, which is a generalization of Theorem 4. to N-dimensions and to forms of all degrees.

21 HARDY SPACE OF EXACT FORMS ON R N 2 Theorem 6.6. Let b L 2 loc (RN, l ), l, m, n 0 and l + m + n + 2 = N. Then sup b du dv b BMOd (R N, ), (6.6) l u,v R N where the supremum is taken over all u and v with u H (R N, m ), } du L 2 (R N, m+ ) ; v H (R N, n ), dv L 2 (R N, n+ ). (6.7) The implicit constants in (6.6) depend only on N. To prove Theorem 6.6 we need the following lemmas. Lemma 6.7. Let B be a ball in R N. () If u satisfies either of the following conditions: ) u L 2 (B, l ), du = 0 in B and n u B = 0 when 0 < l < N; 2) u L 2 (B, l ) with u = 0, where l = N, then there exists ϕ H 0 (B, l ) and a constant C independent of u and B such that u = dϕ, Dϕ L 2 (B, l ) C u L 2 (B, l ) (6.8) and ϕ L 2 (B, l ) C r(b) u L 2 (B, l ). (6.9) (2) If u satisfies either of the following conditions: ) u L 2 (B, l ), δu = 0 in B and n u B = 0 when 0 < l < N; 2) u L 2 (B, l ) with u = 0, where l = 0, then there exists ψ H 0 (B, l+ ) and a constant C independent of u and B such that u = δψ and Dψ L 2 (B, l+ ) C u L 2 (B, l ). Proof. () When u satisfies the conditions of 2), Lemma 6.7 () is a special case of Nečas result [N, Lemma 7., Chapter 3]. So we only prove ). From Theorem in Chapter 3 of [Sc], there exists ϕ H (B, l ) such that u = dϕ, ϕ B = 0 and ϕ H (B, l ) C u L2 (B, l ) for some constants C independent of u. So the only thing we need to check is that the constants in (6.8) and (6.9) are independent of balls B. We only prove this for (6.9). Let B = B(x 0, r), x 0 be the center of B, r = r(b). It is easy to check that u(x 0 +ry), y is in the

22 22 ZENGJIAN LOU ALAN M c INTOSH unit ball B 0, satisfies the conditions of ) for B 0. So there exists ϕ(x 0+ry) r H 0 (B 0, l ) and a constant C independent of u and r such that and u(x 0 + ry) = d ϕ(x 0 + ry) r r ϕ(x 0 + ry) L 2 (B 0, l ) C u(x 0 + ry) L 2 (B 0, l ). This is equivalent to (6.9) by a simple computation. (2) can be derived from Corollary of [Sc, Chapter 3], Nečas lemma and a similar discussion to (). Remark. Lemma 6.7 holds when B is replaced by any smooth and contractible domain in R N, though with constants C which depend on the domain. Lemma 6.8 [GR, Theorem 2.7]. For a ball B R N and 0 l N, let H = {u L 2 (B, l ) : du = 0 in B, n u B = 0} and H = {u L 2 (B, l ) : δu = 0 in B, n u B = 0}. Then L 2 (B, l ) = H {δw : w H (B, l+ )} = H {dw : w H (B, l )}. The following result is a generalized version of the div-curl lemma by Coifman, Lions, Meyer and Semmes in [CLMS, Theorem II.]. When m+l = N, it can be found in [HLMZ, Proposition 4.]. Lemma 6.9. If < p <, p + q =, 0 < m + l N, u Lp (R N, m ), v L q (R N, l ), du = 0, dv = 0 in R N. Then u v H (R N, m+l ) and there exists a constant C independent of u and v such that u v H (R N, m+l ) C u L p (R N, m ) v L q (R N, l ). Proof. Suppose m + l = N. When l =, Lemma 6.9 becomes Theorem II. of [CLMS]. The proof of the case l is completely similar to the case of l =. If m + l < N, we know that u v H (R N, m+l ) if and only if u v dx im+l+ dx in H (R N, N ). Set V = v dx im+l+ dx in. Then v L q (R N, l ) and dv = 0. This implies that V L q (R N, N m ) and dv = 0. For u and V, applying the result when m + l = N, we have u V H (R N, N ) and This proves the result. u V H (R N, N ) C u L p (R N, m ) V L q (R N, N m ) = C u Lp (R N, m ) v L q (R N, l ). We are now ready to prove Theorem 6.6. There are some similarities between its proof and that of Theorem 4..

