NOTES ON RECENT ADVANCES ON BERGMAN-TYPE PROJECTIONS IN FUNCTION SPACES OF SEVERAL VARIABLES

Abstract

The intention of this survey is to collect in a single paper many recent results and advances related to Bergman-type projections acting on various spaces of analytic functions of several complex variables in the unit ball, tubular domains over symmetric cones, and bounded strongly pseudoconvex domains.

Various new and interesting extensions of classical results on Bergman projections are presented in our survey.
All these results were previously obtained  by the first author in various papers. Bergman-type projections have many important applications in the complex function theory of several complex variables in tubular domains over symmetric cones and in bounded strongly pseudoconvex domains. Our results can be seen as direct extensions of previously known results of E. Stein, D. Bekolle, D. Debertol, B. F. Sehba, W. S. Cohn, C. Nana, L. Chen, Sh. Zhang and others, who related function spaces of the same dimension.

In this paper, various new and interesting problems in this research area will also be formulated and posed by the authors. Theorems on Bergman-type projections between analytic function spaces of different dimensions over domains in $\mathbb{C}^n$ may be viewed as a new research area and may have important applications in complex function theory.