Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems
type: Item
instance of: scholarly article
title: Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems
zbMATH Open document ID: 1263.35080
DOI: 10.4171/RMI/718
author: Xuan Thinh Duong
author: Dorina Mitrea
author: Marius Mitrea
author: Lixin Yan
author: Steve Hofmann
published in: Revista Matemática Iberoamericana
publication date: 2013-03-11T00:00:00Z
review text: Summary: This article has three aims. First, we study Hardy spaces, \(h^p_L(\Omega )\), associated with an operator \(L\) which is either the Dirichlet Laplacian \(\Delta _{D}\) or the Neumann Laplacian \(\Delta _{N}\) on a bounded Lipschitz domain \(\Omega\) in \({\mathbb{R}}^n\), for \(0 < p \leq 1\). We obtain equivalent characterizations of these function spaces in terms of maximal functions and atomic decompositions. Second, we establish regularity results for the Green operators, regarded as the inverses of the Dirichlet and Neumann Laplacians, in the context of Hardy spaces associated with these operators on a bounded semiconvex domain \(\Omega\) in \({\mathbb{R}}^n\). Third, we study relations between the Hardy spaces associated with operators and the standard Hardy spaces \(h^p_r(\Omega )\) and \(h^p_z(\Omega )\), then establish regularity of the Green operators for the Dirichlet problem on a bounded semiconvex domain \(\Omega\) in \({\mathbb{R}}^n\), and for the Neumann problem on a bounded convex domain \(\Omega\) in \({\mathbb{R}}^n\), in the context of the standard Hardy spaces \(h^p_r(\Omega)\) and \(h^p_z(\Omega)\). This gives a new solution to the conjecture made by \textit{D.-C. Chang, S. Krantz} and \textit{E. M. Stein} [in: Complex analysis, Proc. Symp., Madison/WI (USA) 1992, Contemp. Math. 137, 119--131 (1992; Zbl 0765.42014)] regarding the regularity of Green operators for the Dirichlet and Neumann problems on \(h^p_r(\Omega)\) and \(h^p_z(\Omega)\), respectively, for all \(\frac{n}{n+1} < p\leq 1\).
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35J08
zbMATH DE Number: 6143292
zbMATH Keywords: Hardy space
zbMATH Keywords: heat semigroup
zbMATH Keywords: atom
zbMATH Keywords: inhomogeneous Dirichlet and Neumann problems
zbMATH Keywords: Green operator
zbMATH Keywords: semiconvex domain
zbMATH Keywords: convex domain
MaRDI profile type: Publication
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Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems (cites work)
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