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Hardy and Hardy-Sobolev spaces on strongly Lipschitz domains and some applications. (English) Zbl 1354.42039

Summary: Let \(\Omega\subset \mathbb R^{n}\) be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, \(H^{p}_{r} (\Omega)\) and \(H^{p}_{z} (\Omega)\), and Hardy-Sobolev spaces, \(H^{1,p}_{r} (\Omega)\) and \(H^{1,p}_{z,0} (\Omega)\) on \(\Omega\), for \(p\in (\frac{n}{n+1}, 1]\). The authors establish grand maximal function characterizations of these spaces. As applications, the authors obtain some div-curl lemmas in these settings and, when \(\Omega\) is a bounded Lipschitz domain, the authors prove that the divergence equation \(\mathrm{div}\mathbf u = f\) for \(f\in H^{p}_{z} (\Omega)\) is solvable in \(H^{1,p}_{z,0} (\Omega)\) with suitable regularity estimates.

MSC:

42B30 \(H^p\)-spaces
42B37 Harmonic analysis and PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B25 Maximal functions, Littlewood-Paley theory
35F15 Boundary value problems for linear first-order PDEs
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