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Hardy Spaces and Potential Theory on $$C^1$$ Domains in Riemannian Manifolds
Martin Dindoš, University of Edinburgh, Scotland
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Memoirs of the American Mathematical Society
2008; 78 pp; softcover
Volume: 191
ISBN-10: 0-8218-4043-6
ISBN-13: 978-0-8218-4043-6
List Price: US$65 Individual Members: US$39
Institutional Members: US\$52
Order Code: MEMO/191/894

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 $$C^{1,1}$$.