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Résolution avec régularité jusqu'au bord de l'équation de Cauchy-Riemann dans des domaines à coins et de l'équation de Cauchy-Riemann tangentielle en codimension quelconque

Abstract : In this work, we are mainly interested in the study of two classical equations : the Cauchy-Riemann equation on some domains of ${\Bbb C}^n$ and the tangential Cauchy-Riemann equation on some domains in a generic $q$-concave CR manifold. For each equation the first step is to obtain local results with solutions having regularity properties up to the boundary of the domain. In the complex case, the method consists in an explicit construction of a solution to the Cauchy-Riemann equation using integral representation theory. This theory revived anew in the seventies with the work of H. Grauert, G.M. Henkin, I. Lieb and E. Ramirez. Thus, we prove ${\cal C}^k$-estimates for the Cauchy-Riemann equation on local $q$-convex and $q$-concave wedges. In the CR case, local results are deduced from the complex case using some tools of homological algebra and sheaf theory. These methods are coming in particular from works of A. Andreotti, G. Fredericks, C.D. Hill and M. Nacinovich. Then we show local results for the tangential Cauchy-Riemann equation for differential forms smooth up to the boundary of the domain. Next, we use the local results and the Grauert's bumping method to prove some vanishing, finiteness and separation theorems for the cohomology groups.
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https://tel.archives-ouvertes.fr/tel-00002226
Contributor : Arlette Guttin-Lombard <>
Submitted on : Monday, January 6, 2003 - 3:27:33 PM
Last modification on : Wednesday, November 4, 2020 - 1:58:06 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 7:10:32 PM

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  • HAL Id : tel-00002226, version 1

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Hélène Ricard. Résolution avec régularité jusqu'au bord de l'équation de Cauchy-Riemann dans des domaines à coins et de l'équation de Cauchy-Riemann tangentielle en codimension quelconque. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2002. Français. ⟨tel-00002226⟩

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