On the LpBergman theory



报告题目:On the Lp Bergman theory


报告时间:2023/11/10  15:00-16:00


报告摘要:In this talk, we’d like to introduce the properties of $L^p$ Bergman kernels on bounded domains in $\mathbb C^n$. To indicate the basic difference between $L^p$ and $L^2$ cases, we show that the $L^p$ Bergman kernel 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 $p=1$.

In the second part, we shall talk about the geometric aspect of the $L^p$ Bergman theory. We 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 $K_p(z,w)$ or $B_p(z;X)$ as $p$ and the domain vary.


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。主要研究多复变函数论与复几何。2008年获得北京市首届优秀博士学位论文奖、2009年获得中科院王宽诚博士后人才工作奖励。近年来,先后主持过国家自然科学基金青年项目、北京市自然科学基金面上项目、国家自然科学基金面上项目。在Adv. Math., TAMS, JFA, JGA等期刊发表学术论文20余篇。