A remark on the div-curl lemma. P. G. Lemarié Rieusset Laboratoire Analyse et Probabilités.

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1 A remark on the div-curl lemma. P. G. Lemarié Rieusset Laboratoire Analyse et Probabilités Université d Évry Val d Essonne plemarie@univ-evry.fr Abstract : We prove the div-curl lemma for a general class of functional spaces, stable under the action of Calderón Zygmund operators. The proof is based on a variant of the renormalization of the product introduced by S. Dobyinsky, on the use of divergence free wavelet bases. Introduction. In 1992, Coifman, Lions, Meyer Semmes published a paper [COILMS 92] where they gave a new interpretation of the compensated compactness introduced by Murat Tartar [MUR 78]. They showed that the functions considered by Murat Tartar had a greater regularity than expected : they belonged to the Hardy space H 1. They gave a new version of the div-curl lemma of Murat Tartar : Theorem 1 If 1 < p <, q = p/(p 1), f (L p (IR d )) d g (L q ) d, then div f = 0 curl g = 0 f. g H 1 There are many proofs of this result. We shall rely mainly on the proof by S. Dobyinsky, based on the renormalization of the product introduced in [DOB 92]. As pointed to me by Prof. Grzegorz Karch, it is easy to see that this result may be extended to a large class of functional spaces. For instance, we have the straightforward consequence of Theorem 1, for the case of weak Lebesgue spaces L p, (better seen as Lorentz spaces L p, ) their preduals L q,1 : Corollary 1 : If 1 < p <, q = p/(p 1), f (L p, (IR d )) d g (L q,1 ) d, then div f = 0 curl g = 0 f. g H 1 div g = 0 curl f = 0 f. g H 1 1

2 Proof : All we need is the projection operators that lead to the Helmhotz decomposition of a vector field : Id = IP + IQ where IQ is the projection onto irrotational vector fields : IQ h = 1 div h IP the projection operator onto solenoidal vector fields. Those projection operators are matrix of singular integral operators thus are bounded on Lebesque spaces L r, 1 < r <,, by interpolation, on Lorentz spaces spaces L r,t, 1 < r <, 1 t +. 1 q Let ɛ > 0 such that ɛ < min 1/p, 1/q. We write 1 p + = 1 p + ɛ, 1 p = 1 p ɛ, 1 q + = 1 q + ɛ = 1 q ɛ. If f (L p, (IR d )) d, we can write, for every A > 0, f = α A + β A with α A L p CA f L p, β A L p + CA 1 f L p,. If div f = 0, we have moreover f = IP f = IP α A + IP β A. On the other h, if g (L q,1 ) d, we can write g = j IN λ j g j with g j L q g L q + 1 j IN λ j C g L q,1. If curl g = 0, we have moreover g = IQ g = j IN λ j IQ g j. Let A j = g j 1/2 L q g j 1/2 L q +. We then write f. g = j IN λ j (IP α Aj. IQ g j + IP β Aj. IQ g j ) get (from the div-curl theorem of f Coifman, Lions, Meyer Semmes) f. g H 1 C j IN λ j ( IP α Aj L p IQ g j L q + + IP β Aj L p + IQ g j L q ) C f L p, j IN λ j (A j g j L q + + A 1 j g j L q ) = C f L p, j IN λ j C f L p, g L q,1 The proof for the case div g = 0 curl f = 0 is similar. In this paper, we aim to find a general class of functional spaces for which the div-curl lemma still holds. As we may see from the proof of the Corollary 1, singular integral operators will play a key role in our result. In section 1, we shall introduce Calderòn Zygmund pairs of functional spaces which will allow us to prove such a general result. In section 2, we recall basics of divergence free wavelet bases (as described in the book [LEM 02]). In section 3, we prove our main theorem. Then, in section 4, we give examples of Calderòn Zygmund pairs of functional spaces. 1. Calderón Zygmund pairs of Banach spaces. We begin by recalling the definition of a Calderón Zygmund operator : 2

