走向现代数学学术报告 - 张利友教授（No.524）

报告题目：On the $L^p$ Bergman theory

报告人：张利友教授（首都师范大学）

报告时间：2022年7月6日，10：00

报告地点：腾讯会议964135979

报告摘要：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 $K_p(z)$ in terms of $K_2(z)$. The talk is based on a joint work with Bo-Yong Chen.

报告人简介：张利友，首都师范大学数学科学学院教授，博士生导师。2007年博士毕业于首都师范大学，导师殷慰萍教授。2007-2009年中科院数学所博士后，合作导师陆启铿院士。主持国家自然科学基金面上项目，青年基金，北京市自然科学基金面上项目。曾获首届北京市优秀博士学位论文奖、中科院王宽诚博士后人才工作奖励。近年来，与合作者相关研究成果发表在Adv. Math., JFA, Trans. AMS等国际期刊上。