Boundary Regularity of Bergman Kernel in H\"older space

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.