Sharp restrictions of analytic function spaces of several variables

TL;DR

This paper reviews recent advances in the study of trace problems for analytic function spaces in several complex variables, focusing on tubular and strongly pseudoconvex domains, and presents new results on trace estimates and Bergman projections.

Contribution

It provides new sharp trace estimates for analytic function spaces in complex domains and explores their relation to Bergman projections, expanding understanding in several complex variables.

Findings

New trace theorems for tubular and strongly pseudoconvex domains.

Connections established between trace estimates and Bergman projections.

Compilation of recent sharp results on traces in product domains.

Abstract

In this expository paper we collect many recent advances in analytic function spaces of several complex variables related with trace problem in tubular domains over symmetric cones and bounded strongly pseudoconvex domains with smooth boundary. We consider various function space of analytic functions of several variables in various domains in $C^n$ and provide or complete descriptions of traces or estimates of traces of various analytic function spaces in various domains obtained in recent years, by various authors. The problem to find sharp estimates of traces of Hardy analytic function spaces in the unit polydisk first was posed by W. Rudin in 1969. Since then many papers appeared in literature. We collect in this expository paper not only already many known results on traces of various analytic function spaces in product domains but also discuss various new interesting results related with this problem. Related to trace problem various results were provided previously by G. Henkin, E. Amar, H. Alexander and various other authors. Finnaly, note that our trace theorems are closely related with the Bergman type projections acting between function spaces with different dimensions. This expository paper contains mainly new results concerning traces in tubular and bounded strongly pseudoconvex domains, proofs of these theorems are based in particular also on various properties of Bergman type projection, in this expository paper we will also shortly discuss some new results obtained by first author on Bergman type projections in these complicated domains in $C^n$. This is the second part of our notes related with trace problem. In the first part we provided a large list of recent sharp results on traces in the polydisk and polyball and was published in [1].