The Bergman projection in $L^p$ for domains with minimal smoothness

TL;DR

This paper establishes $L^p$ regularity of the Bergman projection on strongly Levi-pseudoconvex domains with minimal boundary smoothness, extending understanding of holomorphic function spaces in complex analysis.

Contribution

It proves $L^p$ regularity for the Bergman projection and related operators on minimally smooth domains, a significant extension of classical results.

Findings

Proves $L^p$ regularity for the Bergman projection $B$.

Shows the density of holomorphic functions near the boundary in $L^p$ spaces.

Extends regularity results to domains with minimal boundary smoothness.

Abstract

Let $D\subset\mathbb C^n$ be a bounded, strongly Levi-pseudoconvex domain with minimally smooth boundary. We prove $L^p(D)$-regularity for the Bergman projection $B$, and for the operator $|B|$ whose kernel is the absolute value of the Bergman kernel with $p$ in the range $(1,+\infty)$. As an application, we show that the space of holomorphic functions in a neighborhood of $\bar{D}$ is dense in $\vartheta L^p (D)$.