主标题：微分方程理论、计算与应用讨论班（5）

报告题目：The Non-existence Results for a Class of Integral equation

报告人：湖南农业大学理学院 许建开（副教授）

报告时间：12月11日下午14:00-15:30

报告地点：数计学院4号楼229会议室

摘要:In this talk, we consider the following integral system

 $$u(x,b)=\\int_{R^n} \\frac{u^q(y)}{(b+|x-y|)^{\\lambda} dy$$

which is related to the weak type convolution-Young’s inequality. Under the assumption of that $\\lambda\\in (0,n)$ and $02n/λ−1, the system does not admit positive solution in $L^q+1(Rn)(n > 2),$ which implies that the maximizing pair of the weak type convolution-Young’s inequality with kernel function (b + |x|)−\\lambdadoes not exist. Meanwhile, for λ \\in (−\\infty, 0) and q = 2n/\\lambda− 1, we also show that the system doesn’t admit non-negative Lebesgue measurable solution. This is distinct from theoriginal conformal invariant integral system.

报告题目: An Extension of Discrete Weighted HLS Inequality in Space Dimension One

报告人：湖南农业大学理学院  许建开（副教授）

报告时间：12月11日下午16:00-17:30

报告地点：数计学院4号楼229会议室

摘要:In this talk, we establish the critical version of discrete weighted Hardy Littlewood Sobolev inequality with $p=q=2$, $\\lambda=1$ and $\\alpha+\\beta=0$ in $\\mathbb{R}^1$:

\\sum_{\\substack{-N\\leq r,s \\leq N;\\\\ r\\neq 0, s\\neq 0;\\\\\neq s}}\\frac{1}{|r|^{\\alpha}}\\frac{a_rb_s|s|^{\\alpha}}{|r-s|}\\leq C_{\\alpha}{\m\\lambda_{N}^{\\alpha}}\\|a\\|_2~\\|b\\|_2,

and as $\\alpha\\geq 1$, we obtain that she sharp estimate ${\m \\lambda_{N}^{\\alpha}}$ is $N^{\\alpha-1/2}$. Namely,

\\sum_{\\substack{-N\\leq r,s \\leq N;\\\\ r\\neq 0, s\\neq 0;\\\\\neq s}}\\frac{1}{|r|^{\\alpha}}\\frac{~~~a_r~~b_s~~|s|^{\\alpha}}{|r-s|}\\leq C_{\\alpha}~ {\m

{N}^{\\alpha-1/2}}\\|a\\|_2~\\|b\\|_2.