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Boundary Hölder and $L^p$ estimates for local solutions of the tangential Cauchy-Riemann equation


Authors: Christine Laurent-Thiébaut and Mei-Chi Shaw
Journal: Trans. Amer. Math. Soc. 357 (2005), 151-177
MSC (1991): Primary 32F20, 32F10, 32F40
DOI: https://doi.org/10.1090/S0002-9947-04-03677-3
Published electronically: July 22, 2004
MathSciNet review: 2098090
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Abstract: We study the local solvability of the tangential Cauchy-Riemann equation on an open neighborhood $\omega$ of a point $z_0\in M$ when $M$ is a generic $q$-concave $CR$ manifold of real codimension $k$ in $\mathbb{C} ^n$, where $1\le k\le n-1$. Our method is to first derive a homotopy formula for $\overline\partial_b$ in $\omega$ when $\omega$ is the intersection of $M$ with a strongly pseudoconvex domain. The homotopy formula gives a local solution operator for any $\overline\partial_b$-closed form on $\omega$ without shrinking. We obtain Hölder and $L^p$ estimates up to the boundary for the solution operator.

RÉSUMÉ. Nous étudions la résolubilité locale de l'opérateur de Cauchy- Riemann tangentiel sur un voisinage $\omega$ d'un point $z_0$d'une sous-variété $CR$ générique $q$-concave $M$ de codimension quelconque de $\mathbb C^n$. Nous construisons une formule d'homotopie pour le $\overline\partial_b$ sur $\omega$, lorsque $\omega$ est l'intersection de $M$ et d'un domaine strictement pseudoconvexe. Nous obtenons ainsi un opérateur de résolution pour toute forme $\overline\partial_b$-fermée sur $\omega$. Nous en déduisons des estimations $L^p$ et des estimations hölderiennes jusqu'au bord pour la solution de l'équation de Cauchy-Riemann tangentielle sur $\omega$.


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Additional Information

Christine Laurent-Thiébaut
Affiliation: Université de Grenoble, Institut Fourier, UMR 5582 CNRS/UJF, BP 74, 38402 St Martin d’Hères Cedex, France
Email: Christine.Laurent@ujf-grenoble.fr

Mei-Chi Shaw
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: mei-chi.shaw.1@nd.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03677-3
Keywords: CR manifolds, H\"older estimates, $L^p$-estimates, tangential Cauchy Riemann equation
Received by editor(s): May 28, 2003
Published electronically: July 22, 2004
Additional Notes: The second author was supported by NSF grant DMS01-00492
Article copyright: © Copyright 2004 American Mathematical Society

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