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Selberg's Integral Formula and Sharp Constants for Hardy-Littlewood-Sobolev Inequality

编辑:xyx 时间:2018年07月14日 访问次数:262

浙江大学数学科学学院九十周年院庆系列活动之五十七

Title: :Selberg's Integral Formula and Sharp Constants for Hardy-Littlewood-Sobolev Inequality
Speaker: Professor . YAN,Dunyan  (中国科学院大学)
Time:  2018-07-14  17:00-18:00
Location: 200-9,  Sir Shaw Run Run Business Administration building,School of Mathematical Sciences, Yuquan Campus

Abstract: In this talk, we investigate some necessary and sufficient conditions which ensure
validity of the Selberg's integral formula. That is,  the Selberg's integral equation is as follows
$$\int_{\mathbb{R}^n}\prod\limits^k_{i=1}|x^i-t|^{-d_i}dt
=C_{d_1,\cdots,d_k,n}\prod\limits_{1\le i<j\le
k}|x^i-x^j|^{-\alpha_{ij}},$$
where $x^{i}\in \mathbb{R}^n$ $d_i$ is  nonzero real number, with $i=1,\cdots,k$.
Actually, we completely answer the question raised by Grafakos in the reference \cite{GM}.
In fact, for some cases, the  constant number $C_{d_1,\cdots,d_k,n}$ is just the sharp bound of the following Hardy-Littlewood -Sobolev inequality
$$\left|{\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{f(x)g(y)}{|x|^\alpha|x-y|^\lambda|y|^\beta}
dxdy}\right|\le
C(p,q,\alpha,\lambda,\beta,n)\|f\|_{L^{p}(\mathbb{R}^n)}\|g\|_{L^{q}(\mathbb{R}^n)}.$$
 
Contact Person:  WANG Meng, (mathdreamcn@zju.edu.cn)