报告主题：具有Hardy位势的高阶和分数阶算子的Kato平滑和Strichartz预计

报 告 人：尧小华 教授 （华中师范大学）

报告时间：2020年11月7日（周六） 9:00

参会方法：腾讯聚会

聚会ID：458 671 203

邀 请 人：赵发友

主理部分：理学院数学系

报告摘要：Let 0<\sigma<n/2 and H=(-\Delta)^\sigma +V(x) be Schr\"odinger type operators with certain scaling-critical potentials V(x), which include the Hardy potential C|x|^{-2\sigma} with a subcritical coupling constant C as typical examples. In this talk, we consider several global estimates for the resolvent and the solution to the time-dependent Schr\"odinger equation associated with H. We first prove the uniform resolvent estimates of Kato-Yajima type for all 0<\sigma<n/2 using a version of Mourre's theory, which turn out to be equivalent to Kato smoothing estimates for the Cauchy problem. Using these estimates, we then obtain Strichartz estimates for \sigma>1/2 and uniform Sobolev estimates of Kenig-Ruiz-Sogge type for \sigma\ge n/(n+1). These completely extend the same properties for the Schr\"odinger operator with the inverse-square potential to the higher-order and fractional cases. Moreover, we point out that these arguments can be further applied to a large class of higher-order inhomogeneous elliptic operators and even to certain long-range metric perturbations of the Laplace operator.

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