# Hardy Spaces and Potential Theory on \(C^1\) Domains in Riemannian Manifolds

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*Martin Dindoš*

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 Laplace-Beltrami operator. He proves that
both Dirichlet and Neumann problem for Laplace-Beltrami 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 \((n-1)/n<p\le
1\).

In the whole work the author works with Riemannian metric
\(g\) with smallest possible regularity. In particular,
mentioned results for the Laplace-Beltrami equation require
Hölder class regularity of the metric tensor; the equivalence
theorem requires \(g\) in \(C^{1,1}\).

#### Table of Contents

# Table of Contents

## Hardy Spaces and Potential Theory on $C^{1}$ Domains in Riemannian Manifolds

- Contents v6 free
- Abstract vi7 free
- Chapter 0. Introduction 18 free
- Chapter 1. Background and Definitions 411 free
- Chapter 2. The Boundary Layer Potentials 916
- Chapter 3. The Dirichlet problem 2128
- Chapter 4. The Neumann problem 2734
- Chapter 5. Compactness of Layer Potentials, Part II; The Dirichlet regularity problem 3138
- §5.1. Preliminaries 3138
- §5.2. Compactness and invertibihty of K on Sobolev space H[sup(1,p)] 3340
- §5.3. Compactness and invertibihty of K on Hardy-Sobolev space H[sup(1,p)] 3845
- §5.4. Dirichlet regularity problem, Sobolev H[sup(1,p)] (1 < p < ∞) data 4148
- §5.5. Dirichlet regularity problem, H[sup(1,p)] (( n 1) / n < p ≤ 1) data 4249

- Chapter 6. The equivalence of Hardy space definitions 4451
- Appendix A. Variable Coefficient Cauchy Integrals 5360
- Appendix B. One Result on the Maximal Operator 6572
- Bibliography 7784