Regularity of the $p-$Bergman kernel

We show that the $p-$Bergman kernel $K_p(z)$ on a bounded domain $\mathrm{\Omega }$ is of locally $C^{1,1}$ for $p\ge 1$.The proof is based on the locally Lipschitz continuity of the off-diagonal $p-$Bergman kernel $K_p(\zeta ,z)$ for fixed $\zeta \in \mathrm{\Omega }$. Global irregularity of $K_p(\zeta ,z)$ is presented for some smooth strongly pseudoconvex domains when $p\gg 1$. As an application of the local $C^{1,1}-$regularity, an upper estimate for the Levi form of $\log K_p(z)$ for $1<p<2$ is provided. Under the condition that the hyperconvexity index of $\mathrm{\Omega }$ is positive, we obtain the log-Lipschitz continuity of $p\mapsto K_p(z)$ for $1\le p\le 2$.