### 报告学术：Selberg’s Integral Formula and Sharp Constants for Hardy-Littlewood-Sobolev Inequality

validity of the Selberg's integral formula. That is,  the Selberg's integral equation
$$\int_{\mathbb{R}^n}\prod\limits^k_{i=1}|x^i-t|^{-d_i}dt=C_{d_1,\cdots,d_k,n}\prod\limits_{1\le i<j\lek}|x^i-x^j|^{-\alpha_{ij}}$$ 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 $$\left|{\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{f(x)g(y)}{|x|^\alpha|x-y|^\lambda|y|^\beta} dxdy}\right|\leC(p,q,\alpha,\lambda,\beta,n)\|f\|_{L^{p}(\mathbb{R}^n)}\|g\|_{L^{q}(\mathbb{R}^n)}.$$  In the final, we we obtained the sharp constants for Hardy-Littlewood-SobolevInequalityby using the Selberg's integral formula