# L'équation de Cauchy-Riemann avec conditions de support dans des domaines à bords Levi-dégénérés

Abstract : In the first part, we consider a domain $\Omega$ with Lipschitz boundary, is relatively compact in an $n$-dimensional Kähler manifold, which satisfies some log \delta-pseudoconvexity'' conditions. We show that the $\overline\partial$ -equation with exact support in $\Omega$ admits a solution in bidegrees $(p,q)$, $1\leq q\leq n-1$. Moreover, the range of $\overline\partial$ acting on smooth $(p,n-1)$-forms with support in $\overline\partial$ is closed. Applications are given to the solvability of the tangential Cauchy-Riemann equations for smooth forms and currents for all intermediate bidegrees on boundaries of weakly pseudoconvex domains in Stein manifolds and to the solvability of the tangential Cauchy-Riemann equations for currents on Levi-flat $CR$ manifolds of arbitrary codimension. In a second part, we study the $\overline\partial$ -equation with zero Cauchy data along a hypersurface with constant signature. Applications to the solvability of the tangential Cauchy-Riemann equations for smooth forms with compact support and currents on the hypersurface are given. In particular the Hartogs phenomenon holds in weakly 2-convex-concave hypersurfaces with constant signature in Stein manifolds.
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Document type :
Theses
Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2002. Français
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https://tel.archives-ouvertes.fr/tel-00001662
Contributor : Arlette Guttin-Lombard <>
Submitted on : Thursday, September 5, 2002 - 3:08:24 PM
Last modification on : Thursday, September 5, 2002 - 3:08:24 PM
Document(s) archivé(s) le : Tuesday, September 11, 2012 - 5:55:17 PM

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• HAL Id : tel-00001662, version 1

### Citation

Judith Brinkschulte. L'équation de Cauchy-Riemann avec conditions de support dans des domaines à bords Levi-dégénérés. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2002. Français. <tel-00001662>

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