On the $p-$Bergman theory

TL;DR

This paper develops a comprehensive $p$-Bergman theory on bounded domains in complex space, exploring kernel regularity, metric convergence, and boundary behavior, highlighting differences from the classical $L^2$ case.

Contribution

It introduces new results on the regularity, stability, and geometric properties of $p$-Bergman kernels and metrics, extending classical theory to the $L^p$ setting.

Findings

$p$-Bergman kernel is not real-analytic on some domains for even $p\, ext{≥}\,4$

Off-diagonal $p$-Bergman kernel is Hölder continuous with specific orders

The $p$-Bergman metric converges to the Carathéodory metric as $p o\, ext{infinity}

Abstract

In this paper we attempt to develop a general $p-$Bergman theory on bounded domains in $\mathbb C^n$. To indicate the basic difference between $L^p$ and $L^2$ cases, we show that the $p-$Bergman kernel $K_p(z)$ is not real-analytic on some bounded complete Reinhardt domains when $p\ge 4$ is an even number. By the calculus of variations we get a fundamental reproducing formula. This together with certain techniques from nonlinear analysis of the $p-$Laplacian yield a number of results, e.g., the off-diagonal $p-$Bergman kernel $K_p(z,\cdot)$ is H\"older continuous of order $\frac12$ for $p>1$ and of order $\frac1{2(n+2)}$ for $p=1$. We also show that the $p-$Bergman metric $B_p(z;X)$ tends to the Carath\'eodory metric $C(z;X)$ as $p\rightarrow \infty$ and the generalized Levi form $i\partial\bar{\partial}\log K_p(z;X)$ is no less than $B_p(z;X)^2$ for $p\ge 2$ and $ C(z;X)^2$ for $p\le 2.$ Stability of $K_p(z,w)$ or $B_p(z;X)$ as $p$ varies, boundary behavior of $K_p(z)$, as well as basic facts on the $p-$Bergman prjection, are also investigated.