Théorie $L^p$ et dualité de Serre pour l’équation de Cauchy-Riemann

Laurent-Thiébaut, Christine

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 2, pp. 251-279.

Dans cet article nous proposons une étude systématique de la théorie $L^p$ pour l’opérateur de Cauchy-Riemann dans les variétés complexes. Dans une première partie nous étudions la théorie $L^p$ locale puis nous développons la théorie d’Andreotti-Grauert dans le cadre $L^p$. La seconde moitié de l’article est consacrée à la dualité de Serre et à ses applications à la résolution avec support exact de l’équation de Cauchy-Riemann dans les espaces $L^p$.

In this paper we propose a systematic study of the Cauchy-Riemann operator in the $L^p$-setting in complex manifolds. We first consider $L_{loc}^p$-theory and then we develop an $L^p$ Andreotti-Grauert theory. In the second half of the paper we consider Serre duality and its applications to the solvability of the Cauchy-Riemann equation with exact support in $L^p$-spaces.

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