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Littlewood-Paley decompositions related to symmetric cones

We obtain a Whitney decomposition of a symmetric cone Ω, analog to that of the positive real line into dyadic intervals [2j , 2j+1). This gives a natural tool for developing a Littlewood-Paley theory for spaces of functions with spectrum in Ω. Such functions extend into holomorphic functions on the tube TΩ. We consider here the mixed norm Bergman spaces Ap,2 ν (TΩ), for which we find a Littlewood-Paley characterization. As a consequence, we obtain optimal results for the boundedness of the Bergman projector Pν in Lp,2 ν (TΩ). When the projector is unbounded, a precise description of Pν(L ν ) is also given, as a space of equivalence classes of holomorphic functions in relation with the dual of Ap ′,2 ν (TΩ).