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. The same remains true in Hardy spaces $\hbar^p(\partial\Omega)$ for $(n-1)/n. 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 $C^{1,1}$.

nexusstc/Hardy Spaces and Potential Theory on C1 Domains in Riemannian Manifolds/badaa809eb90270a6393d4c4796b4720.pdf

lgli/DINDOS - Hardy Spaces.pdf

lgrsnf/DINDOS - Hardy Spaces.pdf

zlib/Mathematics/Mathematical Theory/Martin Dindos/Hardy Spaces and Potential Theory on C1 Domains in Riemannian Manifolds_24385953.pdf

Hardy Spaces and Potential Theory on $C^1$ Domains in Riemannian Manifolds (Memoirs of the American Mathematical Society)

Hardy spaces and potential theory on Cp1s domains in Riemannian manifolds

Dindoš, Martin

Martin Dindoš

Memoirs of the American mathematical society -- No 894. vol. 191, Providence (R. I.), United States, 2008

Memoirs of the American Mathematical Society -- no. 894, Providence, RI, Rhode Island, 2008

United States, United States of America

December 28, 2007

producers:
3-Heights(TM) PDF Producer 2.1.15.0 (http://www.pdf-tools.com)

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Includes bibliographical references

Библиогр.: с. 77-78

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<p>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 Riviere 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 $1&lt;p&lt;\infty$. The same remains true in Hardy spaces $\hbar^p(\partial\Omega)$ for $(n-1)/n&lt;p\le 1$. 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 Holder class regularity of the metric tensor; the equivalence theorem requires $g$ in $C^{1,1}$.</p>

2022-12-28