和山数学论坛第73讲


一、主 题：Sharp Bound for Hardy-Littlewood-Sobolev Inequality and Selberg's Integral Formula
二、主讲人：吴迪博士（后） 中山大学
三、时 间：2015年9月4日（周五上午）10:30—11:00
四、地 点：闻理园A4-305室

内容摘要：
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
$$\int_{\mathbb{R}^n}\prod\limits^k_{i=1}|x^i-t|^{-d_i}dt=C_{d_1,\cdots,d_k,n}\prod\limits_{1\le iCondition I is that $k=2$ and $\max\{d_1,d_2\}Condition II is that $k=3$, $\max\{d_1,d_2,d_3\}Actually, we completely answer the question raised by Grafakos in the reference
In fact, for some cases, the constant number $C_{d_1,\cdots,d_k,n}$ is just the sharp bound of the following Hardy-Littlewood Sobolevnequality$$\left|{\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{f(x)g(y)}{|x|^\alpha|x-y|^\lambda|y|^\beta}dxdy}\right|\leC(p,q,\alpha,\lambda,\beta,n)\|f\|_{L^{p}(\mathbb{R}^n)}\|g\|_{L^{q}(\mathbb{R}^n)}.$$

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                                     2015年8月31日