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.
Categories: math.CV