Invited by Prof. Jun Geng and Sibei Yang of School of Mathematics and Statistics, Prof. Xiaohua Yao from Central China Normal University of Science and Technology will give lectures online.

Title：Kato smoothing and Strichartz estimates for fractional operators with Hardy potentials

Time：2:30 p.m.,Nov. 24th, 2021

Conference ID：445 850 497( Tencent Conference)

Abstract：Let $0<\sigma<n/2$ and $H=(-\Delta)^\sigma+a|x|^{-2\sigma}$ be Schrodinger type operators on $\R^n$ with a sharp coupling constant $a\le -C_{\sigma,n}$ ( $C_{\sigma,n}$ is the best constant of Hardy's inequality of order $\sigma$). In the present talk, we will address that sharp global estimates for the resolvent and the solution to the time-dependent Schrodinger equation associated with $H$. In the case of the subcritical coupling constant $a>-C_{\sigma,n}$, we first prove the uniform resolvent estimates of Kato--Yajima type for all $0<\sigma<n/2$, which turn out to be equivalent to Kato smoothing estimates for the Cauchy problem. We then establish Strichartz estimates for $\sigma>1/2$ and uniform Sobolev estimates of Kenig--Ruiz--Sogge type for $\sigma\ge n/(n+1)$. In the critical coupling constant case $a=-C_{\sigma,n}$ , we show that the same results as in the subcritical case still hold for functions orthogonal to radial functions. This is a joint-work (To appear in CMP) with Haruya Mizutani.