On the $L^p$ Bergman theory

讲座时间 Datetime: 2022年9月24日，星期六，16:00-17:00 

地点 Venue: 腾讯线上会议：325-407-632

主持人 Host：邵国宽

报告人 Speaker: 张利友 

单位 Affiliation: 首都师范大学

报告摘要 Abstract:

In this talk, we’d like to introduce a general $L^p$ 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 $p > 4$ 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，yields a number of results. For instance, the off-diagonal $L^p$ 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 $L^p$ 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 $p\le 2.$ If time permits, we will also talk about the stability of $K_p(z)$ or $B_p(z;X)$ as $p$ varies and the boundary behavior of $K_p(z)$. The talk is based on a joint work with Bo-Yong Chen.