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Resolvent estimates for the Laplacian

Date:2019-02-28 | Visitcount:412
Title: Resolvent estimates for the Laplacian
Speaker:Sanghyuk Lee (Seoul National University)
Time:15:10-16:10, Feb. 4, 2019 (Thursday)
Location:105,Sir Shaw Run Run Business Administration building

Abstract: 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.

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