Boundary Regularity of Bergman Kernel in Hölder 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 $0<\alpha \le 1$, we show that 1) the Bergman kernel $B(\cdot ,w_0)\in C^{k+\min \{\alpha ,\frac{1}{2}\}}(\bar{D})$, for any $w_0\in D$; 2) The Bergman projection on $D$ is a bounded operator from $C^{k+\beta }(\bar{D})$ to $C^{k+\min \{\alpha ,\frac{\beta }{2}\}}(\bar{D})$ for any $0<\beta \le 1$. Our results both improve and generalize the work of E. Ligocka.