05-11-2018
799

数学学院学术报告-11月8日

$$\int_{\mathbb{R}^n}\prod\limits^k_{i=1}|x^i-t|^{-d_i}dt=C_{d_1,\cdots,d_k,n}\prod\limits_{1\le i<j\le k}|x^i-x^j|^{-\alpha_{ij}}$$

holds for any $x^{i}\in \mathbb{R}^n$ and some nonzero real numbers $d_i$ with $i=1,\cdots,k$ if and only if one of the following two conditions holds.

Condition I is that $k=2$ and $\max\{d_1,d_2\}<n<d_1+d_2$;

Condition II is that$k=3$, $\max\{d_1,d_2,d_3\}<n$ and $d_1+d_2+d_3=2n$.

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 -Sobolev inequality

$$\left|{\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{f(x)g(y)}{|x|^\alpha|x-y|^\lambda|y|^\beta}dxdy}\right|\le C(p,q,\alpha,\lambda,\beta,n)\|f\|_{L^{p}(\mathbb{R}^n)}\|g\|_{L^{q}(\mathbb{R}^n)}.$$

In the final, we obtained the sharp constants for Hardy-Littlewood-Sobolev Inequality by using the Selberg's integral formula.

１．Fourier分析与PDE的求解方法

２．离散调和分析与PDE的求解

３．PDE的经典研究方法-调和分析观点

４．PDE的经典研究方法与现代调和分析方法的比较

５．振荡积分、格点估计与Weyl定理

E-mail：mathruc@ruc.edu.cn