Bergman subspaces and subkernels: Degenerate $L^p$ mapping and zeroes

TL;DR

This paper investigates the behavior of the Bergman projection on $L^p$ spaces across a family of pseudoconvex domains, revealing that irrational parameters restrict boundedness to $p=2$, highlighting irregularity in certain cases.

Contribution

It introduces a new family of domains parameterized by $b3$ and analyzes the $L^p$ regularity of the Bergman projection, showing novel irrationality-dependent boundedness properties.

Findings

Bergman projection's boundedness depends on the domain parameter b3.

For irrational b3, boundedness occurs only at p=2.

The analysis links domain geometry with operator regularity.

Abstract

Regularity and irregularity of the Bergman projection on $L^p$ spaces is established on a natural family of bounded, pseudoconvex domains. The family is parameterized by a real variable $\gamma$. A surprising consequence of the analysis is that, whenever $\gamma$ is irrational, the Bergman projection is bounded only for $p=2$.