On the $p$-essential normality of principal submodules of the Bergman module on strongly pseudoconvex domains

TL;DR

This paper proves that principal submodules of the Bergman module on strongly pseudoconvex domains are $p$-essentially normal for all $p>n$, extending previous results from the unit ball to more general domains.

Contribution

It generalizes the $p$-essential normality of principal submodules from the unit ball to strongly pseudoconvex domains with smooth boundary.

Findings

Principal submodules are $p$-essentially normal for all $p>n$ on these domains.

Extension of previous results from the unit ball to strongly pseudoconvex domains.

Submodules vanishing on analytic subsets of codimension 1 are also $p$-essentially normal.

Abstract

In this paper, we show that under a mild condition, a principal submodule of the Bergman module on a bounded strongly pseudoconvex domain with smooth boundary in $\mathbb{C}^n$ is $p$-essentially normal for all $p>n$. This improves a previous result by the first author and K. Wang, in which it was shown that any polynomial-generated principal submodule of the Bergman module on the unit ball $\mathbb{B}_n$ is $p$-essentially normal for all $p>n$. As a consequence, we show that the submodule of $L_a^2(\mathbb{B}_n)$ consisting of functions vanishing on an analytic subset of pure codimension $1$ is $p$-essentially normal for all $p>n$.