Recent advances in Bergman type projections in bounded pseudoconvex and tubular domains

TL;DR

This survey consolidates recent advances in Bergman type projections across various complex domains, extending classical results and exploring their applications in complex function theory with new problems posed.

Contribution

It provides new extensions of classical Bergman projection results to higher dimensions and complex domains, including some results applicable even to the simplest unit disk and polydisk.

Findings

Extended classical results to higher-dimensional domains

Identified new applications in complex function theory

Formulated new open problems in the area

Abstract

The intention of this survey to collect in one paper many recent results and advances related with Bergman type projection acting in various spaces of analytic functions in several complex variables in the unit ball, tubular domains over symmetric cones and bounded strongly pseudoconvex domains between function spaces of different dimensions. Various new interesting extentions of old, classical results on Bergman projections will be provided in our survey. Previously all these results were given in various papers of the first author. Bergman type projections have many nice applications in 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 provided earlier by E.Stein, D.Bekolle, D.Debertol, B.F.Sehba, W.S.Cohn, C.Nana, L.Chen, Sh.Zhang and others which relate function spaces with the same dimension. Practically all results on Bergman type projections acting between function spaces with different dimension may be valid both in context of tubular and pseudoconvex domains with similar proofs though we formulate some results only in tube or pseudoconvex domains. Some our results are new even in case of simplest unit disk and polydisk. Problems which are related with Bergman type projection acting between various analytic and measurable function spaces of different dimensions in various complicated domains in C^n are as far as we know completely new and may have various new interesting applications in compex function theory. In this paper various new interesting problems in this research area will also be formulated and posed by authors. Our results may be valid also with very similar proofs in various Siegel domains of second type, matrix domains, bounded symmetric domains and various other domains in C^n with complicated structure.