Boundedness of Bergman projectors on homogeneous Siegel domains

Abstract

In this paper we study the boundedness of Bergman projectors on weighted Bergman spaces on homogeneous Siegel domains of Type II. As it appeared to be a natural approach in the special case of tube domains over irreducible symmetric cones, we study such boundedness on the scale of mixed-norm weighted Lebesgue spaces. The sharp range for the boundedness of such operators is essentially known only in the case of tube domains over Lorentz cones.In this paper we prove that the boundedness of such Bergman projectors is equivalent to variuos notions of atomic decomposition, duality, and characterization of boundary values of the mixed-norm weighted Bergman spaces, extending results moslty known only in the case of tube domains over irreducible symmetric cones. Some of our results are new even in the latter simpler context.We also study the simpler, but still quite interesting, case of the "positive" Bergman projectors, the integral operator in which the Bergman kernel is replaced by its absolute value. We provide a useful characterization which was previously known for tube domains. s = {0} if s < 0, and also if s = 0 and q < ∞. In addition, A p,∞ 0 is the classical Hardy space H p (C + ). In order to avoid trivialities, from now on we shall assume s > 0. In addition, for simplicity, in this introductory section we shall generally limit ourselves to the case p, q 1. Then, the