江蘇省應用數學（中國礦業大學）中心系列學術報告
報告題目：On the Lp Bergman theory
報告人：張利友教授
報告時間：2023/11/10 15:00-16:00
報告地點：理A321
報告摘要：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.
報告人簡介：張利友，首都師範大學數學科學學院教授、博士生導師。主要研究多複變函數論與複幾何。2008年獲得北京市首屆優秀博士學位論文獎、2009年獲得中科院王寬誠博士後人才工作獎勵。近年來，先後主持過國家自然科學基金青年項目、北京市自然科學基金面上項目、國家自然科學基金面上項目。在Adv. Math., TAMS, JFA, JGA等期刊發表學術論文20餘篇。