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For certain annuli in $\mathbb{C}^n$, $n\geq 2$, with non-smooth holes, we show that the $\bar{\partial}$-operator from $L^2$ functions to $L^2$ $(0,1)$-forms has closed range. The holes admitted include products of pseudoconvex domains and certain intersections of smoothly bounded pseudoconvex domains. As a consequence, we obtain estimates in the Sobolev space $W^1$ for the $\bar{\partial}$-equation on the non-smooth domains which are the holes of these annuli., Small typos corrected; Final Version; To appear in Indiana University Mathematics Journal

We study the stability of the embeddability of compact 2 2 -concave CR manifolds in complex manifolds under small horizontal perturbations of the CR structure.

We prove sharp anisotropic H\"older estimates for the local solutions of the tangential Cauchy-Riemann equation in q-concave CR manifolds and we derive the same kind of estimates for global solutions when the manifold is compact.

International audience; Using functional analysis, we derive from local homotopy formulas for the tangential Cauchy-Riemann operator a global homotopy formula for compact CR manifolds without loss of regularity.

LetX be a complex manifold with finitely many ends such that each end is either q-concave or (n−q)-convex. If $q< \tfrac{1}{2}n$ , then we prove that Hpn−q(X) is Hausdorff for all p. This is not true in general if $q \geqslant \tfrac{1}{2}n$ (Rossi’s example with n=2 and q=1). If all ends are q-concave, then this is the classical Andreotti-Vesentini separation theorem (and holds also for $q \geqslant \tfrac{1}{2}n$ ). Moreover the result was already known in the case when the q-concave ends can be ‘filled in’ (again also for $q \geqslant \tfrac{1}{2}n$ ). To prove the result we first have to study Serre duality for the case of more general families of supports (instead of the family of all closed sets and the family of all compact sets) which is the main part of the paper. At the end we give an application to the extensibility of CR-forms of bidegree (p, q) from (n−q)-convex boundaries, $q< \tfrac{1}{2}n$ .