3 Definition 1 : A) A singular integral operator is a continuous linear mapping from D(IR d ) to D (IR d ) whose distribution kernel K(x, y) D (IR d IR d ) (defined formally by the formula T f(x) = K(x, y)f(y) dy) has its restriction outside the diagonal x = y defined by a locally Lipschitz function with the following size estimates : i) sup x y K(x, y) x y d < + ii) sup x y x K(x, y) x y d+1 < + iii) sup x y y K(x, y) x y d+1 < + For such an operater T, we define T SIO = K(x, y) x y d L (Ω)+ x K(x, y) x y d+1 L (Ω)+ y K(x, y) x y d+1 L (Ω) where K is the distribution kernel of T Ω = IR d IR d {(x, y) / x = y} B) A Calderón Zygmund operator is a singular integral operator T which may be extended as a bounded operator on L 2 : sup ϕ D, ϕ 2 1 T (ϕ) 2 < +. We define CZO as the space of Calderón Zygmund operators, endowed with the norm : T CZO = T L(L 2,L 2 )+ T SIO. We may now define our main tool : Definition 2 : A Calderón Zygmund pair of Banach spaces (X, Y ) is pair of Banach spaces such that : i) we have the continuous embedding : D(IR d ) X D D(IR d ) Y D iii) Let X 0 be the closure of D in X; then the dual spacex0 of X 0 (i.e. the space of bounded linear forms on X 0 ) coincides with Y with equivalence of norms : a distribution T belongs to Y if anly if there exist a constant C T such that for all ϕ D we have T ϕ D,D C T ϕ X iiii) Let Y 0 be the closure of D in Y ; then the dual space Y0 of Y 0 coincides with X with equivalence of norms iv) Every Calderón Zygmund operator may be extended as a bounded operator on X 0 on Y 0 : there exists a constant C 0 such that, for every T CZO every ϕ D, we have T (ϕ) X 0 Y 0 T (ϕ) X C 0 T CZO ϕ X T (ϕ) Y C 0 T CZO ϕ Y By duality, we find that every Calderón Zygmund operator may be extended as a bonded operator on X Y : if T is defined by the formula T (ϕ) ψ D,D = ϕ T (ψ) D,D, 3

4 then T CZO implies T CZO we may define T (f) on X as the distribution ϕ f T (ϕ) Y 0,Y 0. The two definitions of T coincides on X 0. For m L, the operator T m : ϕ mϕ belongs to CZO (with kernel K(x, y) = m(x)δ(x y)). The stability of X Y through multiplication by bounded smooth functions (with the inequalities mf X C 0 m f X mf Y C 0 m f Y ) shows that elements of X Y are (complex) local measures that X 0 Y 0 are embedded into L 1 loc. Our main result is then the following one (to be proved in Section 3) : Theorem 2 : Let (X, Y ) be a Calderón Zygmund pair of Banach spaces (X, Y ). If f X0 d g Y d, then div f = 0 curl g = 0 f. g H 1 div g = 0 curl f = 0 f. g H 1 Remark : The distribution f. g is well-defined, since f X0 d : if ϕ D, then we have ϕf X0 d g (X0 ) d. 2. Divergence free wavelet bases. In this section, we give a short review of properties of divergence-free wavelet bases. Wavelet theory was introduced in the 1980 s as an efficient tool for signal analysis. Orthonormal wavelet bases were first constructed by Y. Meyer [LEMM 86], G. Battle [BAT 87] P.G. Lemarié-Rieusset; a major advance was done with the construction of compactly supported orthonormal wavelets by I. Daubechies [DAU 92]. Then bi-orthogonal bases were introduced by A. Cohen, I. Daubechies J.C. Feauveau [COHDF 92]. Divergence-free wavelets were introduced by Battle Federbush [BATF 95]. Compactly divergence-free wavelets were introduced by P.G. Lemarié Rieusset [LEM 92]; they are not orthogonal wavelets [LEM 94], but have been explored for the numerical analysis of the Navier Stokes equations [URB 95] [DER 06]. Let H div=0 H curl=0 be defined as H div=0 = { f (L 2 ) d / div f = 0} H curl=0 = { f (L 2 ) d / curl f = 0}. For a function f (L 2 ) d, j ZZ k ZZ d, we define f j,k as f j,k (x) = 2 jd/2 f(2 j x k). Let us recall the mains results of [LEM 92] (described as well in the book [LEM 02]). The idea is to begin with an Hilbertian basis of compactly supported wavelets, associated to 4

