Generalizations of the higher dimensional Suita conjecture and its relation with a problem of Wiegerinck

TL;DR

This paper extends the Suita conjecture to higher dimensions using generalized Bergman kernels, explores the Bergman space dimension in pseudoconvex domains, and relates it to Wiegerinck's problem, providing new partial results and domain classes.

Contribution

It introduces higher order Bergman kernel generalizations, links the Bergman space dimension problem to Wiegerinck's conjecture, and identifies domain classes with positive answers.

Findings

New partial results on Bergman space dimensions in pseudoconvex domains

Relation established between Wiegerinck's problem and balanced domains

Regularity properties of functions involving Azukawa indicatrices

Abstract

We generalize the inequality being a counterpart of the several complex variables version of the Suita conjecture. For this aim higher order generalizations of the Bergman kernel are introduced. As a corollary some new partial results on the dimension of the Bergman space in pseudoconvex domains are given. A relation between the problem of Wiegerinck on possible dimension of the Bergman space of unbounded pseudoconvex domains in general case and in the case of balanced domains is also shown. Moreover, some classes of domains where the answer to the problem of Wiegerinck is positive are given. Additionally, regularity properties of functions involving the volumes of Azukawa indicatrices are shown.