On the Hardy Space Theory of Compensated Compactness Quantities

2016/11/30 by Sauli Lindberg · 17 citations
Mathematics · #Advanced Harmonic Analysis Research #Class (philosophy) #Combinatorics #Compact space #Computer science #Discrete mathematics #Hardy space #Holomorphic and Operator Theory #Jacobian matrix and determinant #Mathematical Analysis and Transform Methods #Mathematics #Operator (biology) #Physics #Pure mathematics #Space (punctuation) #math.CA #math.FA

paper · pdf · doi:10.1007/s00205-017-1087-2

published in Archive for Rational Mechanics and Analysis 224(2), 709-742 (Springer Science+Business Media) · 29 pages; added one reference and an acknowledgement, changed an inequality on p. 27 into a two-sided estimate

arxiv created 2016/12/05 · openalex publication_date 2017/02/07 · arxiv updated 2017/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 Rⁿ,\mathbb Rⁿ) onto the Hardy space H¹(\mathbb Rⁿ) for any n ≥ 2. The related question about surjectivity of J \colon W^1,n(\mathbb Rⁿ,\mathbb Rⁿ) → H¹(\mathbb Rⁿ) is still open. The second main result and its variants reduce the proof of H¹ 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 H¹ regularity, shedding light on another problem of Coifman, Lions, Meyer, and Semmes.

Citations

• Harmonic Maps, Conservation Laws and Moving Frames
• An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs
• Hp spaces of several variables

Cited by

• Sharp commutator estimates via harmonic extensions