A sufficient condition for $L^p$ regularity of the Berezin transform

TL;DR

This paper investigates the conditions under which the Berezin transform exhibits $L^p$ regularity across various complex domains, establishing positive results for many and a specific negative result for the Hartogs triangle.

Contribution

It provides a sufficient condition for $L^p$ regularity of the Berezin transform on broad classes of domains and demonstrates its failure on the Hartogs triangle.

Findings

Berezin transform is $L^p$ regular on many domains

Fails to be $L^2$ regular on the Hartogs triangle

Identifies a broad class of domains with $L^p$ regularity

Abstract

We prove that the Berezin transform is $L^p$ regular on a large class of domains in $\mathbb{C}^n$ and not $L^2$ regular on the Hartogs triangle.