Hardy spaces of exact forms on Lipschitz domains in R^n

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: 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

Hardy Spaces of Differential Forms on Riemannian Manifolds

“…Our aim is, therefore, to develop an appropriate theory of Hardy spaces on Riemannian manifolds (whether or not compact) and to apply this to the Riesz transform (for us, only the "geometric" Riesz transform matters so that we drop the "s" in transform). This will indeed generalize the theory in [32] to a geometric context. In particular, their Hardy spaces will be our spaces H 1 d (Λ k T * R n ) for 0 ≤ k ≤ n (see Section 8.2.2).…”

“…Conversely, let a = Db be a 1-molecule, B a ball and (χ k ) k≥0 a sequence of C ∞ (M) functions adapted to B, such that (6.2) holds with N = 1. Notice that, for some universal constant C ′ > 0, one has [32], as was claimed in the introduction.…”

“…Moreover, f H 1 (R n ) is comparable with the infimum of |λ k | over all such decompositions. In [32], Lou and the second author establish an atomic decomposition for…”

Hardy-Sobolev Spaces Decomposition in Signal Analysis

“…Hardy-Sobolev spaces in recent years have undergone a new phase of development ( [2], [11]). To the authors' knowledge, no applications in signal analysis have been noted.…”

On the Product of Functions in BMO and H^\text{1}

“…Although it is not immediate from this definition, for sufficiently regular domains, the Hardy space H 1 (Ω) consists of restrictions to Ω of functions in H 1 (R n ) [45], see also [9,43] for Lipschitz domains. Obviously, these are regular distributions; actually we have the inclusion…”