5 a multi-resolution analysis (V j ) j Z of L 2 (IR). Associated to this multi-resolution analysis (with orthogonal projection operator Π j onto V j ), there is a bi-orthogonal multi-resolution analysis (V + j ) (V j ) with projection Π (j) onto V j orthogonally to V + j such that d dx Π j = Π (j) d dx. Starting from this one-dimensional setting, we now consider a bi-orthogonal multiresolution analysis of (L 2 (IR d )) d (V j,1,..., V j,d ) (Vj,1,..., V j,d ) where V j,k = V j,k,1... V j,k,d with V j,k,l = V j for k l V j,k,k = V j with Vj,k,l = V j for k l Vj,k,k = V + (V j,1,..., V j,d ) orthogonally to (V j,1,..., V j,d ). Its adjoint P j Vj,k j,k,1 j,k,d j Let P j be the projection operator onto is the projection operator onto (Vj,1,..., V j,d ) orthogonally to (V j,1,..., V j,d ). The point is that we have P j ( f) = (Π j f) div (Pj f) = Π j (div f). Those projection operators P j Pj can give an accurate description of H div=0 H curl=0 : Proposition 1 : (Multi-resolution analysis for divergence-free or irrotational vector fields) Let N IN. Then there exists a compact set K N IR d such that : A) Multi-resolution analysis : There exists *) functions ϕ ξ ϕ ξ in (L2 ) d, 1 ξ d *) functions ψ χ ψ χ in (L 2 ) d, 1 χ d(2 d 1) such that i) the functions ϕ ξ, ϕ ξ, ψ χ ψ χ are supported in the compact K N ii) the functions ϕ ξ, ϕ ξ, ψ χ ψ χ are of class C N iii) for l IN d with d i=1 l i N, we have x l ψχ dx = x l ψ χ dx = 0 iv) for j, j, in ZZ, k, k in ZZ d, ξ, ξ in {1,..., d}, χ, χ in {1,..., d(2 d 1)} ϕ ξ,j,k. ϕ ξ,j,k dx = δ k,k δ ξ,ξ ψ χ,j,k. ψ χ,j,k dx = δ j,j δ k,k δ χ,χ v) The projection operators P j can be defined on (L 2 ) d by P j ( f) = f ϕ ξ,j,k ϕ ξ,j,k. They are bounded on (L 2 ) d satisfy k Z d 1 ξ d P j P j+1 = P j+1 P j = P j, lim P jf 2 = 0 j lim f P jf 2 = 0. j + vi) The operators Q j defined on (L 2 ) d by Q j ( f) = f ψ χ,j,k ψ χ,j,k k Z d 1 χ d(2 d 1) 5

6 are bounded on (L 2 ) d satisfy Q j = P j+1 P j f 2 Q jf 2 2 f ψ χ,j,k 2 j Z j Z k Z d 1 χ d(2 d 1) B) Irrotational vector fields : The projection operators P j satisfy : f (L 2 ) d curl f = 0 curl P j ( f) = 0 Moreover, there exists *) 2 d 1 functions γ η (L 2 ) d, 1 η 2 d ) with curl γ η = 0 *) 2 d 1 functions γ η (L 2 ) d, 1 η 2 d 1 such that i) the functions γ η γ η are supported in the compact K N ii) the functions γ η γ η are of class C N iii) for l IN d with d i=1 l i N, we have x l γ η dx = x l γ η dx = 0 iv) for j, j, in ZZ, k, k in ZZ d η, η in {1,..., 2 d 1}, γ η,j,k. γ η,j,k dx = δ j,j δ k,k δ η,η vi) The operators S j defined on (L 2 ) d by S j ( f) = f γ η,j,k γ η,j,k are bounded on (L 2 ) d satisfy k Z d 1 η 2 d 1 f H curl=0 S j f = Qj f f H curl=0 f 2 f γ η,j,k 2 j Z k Z d 1 η 2 d 1 C) Divergence-free vector fields : The projection operators P j satisfy : f (L 2 ) d div f = 0 div P j ( f) = 0 Moreover, there exists *) (d 1)(2 d 1) functions α ɛ (L 2 ) d, 1 ɛ (d 1)(2 d 1) with div α ɛ = 0 *) (d 1)(2 d 1) functions α ɛ (L 2 ) d, 1 ɛ (d 1)(2 d 1) 6

