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arXiv · 1810.09740

Sharp resolvent estimates outside of the uniform boundedness range

Yehyun Kwon·Sanghyuk Lee
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Abstract

In this paper we are concerned with resolvent estimates for the Laplacian $\Delta$ in Euclidean spaces. Uniform resolvent estimates for $\Delta$ were shown by Kenig, Ruiz and Sogge \cite{KRS} who established rather a complete description of the Lebesgue spaces allowing such estimates. However, the problem of obtaining sharp $L^p$--$L^q$ bounds depending on $z$ has not been considered in a general framework which admits all possible $p,q$. In this paper, we present a complete picture of sharp $L^p$--$L^q$ resolvent estimates, which may depend on $z$. We also obtain the sharp resolvent estimates for the fractional Laplacians and a new result for the Bochner--Riesz operators of negative index.

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Yehyun Kwon, Sanghyuk Lee. 2018-10-23. Sharp resolvent estimates outside of the uniform boundedness range. https://doi.org/10.1007/s00220-019-03536-y

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