23 HARDY SPACE OF EXACT FORMS ON R N 23 Proof of Theorem 6.6. Suppose that u and v satisfy (6.7). Lemma 6.9 yields that du dv H d (RN, m+n+2 ) and du dv H d (R N, N l ) = du dv H (R N, m+n+2 ) C du L 2 (R N, m+ ) dv L 2 (R N, n+ ) C. Let b BMO d (R N, l ) and h Hd (RN, N l ). By using a similar argument as in Theorem 3.3, we have b h C b BMOd (R N, ) h l H R N d (R N, ). N l Therefore b du dv b BMOd (R N, ) du dv l H R N d (R N, N l ) C b BMOd (R N, ). l We now prove the reversed inequality in (6.6). By scaling we need only to show that for the unit ball B 0, there exist u 0 and v 0 with such that inf g 0 } u 0 H0 (B 0, m ), du 0 L 2 (R N, m+ ), v 0 H 0 (2B 0, n ), dv 0 L 2 (R N, n+ ) (6.0) ( ) /2 b g 0 2 dx C b du 0 dv 0, (6.) B 0 B 0 where the infimum is taken over all g 0 L 2 (B 0, l ) with dg 0 = 0 in B 0 and C is a constant independent of b, u 0 and v 0. Let b L 2 (B 0, l ), Lemma 6.8 gives b = h + dq for h H = {h L 2 (B 0, l ) : δh = 0, n h B0 = 0} and q H (B 0, l ). We have and b du 0 dv 0 = h du 0 dv 0 B 0 B 0 inf g 0 ( ) /2 b g 0 2 dx h L2(B0, l). B 0 Thus to prove (6.) it is sufficient to show that there exist u 0 and v 0 satisfying (6.0) such that h L 2 (B 0, l ) C h du 0 dv 0 (6.2) B 0 for all h H, where C does not depend on h, u 0 and v 0. Applying Lemma 6.7 (2) to h H, there exists ϕ H 0 (B 0, l+ ) and a constant C 0 independent of h such that h = δϕ

24 24 ZENGJIAN LOU ALAN M c INTOSH and Dϕ L 2 (B 0, l+ ) C 0 h L 2 (B 0, l ). (6.3) Let ϕ = I ϕ Idx I, where I = (i,, i l+ ), i < < i l+ N. Then h = δϕ = ±d ϕ = ± I dϕ I ( dx I ). Denote dx I by dx J, where J = {j,, j m+n+ }, j < < j m+n+ N. Thus h 2 L 2 (B 0, l ) h dϕ I dx J I B 0 C max h dϕ I dx J I B 0 := C h dϕ I0 dx J0 (6.4) B 0 for some choice of I 0, where dx J0 = dx I0 and C = ( N l+). Let J0 = {j 0,, j 0m+n+ }. Define u 0 = ϕ I 0 dx j0 dx j0 m C 0 h L2 (B 0, l ) It is easy to check that u 0 H 0 (B 0, m ) and du 0 L 2 (R N, m+ ) by (6.3). Now we construct v 0. Let ψ 0 C 0 (R N ), ψ 0 equals in B 0 and 0 outside 2B 0. Setting v 0 = γψ 0 x j0m+ dx j0m+2 dx j0m+n+,. where γ > 0 is a constant so that dv 0 L 2 (R N, n+ ). So v 0 C 0 (2B 0, n ) and dv 0 = γdx j0m+ dx j0m+n+ in B 0. Combining (6.4) with the construction of u 0 and v 0, we obtain h L 2 (B 0, l ) C h du 0 dv 0, B 0 where C = γ ( N l+) C0. This proves (6.2). The proof of Theorem 6.6 is completed. Remarks. () From the proof of Theorem 6.6, we see that the equivalence in (6.7) is also true if the supremum is taken over all u H (R N, m ), v H (R N, n ) with Du L 2 (R N, m+ ), Dv L 2 (R N, n+ ). (2) The case l = m = n = 0 and N = 2 in Theorem 6.6 yields the Jacobian determinant estimate (4.9). Let l = m = 0, n = N 2. Theorem 6.6 becomes

