Boundary regularity of Bergman kernel in Hölder space

Ziming Shi

Vol. 324 (2023), No. 1, 157–206

DOI: 10.2140/pjm.2023.324.157

Abstract

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

Keywords

Bergman kernel, Bergman projection, strictly pseudoconvex domain

Mathematical Subject Classification

Primary: 32A25

Secondary: 32T15

Milestones

Received: 7 August 2022

Revised: 8 February 2023

Accepted: 14 April 2023

Published: 22 June 2023