Resolvent estimates for the Laplacian

题目：Resolvent estimates for the Laplacian
报告人：Sanghyuk Lee (首尔大学)
时间：2019年2月28日 （星期四）下午2：00-5：00
地点：工商楼105教室

摘要： In this talk we are concerned with the resolvent estimates for the Laplacian $/Delta$ in Euclidean spaces:
/[ /| (-/Delta-z)^{-1}f /|_{L^q}/le C(z)/| f/|_{L^p}, / / z /in /mathbb C/setminus (0,/infty)./]
The resolvent estimates and its variants have applications to various related problems. Among them are uniform Sobolev estimates, unique continuation properties, limiting absorption principles, and  eigenvalue bounds for the Schr/"odinger operators. Uniform resolvent estimates for $/Delta$ (i.e., the estimate with $C(z)$ independent of $z$) were first obtained  by Kenig, Ruiz and Sogge, who completely characterized the Lebesgue spaces  which allow such uniform resolvent estimates.  But the precise bound $C(z)$ depending on $z$ has not been considered in general framework which admits all possible $p,q$.
In this talk, we present a complete picture of sharp $L^p$--$L^q$ resolvent estimates, which may depend on $z$ and draw a connection to the Bochner-Riesz conjecture.  The resolvent estimates in Euclidean space seem to be expected to behave in a simpler way compared with those on manifolds. However, it turns out that,  for some $p,q$, the estimates exhibit unexpected  behavior which is  similar to those on compact Riemannian manifolds. We also obtain the non-uniform sharp resolvent estimates for the fractional Laplacians and discuss applications to related problems.

联系人：王成波老师（wangcbo@zju.edu.cn)