7 such that i) the functions α ɛ α ɛ are supported in the compact K N ii) the functions α ɛ α ɛ are of class C N iii) for l IN d with d i=1 l i N, we have x l α ɛ dx = x l α ɛ dx = 0 iv) for j, j, in ZZ, k, k in ZZ d ɛ, ɛ in {1,..., (d 1)(2 d 1)}, α ɛ,j,k. α ɛ,j,k dx = δ j,j δ k,k δ ɛ,ɛ vi) The operators R j defined on (L 2 ) d by R j ( f) = f α ɛ,j,k α ɛ,j,k are bounded on (L 2 ) d satisfy k Z d 1 ɛ (d 1)(2 d 1) f H div=0 R j f = Q j f f H div=0 f 2 f α ɛ,j,k 2 j Z k Z d 1 ɛ (d 1)(2 d 1) We would like now to use those special functions with our spaces X Y. We begin with the following lemma : Lemma 1 : a) If f X d 0, then P j f P j f converge strongly to 0 in X d as j converge strongly to f in X d as j +. b) If f Y d 0, then P j f P j f converge strongly to 0 in Y d as j converge strongly to f in Y d as j +. c) If f X d, then P j f P j f converge *-weakly to 0 in X d as j converge *-weakly to f in X d as j +. d) If f Y d, then P j f P j f converge *-weakly to 0 in Y d as j converge *-weakly to f in Y d as j +. Proof : First, we check that the operators are well defined. If f C N has a compact support, then we may write f = fθ, with θ D equal to 1 on a neighborhood of the support of f. Thus, f = T f (θ) we find that f X 0 Y 0. Thus, f g X0,Y is well 7

8 defined for every g Y, f h Y0,X is well defined for every h X. We may then consider the operators on X d : P j ( f) = P j ( f) = k Z d 1 ξ d k Z d 1 ξ d f ϕ ξ,j,k X,Y0 ϕ ξ,j,k f ϕ ξ,j,k X,Y0 ϕ ξ,j,k. We have sup j Z P j CZO = sup j Z P j CZO < +. Thus, those operators are equicontinuous on X d. To prove a), we need to check the limits only on a dense subspace of X d 0. X 0 cannot be embedded into L 1 : if f D with ˆf(0) 0, then the Riesz transforms R j f are not in L 1 but belong to X 0. It means that f D f 1 is not continuous for the X 0 norm. We may find a sequence of functions f n such that f n X converge to 0 f n 1 = 1. Since f n is Lipschitz compactly supported, we can regularize f n find a sequence of smooth compactly supported functions f n,k such that all the f n,k, k IN, are supported in a compact neighborhood of the support of f n converge, as k +, uniformly to f n ; then, we have convergence in X (since Y 0 L 1 loc ) in L1. Thus, we can find a sequence of functions f n which are in D, with f n dx = 1 lim n + f n X = 0. This gives that the set of function f D with f dx = 0 is dense in X 0. We now consider Q j = P j+1 P j Q j = P j+1 P j : Q j ( f) = k Z d 1 χ d(2 d 1) f ψ χ,j,k X,Y0 ψχ,j,k Q j ( f) = f ψ χ,j,k X,Y0 ψ χ,j,k. k Z d 1 χ d(2 d 1) If f D d f dx = 0, we have f = 1 l d l f l for some f l D d. Similarly, we have ψ χ = 1 l d l Ψ χ,l ψ χ = 1 l d l Ψ χ,l for some compactly supported functions of class C N. Thus, we find that, for f D d with f dx = 0, P j+1 f Pj f X + P j+1 f P j f X C min( l=1 d l f X 2 j, d f l X 2 j ). Thus, P j f P j f have strong limits in X d 0 when j goes to or +. If g D d, we write f (L 2 ) d g (L 2 ) d, see that lim P jf g X,Y0 = lim P j f g X,Y0 = 0 j j 8 i=1

