Lecture Notes on Bergman projectors in tube domains over cones : an analytic and geometric viewpoint

Abstract

In these notes we present in printed form the content of a se ries of lectures given by ve of the authors at the International Workshop in Classical Analysis held in Yaounde in December. Our purpose is to introduce the problem of Lppq boundedness of weighted Bergman projectors on tube domains over symmetric cones and show some of the latest progress obtained in this subject. We begin with a complete description of the situation on the upper halfplane. Next we introduce the geometric machinery necessary to study the problem in higher dimensions. This includes the riemannian structure of symmetric cones the induced Whitney decomposition and the introduction of a wider class of spaces with mixed Lp,q norms. Our main result is the boundedness of the weighted Bergman projector on the weighted mixed norm spaces Lp,q for an appropriate range of indices p q. Finally we conclude by discussing various applications further results and open questions.