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

Th\'eorie $L^p$ pour l'\'equation de Cauchy-Riemann