Boundary Regularity of Bergman Kernel in H\"older space

TL;DR

This paper investigates the boundary regularity of the Bergman kernel and projection on strictly pseudoconvex domains with Hölder continuous boundaries, establishing new regularity results that extend previous work by Ligocka.

Contribution

It provides new regularity estimates for the Bergman kernel and projection on domains with Hölder boundary regularity, generalizing prior results by Ligocka.

Findings

Bergman kernel is in C^{k+min{α,1/2}}(ar D) for points inside D.

Bergman projection maps C^{k+β}(ar D) to C^{k+min{α, β/2}}(ar D).

Results improve and extend Ligocka's earlier work.

Abstract

Let $D$ be a bounded strictly pseudoconvex domain in $\mathbb{C}^n$. Assuming $bD \in C^{k+3+\alpha}$ where $k$ is a non-negative integer and $0 < \alpha \leq 1$, we show that 1) the Bergman kernel $B(\cdot, w_0) \in C^{k+ \min\{\alpha, \frac12 \} } (\overline D)$, for any $w_0 \in D$; 2) The Bergman projection on $D$ is a bounded operator from $C^{k+\beta}(\overline D)$ to $C^{k + \min \{ \alpha, \frac{\beta}{2} \}}(\overline D) $ for any $0 < \beta \leq 1$. Our results both improve and generalize the work of E. Ligocka.