Littlewood-Paley decompositions related to symmetric cones

Abstract

We obtain a Whitney decomposition of a symmetric cone Ω, analogtothatofthepositivereallineintodyadicintervals[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(T ), for which we find νΩ a Littlewood-Paley characterization. As a consequence, we obtain optimal results for the boundedness of the Bergman projector Pν in Lp,2(T ). When the projector is unbounded, a precise descrip- νΩ tion of P (Lp,2) is also given, as a space of equivalence classes of νν holomorphic functions in relation with the dual of Ap′,2(T ).