Hardy space of exact forms on $\mathbb {R}^N$

Abstract

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

Atomic decomposition and weak factorization in generalized Hardy spaces of closed forms

“…The index d means that we restrict to closed forms. This may be seen as an H 1 − BMO version of the generalized div-curl lemmas for wedge products of closed forms that have been obtained by Lou and McIntosh (see [19]): they prove that, for L 2 data the wedge product has its coefficients in the Hardy space H 1 (R n ). A weaker H 1 −BMO statement had been obtained in [2].…”

“…We now prove the converse part of Theorem 2.12. The proof follows the lines of [19] but requires to be careful since we do not deal anymore with integrable functions. Let ϕ ∈ C ∞ (R n ) be a radial real-valued function, supported in the unit ball B centered at 0, and satisfying This implies that, for k = 1, · · · , n and for any multi-index γ with |γ| ≤ s + 1,…”

“…Remark 2. 19. In order to decompose f ∈ H ℘ d (R n , Λ ℓ ) it is also possible to copy the proof given for Hardy spaces of Musielak-Orlicz type of holomorphic functions [3].…”

“…which we wanted to prove. Details are given in [19], where the particular case ℘(x, t) = t is considered.…”

“…Proof. The proof follows the same scheme as in [19] but requires first to build adapted functions of bounded mean oscillation. We consider separately small balls or balls that are far from the origin on one hand, large balls that contain the origin or are close to contain it, on the other hand.…”