9 lim P jf g X,Y0 = lim P j f g X,Y0 = f g X,Y0. j + j + Thus, we have, for f D d with f dx = 0, lim P jf X = lim P j f X = 0 j j lim P jf f X = lim P j f f X = 0 j + j + Thus a) is proved. b) is proved in a similar way. By duality, we get c) d). We may now consider the operators : R j ( f) = f α ɛ,j,k X,Y0 α ɛ,j,k k Z d 1 ɛ (d 1)(2 d 1) S j ( f) = f γ η,j,k X,Y0 γ η,j,k. k Z d 1 η 2 d 1 From the identities IP R j = IP Q j IQ S j = IQ Q j which are valid from Dd to Y d 0, we find by duality that R j IP = Q j IP S j IQ = Q j IQ on X d. To be able to use those identities, we shall need the following lemma : Lemma 2 : Let f X d. Then : i) IP f = f div f = 0 ii) IQ f = f curl f = 0 Proof : First, we check that f X f = 0 f = 0. Take θ D such that 1 1 θ 0, θ 0, define γ = θ. Convolution with the kernel is a (1+x 2 ) n+1 2 (1+x 2 ) n+1 2 Calderón Zygmund operator, so we get that γ Y 0. Moreover, if g is a function such that (1+x 2 ) n+1 2 g L, we find that g = γ 1 gγ = T γ 1 g(γ), where the pointwise multiplication operator T γ 1 g is a Calderón Zygmund operator, so we get that g Y 0. This proves that X S. Thus, if f X f = 0, we find that f is a harmonic polynomial. Moreover f γ dx = f T f (γ) X,Y0, hence the integral f γ dx must be finite, f must be f constant. As the smooth functions with vanishing integral are dense in Y 0, we find that the constant is equal to 0. Now, we have for a distribution f that div f = 0 ϕ D d with curl ϕ = 0, f ϕ = 0; 9

10 thus, we have on X d that div IP f = 0. Similarly, we have for a distribution f that curl f = 0 ϕ D d with div ϕ = 0, f ϕ = 0; thus, we have on X d that curl IQ f = 0. Conversely, we start from the decomposition Id = IP + IQ valid on X d. If div f = 0, then we find that h = f IP f = IQ f satisfies. div h = 0 curl h = 0. But this implies that h = 0, hence h = 0 We prove similarly that curl f = 0 implies that f = IQ f. 3. The proof of the div-curl lemma. As in [LEM 02], we prove Theorem 2 by adapting the proof given by Dobyinsky [DOB 92]. This proof uses the renormalization of the product through wavelet bases. If f X d, g Y d if moreover f X d 0 or g Y d 0, we use lemma 1 to get that, in the distribution sense, we have f. g = lim P j. f.p j g P j. f.p j g j + thus f. g = j Z P j. f.q j g + Q j f.p j g + Q j f.q j g. If moreover div f = 0 curl g = 0, we use lemma 2 to get that f. g = j Z P j f.s j g + R j f.pj g + R j f.sj g. We shall prove that the three terms A( f, g) = j Z P j f.s j g, B( f, g) = j Z R j f.pj g C( f, g) = j Z R j f.sj g belong to H 1. We make the proof in the case f X d 0 (the proof is similar in the case g Y d 0 ). We first check that A B map (X 0 ) d Y d to H 1 : we use the duality of H 1 CMO (the 10

