Selberg's Integral Formula and Sharp Constants for Hardy-Littlewood-Sobolev Inequality
来源：太阳成集团tyc411
发布时间：2018-07-14
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太阳成集团tyc411(中国)有限公司-百度百科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
$$/int_{/mathbb{R}^n}/prod/limits^k_{i=1}|x^i-t|^{-d_i}dt
=C_{d_1,/cdots,d_k,n}/prod/limits_{1/le i
where $x^{i}/in /mathbb{R}^n$ $d_i$ is nonzero real number, with $i=1,/cdots,k$.
Actually, we completely answer the question raised by Grafakos in the reference /cite{GM}.
In fact, for some cases, the constant number $C_{d_1,/cdots,d_k,n}$ is just the sharp bound of the following Hardy-Littlewood -Sobolev inequality
$$/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)}.$$
validity of the Selberg's integral formula. That is, the Selberg's integral equation is as follows
$$/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
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)