$L^p$ estimates for the Bergman projection on some Reinhardt domains

TL;DR

This paper establishes $L^p$ regularity for the Bergman projection on certain Reinhardt domains, showing that under specific kernel estimates, the boundedness extends to higher-dimensional successor domains, including non-smooth and non-strictly pseudoconvex cases.

Contribution

It demonstrates that $L^p$ boundedness of the Bergman projection on initial domains extends to successor Reinhardt domains under kernel estimate conditions, broadening the class of domains with known $L^p$ regularity.

Findings

Bergman projection is $L^p$ bounded on successors of strictly pseudoconvex domains.

Boundedness extends to domains without smooth boundary or strict pseudoconvexity.

Kernel estimates on initial domains imply $L^p$ regularity on higher-dimensional domains.

Abstract

We obtain $L^p$ regularity for the Bergman projection on some Reinhardt domains. We start with a bounded initial domain $\Omega$ with some symmetry properties and generate successor domains in higher {dimensions}. We prove: If the Bergman kernel on $\Omega$ satisfies appropriate estimates, then the Bergman projection on the successor is $L^p$ bounded. For example, the Bergman projection on successors of strictly pseudoconvex initial domains is bounded on $L^p$ for $1<p<\infty$. The successor domains need not have smooth boundary nor be strictly pseudoconvex.