Littlewood-Paley decompositions related to symmetric cones and Bergman projections in tube domains

Let \Omega be an irreducible symmetric cone in a Euclidean vector space V of dimension n, endowed with an inner product for which the cone is self-dual. We denote by T_\Omega= V + i\Omega the corresponding tube domain in the complexification of V. The goal of this paper is to present, in the general setting of symmetric cones, a special Littlewood-Paley decomposition adapted to the geometry of \Omega. This will be applied to analytic problems, such as the boundedness of Bergman projectors and the characterization of boundary values for Bergman spaces in the tube domain T_\Omega. This theory is applied to two open problems: (1) the characterization of boundary values of functions in the weighted Bergman spaces A^{p,q}_\nu as distributions in the Besov spaces B^{p,q}_\nu; (2) the boundedness of Bergman projectors P in L^{p,q}_\nu spaces, where P_\nu is the orthogonal projection from L^2_\nu onto A^2_\nu.