Chen, Xiaming; Jiang, Renjin; Yang, Dachun Hardy and Hardy-Sobolev spaces on strongly Lipschitz domains and some applications. (English) Zbl 1354.42039 Anal. Geom. Metr. Spaces 4, 336-362 (2016). 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. Cited in 5 Documents 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 Keywords:Hardy space; Hardy-Sobolev space; grand maximal function; div-curl formula; divergence equation PDFBibTeX XMLCite \textit{X. Chen} et al., Anal. Geom. Metr. Spaces 4, 336--362 (2016; Zbl 1354.42039) Full Text: DOI References: [1] [1] G. Acosta, R. G. Durán, M. A. Muschietti, Solution of the divergence operator on John domains, Adv. Math. 206 (2006), 373-401.; · Zbl 1142.35008 [2] P. Auscher, E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of Rn, J. Funct. Anal. 201 (2003), 148-184.; · Zbl 1033.42019 [3] P. Auscher, E. Russ, P. Tchamitchian, Hardy Sobolev spaces on strongly Lipschitz domains of Rn, J. Funct. Anal. 218 (2005), 54-109.; · Zbl 1073.46022 [4] P. Auscher, E. 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