25 Corollary 6.0. For b L 2 loc (RN ), HARDY SPACE OF EXACT FORMS ON R N 25 b BMO(R N ) sup b α β, α,β R N where the supremum is taken over all α L 2 (R N, ), β L 2 (R N, N ) with dα = dβ = 0 and α L 2 (R N, ), β L 2 (R N, N ). It is easy to see that Corollary 6.0 is equivalent to the following result by Coifman, Lions, Meyer and Semmes in [CLMS, page 262] ( Div-Curl Lemma): for b L 2 loc (RN ) b BMO(R N ) sup b E F dx, E,F R N the supremum being taken over all E, F L 2 (R N, R N ) with div E = 0, curl F = 0 and E L 2 (R N,R N ), F L 2 (R N,R N ). 6.4 A Decomposition Theorem In [CLMS, Theorem III.2], Coifman, Lions, Meyer and Semmes proved a decomposition of H (R N ) into div-curl quantities. We now give a similar decomposition for Hardy spaces H d (RN, l ). Theorem 6.. Let l N and 0 m l 2. Then any f Hd (RN, l ) can be written as f = λ k du k dv k with λ k C f H d (R N, l ) for some constants C depend only on N, where u k H (R N, m ) and v k H (R N, l m 2 ) with du k L 2 (R N, m+ ), dv k L 2 (R N, l m ). Proof. By Theorem 6.4, any f H d (RN, l ) has a decomposition f = µ i a i, (6.5) i=0 where the a i s are H d (RN, l )-atoms and µ i C f H d (R N, l ) i=0 for constants C depending only on N. For simplicity we drop the subscript i of a i temporarily. Since a := a i is an H d (RN, l )-atom, i.e. there exists b L 2 (R N, l ) supported

26 26 ZENGJIAN LOU ALAN M c INTOSH in a ball B such that a = db and a L 2 (B, l ) B /2. Applying Lemma 6.7 (), there exists ϕ H 0 (B, l ) and a constant C 0 independent of a and B such that and a = dϕ (6.6) Dϕ L 2 (B, l ) C 0 a L 2 (B, l ). Let ϕ = I ϕ Idx I, I = (i,, i l ), i < < i l N. From (6.26) the atom a can be written as a = I = I dϕ I dx i dx il d(c 0 B /2 ϕ I ) dx i dx im d(c 0 B /2 x im+ ) dx il. For any I, define u (I) = C 0 B /2 ϕ I dx i dx im. Then u (I) H 0 (B, m ) and du (I) L 2 (B, m+ ). As in the proof of Theorem 6.6, define ψ 0 C 0 (R N ). Let v (I) = γc 0 B /2 ψ B (x im+ x 0 i m+ )dx im+2 dx il, ( ) x x where ψ B (x) = ψ 0 0 r, x 0 denotes the center of the ball B, r = r(b), γ is a constant independent of x 0 and r such that dv (I) L 2 (B, l m ). We see that v (I) C0 (2B, l m 2 ) and Thus any atom a can be written as dv (I) = γc 0 B /2 dx im+ dx il in B. a = γ I du (I) dv (I). Combining this with the atomic decomposition of f in (6.5). 6.. We proved Theorem Let l = N, m = 0 in Theorem 6. we get Corollary 6.2 ([CLMS, Theorem III.2]). Any f H (R N ) can be written as f = λ k E k F k with λ k C f H (R N )

