Invited by Prof. Sibei Yang and Prof. Jun Gen from School of Mathematics and Statistics, Prof.Xiaohua Yao fromCentral China Normal University will give a lecture 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.