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On a smooth domain Ω ⊂⊂ C n , we consider the Dirichlet problem for the complex Monge-Ampère equation ((dd c u) n = f dV, u| bΩ ≡ ϕ). We state the Hölder regularity of the solution u when the boundary value ϕ is Hölder continuous and the density f is only

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Abstract. The goal of this paper is to develop a theory of nonsmooth analytic discs attached to domains with Lipschitz boundary in real submanifolds of C n . We then apply this technique to establish a propagation principle for wedge extendibility of CR-functions on these domains along CR-curves and along boundaries of attached analytic discs. The technique from this paper has been also extensively used by the authors recently to obtain sharp results on wedge extension of CR-functions on wedges in prescribed directions extending results of Boggess-Polking and Eastwood-Graham.

This note is aimed at simplifying current literature about compactness estimates for the Kohn-Laplacian on CR manifolds. The approach consists in a tangential basic estimate in the formulation given by the first author in [Kh10] which refines former work by Nicoara [N06]. It has been proved by Raich [R10] that on a CR manifold of dimension 2n − 1 which is compact pseudoconvex of hypersurface type embedded in C n and orientable, the property named "(CR − P q )" for 1 ≤ q ≤ n−1 2 , a generalization of the one introduced by Catlin in [C84], implies compactness estimates for the Kohn-Laplacian b in degree k for any k satisfying q ≤ k ≤ n − 1 − q. The same result is stated by Straube in [S10] without the assumption of orientability. We regain these results by a simplified method and extend the conclusions in two directions. First, the CR manifold is no longer required to be embedded. Second, when (CR − P q ) holds for q = 1 (and, in case n = 1, under the additional hypothesis that∂ b has closed range on functions) we prove compactness also in the critical degrees k = 0 and k = n − 1. MSC: 32F10, 32F20, 32N15, 32T25

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We describe along the guidelines of Kohn [11], the constant E s which is needed to control the commutator of a totally real vector field T E with∂ * in order to have H s a-priori estimates for the Bergman projection B k , k ≥ q − 1, on a smooth qpseudoconvex domain D ⊂⊂ C n . This statement, not explicit in [11], yields regularity results for B k in specific Sobolev degree s. Next, we refine the pseudodifferential calculus at the boundary in order to relate, for a defining function r of D, the operators (T + ) − δ 2 and (−r) δ 2 . We are thus able to extend to general degree k ≥ 0 of B k , the conclusion of [11] which only holds for q = 1 and k = 0: if for the Diederich-Fornaess index δ of D, we have (1 − δ) 1 2 ≤ E s , then B k is H s -regular. MSC: 32F10, 32F20, 32N15, 32T25