Estimates for the Bergman and Szegö Projections for Pseudoconvex Domains of Finite Type with Locally Diagonalizable Levi Form

This paper deals with precise mapping properties of the Bergman and Szegö projections of pseudo-convex domains of finite type in C whose Levi form are locally diagonalizable at every point of the boundary (see Section 2 for a precise definition). We obtain sharp estimates for these operators for usual Lpk Sobolev spaces, classical Lipschitz spaces Λα and nonisotropic Lipschitz spaces Γα related to the geometry of the domain. Our results, in the present paper, are analog to those obtained for convex domains of finite type in [MS94] and [MS97] and extend previously known results for the strictly pseudoconvex case ([AS79] and [PS77]), for the finite type domains of C ([NRSW89], see also [Chr88] and [FK88]) and in the case of pseudoconvex domains of finite type of C having a Levi form of rank n− 1 ([AC99], see also [Mac88]). Similar results where obtained for pseudoconvex domains in C whose Levi form have comparable eigenvalues (see [Koe02], [Cho02b] and [Cho03]).