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

TL;DR

This survey gathers recent advances in Bergman-type projections acting between analytic function spaces on tubular domains $T_\Omega$ (over symmetric cones) and on bounded strongly pseudoconvex domains, including mixed-norm and Herz-type spaces. It extends classical results of Stein, Bekolle, Debertol, Sehba, Cohn, Nana, Chen, Zhang and others to projections between spaces of the same and different dimensions via operators such as $P_\nu$, $S_{a,b}$, and $T_\beta$. It frames trace problems (X=Y) for Bergman-type projections, develops product-domain and multi-variable extensions, and presents numerous new theorems on boundedness and mapping properties across tube domains, bounded pseudoconvex domains, and related analytic spaces. The work also introduces new analytic function spaces (e.g., Herz-type, BMOA-type) in these domains and outlines open problems with potential applications in complex function theory and higher-dimensional trace theory.

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.

Theorems & Definitions (31)

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