Electronic ISBN:  9781470405007 
Product Code:  MEMO/191/894.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 191; 2008; 78 ppMSC: Primary 42; 35; Secondary 58;
The author studies Hardy spaces on \(C^1\) and Lipschitz domains in Riemannian manifolds. Hardy spaces, originally introduced in 1920 in complex analysis setting, are invaluable tool in harmonic analysis. For this reason these spaces have been studied extensively by many authors.
The main result is an equivalence theorem proving that the definition of Hardy spaces by conjugate harmonic functions is equivalent to the atomic definition of these spaces. The author establishes this theorem in any dimension if the domain is \(C^1\), in case of a Lipschitz domain the result holds if dim \(M\le 3\). The remaining cases for Lipschitz domains remain open. This result is a nontrivial generalization of flat (\({\mathbb R}^n\)) equivalence theorems due to Fefferman, Stein, Dahlberg and others.
The material presented here required to develop potential theory approach for \(C^1\) domains on Riemannian manifolds in the spirit of earlier works by Fabes, Jodeit and Rivière and recent results by Mitrea and Taylor. In particular, the first part of this work is of interest in itself, since the author considers the boundary value problems for the LaplaceBeltrami operator. He proves that both Dirichlet and Neumann problem for LaplaceBeltrami equation are solvable for any given boundary data in \(L^p(\partial\Omega)\), where \(1<p<\infty\). The same remains true in Hardy spaces \(\hbar^p(\partial\Omega)\) for \((n1)/n<p\le 1\).
In the whole work the author works with Riemannian metric \(g\) with smallest possible regularity. In particular, mentioned results for the LaplaceBeltrami equation require Hölder class regularity of the metric tensor; the equivalence theorem requires \(g\) in \(C^{1,1}\). 
Table of Contents

Chapters

0. Introduction

1. Background and definitions

2. The boundary layer potentials

3. The Dirichlet problem

4. The Neumann problem

5. Compactness of layer potentials, part II; the Dirichlet regularity problem

6. The equivalence of Hardy space definitions


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The author studies Hardy spaces on \(C^1\) and Lipschitz domains in Riemannian manifolds. Hardy spaces, originally introduced in 1920 in complex analysis setting, are invaluable tool in harmonic analysis. For this reason these spaces have been studied extensively by many authors.
The main result is an equivalence theorem proving that the definition of Hardy spaces by conjugate harmonic functions is equivalent to the atomic definition of these spaces. The author establishes this theorem in any dimension if the domain is \(C^1\), in case of a Lipschitz domain the result holds if dim \(M\le 3\). The remaining cases for Lipschitz domains remain open. This result is a nontrivial generalization of flat (\({\mathbb R}^n\)) equivalence theorems due to Fefferman, Stein, Dahlberg and others.
The material presented here required to develop potential theory approach for \(C^1\) domains on Riemannian manifolds in the spirit of earlier works by Fabes, Jodeit and Rivière and recent results by Mitrea and Taylor. In particular, the first part of this work is of interest in itself, since the author considers the boundary value problems for the LaplaceBeltrami operator. He proves that both Dirichlet and Neumann problem for LaplaceBeltrami equation are solvable for any given boundary data in \(L^p(\partial\Omega)\), where \(1<p<\infty\). The same remains true in Hardy spaces \(\hbar^p(\partial\Omega)\) for \((n1)/n<p\le 1\).
In the whole work the author works with Riemannian metric \(g\) with smallest possible regularity. In particular, mentioned results for the LaplaceBeltrami equation require Hölder class regularity of the metric tensor; the equivalence theorem requires \(g\) in \(C^{1,1}\).

Chapters

0. Introduction

1. Background and definitions

2. The boundary layer potentials

3. The Dirichlet problem

4. The Neumann problem

5. Compactness of layer potentials, part II; the Dirichlet regularity problem

6. The equivalence of Hardy space definitions