27 HARDY SPACE OF EXACT FORMS ON R N 27 for a constant C depending only on N, where E k, F k L 2 (R N, R N ) with div E k = curl F k = 0 and E k L 2 (R N,R N ), F k L 2 (R N,R N ). The proof of Corollary 6.2 in [CLMS] is based on two results from functional analysis ([CLMS, Lemmas III. and III.2]). The proof we have given is more natural in the context of the theory of Hardy spaces. APPENDIX A. SURJECTIVITY OF THE curl OPERATOR In this appendix we present an unpublished result of Costabel [C], that in three dimensions, the operator curl is surjective from H 0 (Ω, R 3 ) to a closed subspace of L 2 (Ω) when Ω is a bounded contractible strongly Lipschitz domain in R 3. For ψ D (Ω), we adopt the notation Dψ = ( ψ, 2 ψ, 3 ψ), while for v = (v, v 2, v 3 ) D (Ω, R 3 ), we define the divergence and curl operators by div v = 3 i v i i= and curl v = ( 2 v 3 3 v 2, 3 v v 3, v 2 2 v ). For a bounded Lipschitz domain Ω R N, the divergence operator is a continuous map from H0 (Ω, R N ) onto L 2 0(Ω), where H0 (Ω, R N ) denotes the Sobolev space H (Ω, R N ) with zero boundary values, and L 2 0(Ω) = {f L 2 (Ω) : f dx = 0}. This is a result by Nečas in Ω [N, Lemma 7., Chapter 3]. We now consider the operator curl : H0 (Ω, R 3 ) L 2 (Ω, R 3 ). The next proposition shows that the operator curl is surjective from H0 (Ω, R 3 ) to a closed subspace of L 2 (Ω, R 3 ). Proposition A.. Let Ω be a bounded contractible strongly Lipschitz domain in R 3, b L 2 (Ω, R 3 ), div b = 0 in Ω and n b Ω = 0. Then there exists u H 0 (Ω, R 3 ) such that and curl u = b u H (Ω,R 3 ) C b L 2 (Ω,R 3 ), where the constant C depends only on the domain Ω. Proof. Let F denote the extension by zero of b to R 3. Then div F = 0 on all of R 3. Therefore there exists V Hloc (R3, R 3 ) such that F = curl V. On letting B be a large ball containing Ω, then F H (B). In the simply connected domain B\Ω, curl V = 0, so there exists ψ H 2 (B\Ω) such that Dψ = V. Let E : H 2 (B\Ω) H 2 (B) be a bounded extension operator [St]. The vector field U = V D(Eψ) H (B, R 3 ) has support in Ω and satisfies curl U = F. Thus the restriction u = U Ω H0 (Ω, R 3 ) solves the equation curl u = b as required. The mapping b u is a bounded linear mapping depending only on E. Remark. In N dimensions, this result applies to solving du = b for u H 0 (Ω, Λ ) when b is a 2 form satisfying db = 0 and n b Ω = 0. The proof does not apply to general

28 28 ZENGJIAN LOU ALAN M c INTOSH k forms. However when the boundary of Ω is smooth, a slight adaptation of the above argument gives an alternative proof of Lemma 6.7. APPENDIX B. REVIEW OF DIFFERENTIAL FORMS The setting of this section is that of forms on open domain Ω R N. We give a brief outline of the basic formalism. Let {e,, e N } denote the basis of Euclidean space R N and l =,, N. The space of all l linear, alternating functions ξ : (R N ) l R is denoted by l (R N ), or just l where there is no possibility of confusion. In particular (R N ) is the dual of R N and 0 (R N ) = R. The dual base to {e,, e N } will be denoted by e,, e N and referred to as the standard base for (R N ). The vector space of all forms (R N ) = N l=0 l (R N ) is equipped with the inner product < α, β >= α i i l β i i l for α = α i i l e i e i l and β = β i i l e i e i l. For w (R N ) the associated norm is denoted by w =< w, w > /2. The inner product induces a dual pairing between l (R N ) and N l (R N ) which results from the action of the Hodge star operator defined by = e e N ; α β =< α, β > e e N for all α, β l (R N ). The exterior and interior multiplication operators on (R N ) are linear operators defined by and µ k : l l+ ; µ k () = e k, µ k (e i ) = e k e i, µ k : l l ; µ k(e i ) = δ ki, µ k(e i e l ) = δ ki e l δ kl e i, respectively. The exterior and interior operators can be written as ([GHL]) d = N k= µ k x k, δ = N k= µ k. x k We define the interior product between a -form α and an l form u by setting α u = ( ) (l )N (α u). Suppose Ω is a bounded Lipschitz domain in R N. We denote by L 2 (Ω, l ) the space of square integrable l forms on Ω. Definition B.. Let 0 l N. For u L 2 (Ω, l ), we say that du = 0 on Ω if u dϕ = 0 for all ϕ C 0 (Ω, N l ). Ω

