Some properties of the $p-$Bergman kernel and metric

The $p-$Bergman kernel $K_p(\cdot)$ is shown to be of $C^{1,1/2}$ for $1<p<\infty$. An unexpected relation between the off-diagonal $p-$Bergman kernel $K_p(\cdot,z)$ and certain weighted $L^2$ Bergman kernel is given for $1\le p\le 2$. As applications, we show that for each $1\le p\le 2$, $K_p(\cdot,z)\in L^q(\Omega)$ for $q< \frac{2pn}{2n-\alpha(\Omega)}$ and $|K_s(z)-K_p(z)| \lesssim |s-p||\log |s-p||$ whenever the hyperconvexity index $\alpha(\Omega)$ is positive. Counterexamples for $2<p<\infty$ are given respectively. An optimal upper bound for the holomorphic sectional curvature of the $p-$Bergman metric when $2\le p<\infty$ is obtained. For bounded $C^2$ domains, it is shown that the Hardy space and the Bergman space satisfy $H^p(\Omega)\subset A^q(\Omega)$ where $q=p(1+\frac1n)$. A new concept so-called the $p-$Schwarz content is introduced. As applications, upper bounds of the Banach-Mazur distance between $p-$Bergman spaces are given, and $A^p(\Omega)$ is shown to be non-Chebyshev in $L^p(\Omega)$ for $0<p\le 1$. For planar domains, we obtain a rigidity theorem for the $p-$Bergman kernel (which is not valid in high dimensional cases), and a characterization of non-isolated boundary points through completeness of the Narasimhan-Simha metric.