Besov spaces and the boundedness of weighted Bergman projections over symmetric tube domains

Abstract

We extend the analysis of weighted Bergman spaces Ap;q/s on symmetric tube domains, contained in [2], to the case where the weights are positive powers [formula] of the principal minors [Delta]1,...,[Delta]r on the symmetric cone [omega]. We discuss the realization of the boundary distributions of functions in Ap;q/s in terms of Besov-type spaces Bp;q/s adapted to the structure of the cone. We give a necessary and a sufficient condition on the values of p, q and s for which this identification between Ap;q/s and Bp;q/s holds. We also present a continuous version of thesse latter spaces which is new even for the case s1 = ... = s1 considered in [2]. We use these results to discuss multipliers between Besov spaces and the boundedness of the weighted Bergman projection Ps: Lp;q/s --> Ap;q/s. The situation in the rank two case is specifically dealt with.