29 HARDY SPACE OF EXACT FORMS ON R N 29 Definition B.2. For u L 2 (Ω, l ) with du = 0 on Ω, we define n u Ω H /2 ( Ω, l+ ) by < n u Ω, ψ > Ω = ( ) l u dψ, where Ψ C ( Ω, N l ) and ψ = Ψ Ω, H /2 ( Ω, l+ ) is the space of (l + )-forms f each of whose components is in H /2 ( Ω). Remark. It is easy to show that the definition of < n u Ω, ψ > Ω is independent of the choice of the extension Ψ. Note that Ω n u Ω H /2 ( Ω, l+ ) C u L 2 (Ω, l ) for all u L 2 (Ω, l ) such that du = 0 (see, for example, [HLMZ]). The Green s formula is as follows: if u L 2 (Ω, l ) with du L 2 (Ω, l+ ) and ϕ H (Ω, N l ) then Ω du ϕ + ( ) l Ω u dϕ =< n u Ω, ϕ > Ω. The formula follows from Stokes theorem and the facts that C 0 ( Ω, l ) is dense in the space {u L 2 (Ω, l ) : du L 2 (Ω, l+ )} (see, for example, [ISS, Corollary 3.6]) and in the Sobolev space H (Ω, k ). Acknowledgment: The authors would like to thank Martin Costabel, Tom ter Elst and Kewei Zhang for helpful discussions and suggestions. References [B] J. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63(977), [CLMS] R. Coifman, P. L. Lions, Y. Meyer, S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures and Appl. 72(993), [CMS] R. Coifman, Y. Meyer, E. M. Stein, Some new function spaces and their application to harmonic analysis, J. Funct. Anal. 62(985), [C] M. Costabel, Personal communication. [CW] R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83(977), [GHL] J. E. Gilbert, J. A. Hogan, J. D. Lakey, Atomic decomposition of divergence-free Hardy spaces, Mathematica Moraviza, Special Volume, 997, Proc. IWAA, [GMT] G. Geymonat, S. Müller, N. Triantafylldis, Homogenization of nonlinear elastic materials, microscopic bifurcation and microscopic loss of rank-one convexity, Arch. Rational Mech. Anal. 22(993), [GR] V. Girault, P-A. Raviart, Finite Element Methods for Navier-Stokes Equations, theory and algorithms, Springer-Verlag, Berlin Heidelberg, 986. [HLMZ] J. A. Hogan, C. Li, A. M c Intosh, K. Zhang, Global higher integrability of Jacobians on bounded domains, Ann. Inst. H. Poincaré Anal. Non linéaire 7(2000), [ISS] T. Iwaniec, C. Scott, B. Stroffolini, Nonlinear Hodge theory on manifolds with boundary, Annali di Matematica pura ed applicata CLXXVII(999), [La] R. H. Latter, A decomposition of H p (R N ) in term of atoms, Studia Math. 62(978), [Le] P. G. Lemarié-Rieusset, Ondelettes vecteurs á divergence nulle, C. R. Acad. Sci. Paris 33(99), [LM] Z. Lou, A. M c Intosh, Hardy spaces of exact forms on Lipschitz domains in R N. [LM2] Z. Lou, A. M c Intosh, Divergence-free Hardy space on R N +, Sci. China, Ser. A, to appear.

30 30 ZENGJIAN LOU ALAN M c INTOSH [LMWZ] C. Li, A. M c Intosh, Z. Wu, K. Zhang, Compensated compactness, paracommutators and Hardy spaces, J. Funt. Anal. 50(997), [N] J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson et Cie, Eds., Paris; Academia, Editeurs, Prague 967. [Sc] G. Schwarz, Hodge Decomposition - A method for solving boundary value problems, Lecture Notes in Mathematics Vol. 607, Springer-Verlag, Berlin Heidelberg, 985. [St] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ Press, Princeton, 970. [St2] E. M. Stein, Harmonic Analysis, real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, New Jersey, 993. [SW] E. M. Stein, G. Weiss, On the theory of harmonic functions of several variables I: The theory of H p spaces, Acta Math. 03(960), [Z] K. Zhang, On the coercivity of elliptic systems in two dimensional spaces, Bull. Austral. Math. Soc. 54(996), Institute of Mathematics Shantou University Shantou Guangdong P. R. China zjlou@stu.edu.cn Center for Mathematics and its Applications Mathematical Sciences Institute The Australian National University Canberra ACT 0200 Australia alan@maths.anu.edu.au