11 closure of C 0 in BMO) (see Coifman Weiss [COIW 77] Bourdaud [BOU 02]) try to prove that the operators A( f, h) = j Z S j (hp j f) B( f, h) = j Z P j (hr j f) map (X 0 ) d CMO to (X 0 ) d. In order to prove this, we shall prove that A(., h) that B(.h) are matrices of singular integral operators when h D that we have the estimates A(., h) CZO C h BMO B(., h) CZO C h BMO. For B, we may as well study the adjoint operator B ( f, h) = j Z R j (hp j f) First, we estimate the size of the kernels of their gradients. The kernels A h (x, y) of A(., h) B h (x, y) of B(., h) are given by A h (x, y) = j Z k Z d 1 ξ d l Z d 1 η 2 d 1 γ η,j,l(x) h ϕ ξ,j,k γ η,j,l ϕ ξ,j,k (y) B h(x, y) = j Z α ɛ,j,l(x) h ϕ ξ,j,l α ɛ,j,k ϕ ξ,j,k(y) k Z d 1 ξ d l Z d 1 ɛ (d 1)(2 d 1) There are only a few terms that interact, because of the localization of the supports : if K N B(0, M), then h ϕ ξ,j,k γ η,j,l = h ϕ ξ,j,l α ɛ,j,k = 0 if l k > 2M. Let C(h) = D(h) = sup h ϕ ξ,j,k γ η,j,l j Z,k Z d,1 ξ d,l Z d,1 η 2 d 1 sup h ϕ ξ,j,l α ɛ,j,k j Z,k Z d,1 ξ d,l Z d,1 ɛ (d 1)(2 d 1) Then we have A h (x, y) CC(h)2 jd 1 B(0,M) (2 j x k)1 B(0,3M) (2 j y k) j Z k Z d thus A h (x, y) CC(h) 2 jd C C(h) x y d 2 j y x 4M 11

12 similarly B h (x, y) CD(h) x y d. In the same way, we have x A h (x, y) + y A h (x, y) CC(h)2 j(d+1) 1 B(0,M) (2 j x k)1 B(0,3M) (2 j y k) j Z k Z d thus similarly x A h (x, y) + y A h (x, y) CC(h) x y d 1 x B h (x, y) + y B h (x, y) CD(h) x y d 1. Moreover, the function ϕ ξ,j,k. γ η,j,l is supported in B(2 j k, M2 j ), ϕ ξ,j,k. γ η,j,l C2 jd ϕ ξ,j,k. γ η,j,l dx = 0 (since Pj ϕ ξ,j,k. = ϕ ξ,j,k. Q j γ η,j,l = γ η,j,l ). Thus, we find that ϕ ξ,j,k. γ η,j,l H 1 C, so that C(h) C h BMO. We have similar estimates for ϕ ξ,j,l. α ɛ,j,k H 1 (since P j ϕ ξ,j,l. = ϕ ξ,j,l. Q j α ɛ,j,k = α ɛ,j,k, thus ϕ ξ,j,l. α ɛ,j,k dx = 0), thus D(h) C h BMO. Thus far, we have proven that A(., h) B(., h) are singular integral operators. To prove L 2 boundedness, we use the T (1) theorem of David Journé [DAVJ 84]. We ve got to check that the operators are weakly bounded (in the sense of the WBP property), to compute the images of the function f = 1 through the operators through their adjoints. Let x 0 IR d, r 0 > 0 let f g be supported in B(x 0, r 0 ). We want to estimate A( f, h) g D,D B( f, h) g D,D. We have A( f, h) g D,D j Z A j where A j = k Z d 1 ξ d l Z d 1 η 2 d 1 similarly B( f, h) g D,D j Z B j where B j = k Z d 1 ξ d l Z d 1 ɛ (d 1)(2 d 1) g γ η,j,l h ϕ ξ,j,k γ η,j,l f ϕ ξ,j,k g ϕξ,j,l h ϕ ξ,j,l α ɛ,j,k f α ɛ,j,k We have A j C(h) k Z d 1 ξ d l k 2M 1 η 2 d 1 g γ η,j,l f ϕ ξ,j,k 12

13 which gives A j C(h) f ϕ ξ,j,k g γ η,j,l CC(h)2 jd f 1 g 1 k Z d 1 ξ d l Z d 1 η 2 d 1 A j C C(h) k Z d 1 ξ d f ϕ ξ,j,k 2 l Z d 1 η 2 d 1 g γ η,j,l 2 thus Finally, we get A j C C(h) S j g 2 P j f 2 C C(h)2 j g 2 f 2. A( f, h) g D,D CC(h)( 2 j r 0 1 2jd r d 0 f 2 g j r 0 >1 2 j g 2 f 2 ) C C(h)( f 2 + r 0 f 2 )( g 2 + r 0 g 2 ). Similar computations (based on the inequality R j ( f) 2 C2 j f 2 ) gives as well B( f, h) g D,D CD(h)( f 2 + r 0 f 2 )( g 2 + r 0 g 2 ). Thus, our operators satisfy the weak boundedness property. We must now compute the distributions T (1) T (1) when T is one component of the matrix of operators A(., h) or of B(., h). We must prove that, if θ D is equal to 1 on a neighborhood of 0, if θ l,r = (θ 1,l,R,..., θ d,l,r ) with θ k,l,r = δ k,l θ( x R ) if ψ D d with ψ dx = 0, then we have lim R + j Z Sj (hpj θ l,r ) (BMO) d (the limit is taken in (D /IR) d ) similarly that lim R + j Z lim R + j Z lim R + j Z P j (hs jθl,r ) (BMO) d Pj (hr jθl,r ) (BMO) d Rj (hp jθl,r ) (BMO) d To check that, we write h l = (h 1,l,..., h d,l ) with h k,l = δ k,l h we consider ψ D d with ψ dx = 0. We have j Z S j( ψ) 1 < + hp j θ l,r h θ thus we get by dominated convergence that 13

