Lebesgue mixed norm estimates for Bergman projectors: from tube domains over homogeneous cones to homogeneous Siegel domains of type II

Abstract

In this paper, the weight functions are fundamental compound functions of the homogeneous cone. First, we use the recent Bourgain-Demeter $$l^2$$ -decoupling theorem to obtain the optimal weighted Lebesgue mixed norm estimates for weighted Bergman projectors on tube domains over Lorentz cones. This settles a conjecture of D. Debertol. Secondly, we present a transference principle of weighted Lebesgue mixed norm estimates for weighted Bergman projectors from tube domains over homogeneous cones to homogeneous Siegel domains of type II associated to the same cones. So results of C. Nana for homogeneous Siegel domains of type II can be deduced from earlier results of C. Nana and B. Trojan for tube domains over homogeneous cones. Combining our two theorems, we improve these estimates for homogeneous Siegel domains of type II associated with Lorentz cones, e.g. the Pyateckii-Shapiro Siegel domain of type II.