On the Hardy space theory of compensated compactness quantities

Abstract

We make progress on a problem of R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes from 1993 by showing that the Jacobian operator $J$ does not map $W^{1,n}({\mathbb{R}}^n,{\mathbb{R}}^n)$ onto the Hardy space ${\mathcal{H}}^1({\mathbb{R}}^n)$ for any $n\ge 2$. The related question about surjectivity of $J:{\dot{W}}^{1,n}({\mathbb{R}}^n,{\mathbb{R}}^n)\to {\mathcal{H}}^1({\mathbb{R}}^n)$ is still open. The second main result and its variants reduce the proof of ${\mathcal{H}}^1$ regularity of a large class of compensated compactness quantities to an integration by parts or easy arithmetic, and applications are presented. Furthermore, we exhibit a class of nonlinear partial differential operators in which weak sequential continuity is a strictly stronger condition than ${\mathcal{H}}^1$ regularity, shedding light on another problem of Coifman, Lions, Meyer, and Semmes.

arXiv comment29 pages; added one reference and an acknowledgement, changed an inequality on p. 27 into a two-sided estimate

References in corpus (2)

• Sharp commutator estimates via harmonic extensions
• A T(P) theorem for Sobolev spaces on domains

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• The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces
• Energy minimisers with prescribed Jacobian
• Off-diagonal estimates for bilinear commutators
• Lower bound of the parabolic Hilbert commutator
• Weighted $L^p\to L^q$-boundedness of commutators and paraproducts in the Bloom setting
• Off-diagonal estimates for commutators of bi-parameter singular integrals
• A Functional Analytic Perspective to the div-curl Lemma