14 lim ψ. Sj (hpj θ l,r ) dx = S jψ. hl dx. R + j Z j Z j Z S j is a matrix of Calderón Zygmund operators T which satisfy T (1) = 0, hence map H 1 to H 1, so that we find j Z S j ψ. hl dx C h BMO ψ H 1 thus lim R + j Z S j (hp j θ l,r ) (BMO) d. Similar estimates prove that lim R + ψ.r j (hp j θl,r ) dx = j Z R j ψ. hl dx. j Z R j ψ. hl dx C h BMO ψ H 1 so that lim R + j Z R j (hp j θ l,r ) (BMO) d. On the other h, we have ψ.pj (hs jθl,r ) dx C h P jψ 1 S jθl,r C ψ h min(1, 2 j ) min( θ, 2 j R 1 θ ) = O(R 1/2 ) so that lim R + j Z P j(hs j θl,r ) = 0. Similarly, we have lim R + j Z P j (hr j θ l,r ) = 0. Thus, we have proved that A B map X d 0 CMO to X d 0, thus that A B map X d 0 Y d to H 1. We still have to deal with C( f, g) = j Z R j f.s j g. We write C( f, g) = j Z k Z d 1 η 2 d 1 l Z d 1 ɛ (d 1)(2 d 1) g γ η,j,k f α ɛ,j,l α ɛ,j,l. γ η,j,k We have α ɛ,j,l. γ η,j,k = 0 for k l > 2M α ɛ,j,l. γ η,j,k H 1 C for k l 2M. Thus, we are lead to prove that : j Z k Z d 1 η 2 d 1 l k 2M 1 ɛ (d 1)(2 d 1) g γ η,j,k f α ɛ,j,l C f X d 0 g Y d. For 1 η 2 d 1, 1 ɛ (d 1)(2 d 1) r ZZ d with r 2M, we consider J a finite subset of ZZ ZZ d for ɛ J = (ɛ j,k ) (j,k) J { 1, 1} J T ɛj the operator T ɛj ( f) = (j,k) J ɛ j,k f α ɛ,j,k+r γ η,j,k 14

15 Using again the T (1) theorem, we see that T ɛj CZO C, so that T ɛj ( f) X d 0 T ɛj ( f). g dx = (j,k) J ɛ j,k f α ɛ,j,k+r g γ η,j,k C f X d 0 g Y d Now, it is enough to choose ɛ j,k as the sign of f α ɛ,j,k+r g γ η,j,k we may conclude. Thus, Theorem 2 has been proved. 4. Examples. We now give some examples of Calderón Zygmund pairs of Banach spaces (according to Defiinition 2) : a) Lebesgue spaces : X = X 0 = L p Y = Y 0 = L q with 1 < p < + 1/p + 1/q = 1. b) Lorentz spaces : X = X 0 = L p,r Y = L q,ρ with 1 < p < +, 1 r < +, 1/p + 1/q = 1 1/r + 1/ρ = 1. c) Weighted Lebesgue spaces : X = X 0 = L p (w dx) Y = Y 0 = L q (w 1 p 1 dx) with 1 < p < + 1/p + 1/q = 1, when the weight w belongs to the Muckenhoupt class A p. d) Morrey spaces : We consider the Morrey space L α,p defined by f L α,p sup RQ( α 1 f(x) p dx) 1/p < Q Q Q Q We are interested in the set of parameters 1 < p < + 0 < α d/p. The Zorko space L α,p 0 is the closure of D in L α,p. Adams Xiao [ADAX 11] have proved that L α,p is the bidual of L α,p 0 : H α,q = (L α,p 0 ) L α,p = (H α,q ) with 1/p + 1/q = 1. One characterization of H α,p is the following one : f H α,q if only if there is a sequence (λ n ) n IN l 1 a sequence of functions f n of cubes Q n such that f n L q, f n is supported in Q n f n q R α+d/q d Q n. The norm f H α,q is then equivalent to inf (λn ),(f n ),f= λ n f n n IN λ n. Our Calderón Zygmund pair is then X = L α,p Y = Y 0 = H α,q with 1 < p < +, 0 < α d/p 1/p + 1/q = 1. e) Multipliers spaces : We can build new examples from the former ones. Indeed, let X be a Banach space such that i) we have the continuous embeddings : X 1 X X 2 for some Calderón Zygmund pairs of Banach spaces (X 1, Y 1 ) (X 2, Y 2 ) 15

