We show that the Hardy space of divergence-free vector fields on R 3 has a divergence-free atomic decomposition, and thus we characterize its dual as a variant of BM O. Using the duality result we prove a "div-curl" type theorem: for b in L 2 loc (R 3 , R 3), sup b · (∇u × ∇v) dx is equivalent to a BM O-type norm of b, where the supremum is taken over all u, v ∈ W 1,2 (R 3) with ∇u L 2 , ∇v L 2 ≤ 1. This theorem is used to obtain some coercivity results for quadratic forms which arise in the linearization of polyconvex variational integrals studied in nonlinear elasticity. In addition, we introduce Hardy spaces of exact forms on R N , study their atomic decompositions and dual spaces, and establish "div-curl" type theorems on R N. 1. Statement of the main theorem In this paper we show that divergence-free vector fields in the Hardy space H 1 (R 3 , R 3) can be decomposed as a sum of divergence-free atoms. This enables us to characterize the dual space of the divergence-free Hardy space H 1 div (R 3 , R 3) as a new variant of BM O. In turn, we obtain a "div-curl" type theorem which generalizes a result proved by Coifman, Lions, Meyer and Semmes in [CLMS]. Our paper answers questions in two separate but closely related areas of analysis: harmonic analysis, and elliptic systems of partial differential equations. In particular we obtain applications to coercivity problems for the class of linear elliptic systems under the polyconvex condition used in the study of nonlinear elasticity by Ball in [B]. The theorem of "div-curl" type which we prove, concerns an estimate for qua-dratic forms on R 3 in terms of a variant of BM O in the following way. Theorem 4.1. Let b ∈ L 2 loc (R 3 , R 3). Then (1.1) sup u,v∈W R 3 b · (∇u × ∇v) dx ∼ ∼b BMO d (R 3 ,R 3) , where W = {w ∈ W 1,2 (R 3) : ∇w L 2 (R 3 ,R 3) ≤ 1} and (1.2) b BMO d (R 3 ,R 3) = sup B⊂R 3 inf g 1 |B| B |b − g| 2 dx

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In this note, we study the boundedness of integral operators $I_{g}$ and $T_{g}$ on analytic Morrey spaces. Furthermore, the norm and essential norm of those operators are given

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Abstract. We show that the Hardy space of divergence-free vector fields on R 3 has a divergence-free atomic decomposition, and thus we characterize its dual as a variant of BM O. Using the duality result we prove a "div-curl" type theorem:This theorem is used to obtain some coercivity results for quadratic forms which arise in the linearization of polyconvex variational integrals studied in nonlinear elasticity. In addition, we introduce Hardy spaces of exact forms on R N , study their atomic decompositions and dual spaces, and establish "div-curl" type theorems on R N . Statement of the main theoremIn this paper we show that divergence-free vector fields in the Hardy space H 1 (R 3 , R 3 ) can be decomposed as a sum of divergence-free atoms. This enables us to characterize the dual space of the divergence-free Hardy space H 1 div (R 3 , R 3 ) as a new variant of BM O. In turn, we obtain a "div-curl" type theorem which generalizes a result proved by Coifman, Lions, Meyer and Semmes in [CLMS].Our paper answers questions in two separate but closely related areas of analysis: harmonic analysis, and elliptic systems of partial differential equations. In particular we obtain applications to coercivity problems for the class of linear elliptic systems under the polyconvex condition used in the study of nonlinear elasticity by Ball in [B].The theorem of "div-curl" type which we prove, concerns an estimate for quadratic forms on R 3 in terms of a variant of BM O in the following way.where W = {w ∈ W 1,2 (R 3 ) : ∇w L 2 (R 3 ,R 3 ) ≤ 1} and