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

TL;DR

This paper investigates the properties of the $p$-Bergman kernel and metric, revealing regularity, relations to weighted kernels, curvature bounds, and applications to function spaces and domain geometry.

Contribution

It introduces new regularity results, relations, bounds, and concepts like the $p$-Schwarz content, advancing understanding of $p$-Bergman kernels and metrics in complex analysis.

Findings

$K_p(ullet)$ is $C^{1,1/2}$ for $1<p< olinebreak \infty$

Relation between off-diagonal $K_p$ and weighted $L^2$ Bergman kernel for $1 extless p extless 2$

Boundedness of $K_p(ullet,z)$ in $L^q$ spaces for $1 extless p extless 2$

Abstract

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.