16 iii) There is a Banach space A such that D is dense in A the dual space A of A coincides with X with equivalence of norms iii) Every Calderón Zygmund operator may be extended as a bounded operator on X : T (f) X C T CZO f X. Then, if X 0 is the closure of D in X Y = X 0, (X, Y ) is a Calderón Zygmund pairs of Banach space ( A = Y 0 ). This is easy to prove. First, let notice that every Calderón Zygmund operator can be extended on X 2, hence de defined on X; the extra information is that it is bounded from X to X. Moreover, we have D X 1,0 X 0 with continuous embeddings, so that t every Calderón Zygmund operator maps X 0 to X 0, hence by duality maps Y to Y. Moreover, from X 1,0 X 0 X 2,0, we get Y 2 Y Y 1. We will conclude if we prove A = Y 0 ; but we see easily (since truncate convolution operators are Calderón-Zygmund operators) that X 0 is *-weakly dense in X that A is embedded into Y with equivalence of norms (due to hahn Banach theorem). Thus, A = Y 0. We may apply this to the space X = X s,p of pointwise multipliers from potential space Ḣs p (1 < p < +, 0 < s < d/p) : i) we have the continuous embeddings for p 1 > p : L s,p 1 X s,p L s,p (Fefferman-Phong inequality) [FEF 83] iii) X s,p is the dual space of Y s,q defined by : f Y s,q if only if there is a sequence (λ n ) n IN l 1 a sequence of functions f n g n with f n Ḣs p, g n L q, f n Ḣs p 1 g n q 1. The norm f Y s,q is then equivalent to inf (λn ),(f n ),(g n ),f= λ n f n g n n IN λ n. iii) Every Calderón Zygmund operator may be extended as a bounded operator on X : T (f) X C T CZO f X. This is due to a theorem of Verbitsky [MAZV 95]. References. [ADAX 11] ADAMS, D.R., XIAO, J., Morrey spaces in harmonic analysis. Ark. Mat., to appear, doi: /s , 30 pp, March 2011 [BAT 87] BATTLE, G., A block spin construction of ondelettes, Part I: Lemarié functions, Comm. Math. Phys. 110 (1987), pp [BATF 95] BATTLE, G. FEDERBUSH, P., Divergence free vector wavelets, Michigan Math. J. 40 (1995), pp [BOU 02] BOURDAUD, G., Remarques sur certains sous-espaces de BMO(IR n ) et de bmo(ir n ), Ann. Inst. Fourier 52 (2002), pp [COHDF 92] COHEN, A., DAUBECHIES, I. FEAUVEAU, J.C., Biorthogonal bases of compactly supported wavelets, Comm. Pure & Appl. Math 45 (1992), pp [COILMS 92] COIFMAN, R.R., LIONS, P.L., MEYER, Y., SEMMES, S., Compensated compactness Hardy spaces, J. Math. Pures et Appl. 72 (1992), pp [COIW 77] COIFMAN, R.R. WEISS, G., Extension of the Hardy spaces their use in analysis, Bull. Amer. Soc. Math. 83 (1977), pp

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