Sharp resolvent estimates outside of the uniform boundedness range

TL;DR

This paper provides a comprehensive analysis of sharp $L^p$--$L^q$ resolvent estimates for the Laplacian and fractional Laplacians, including new results for Bochner--Riesz operators, extending previous uniform bounds.

Contribution

It introduces a complete framework for sharp $L^p$--$L^q$ resolvent estimates that depend on the spectral parameter $z$, covering all possible $p,q$ ranges.

Findings

Established sharp $L^p$--$L^q$ resolvent bounds depending on $z$.

Derived sharp estimates for fractional Laplacians.

Presented new results for Bochner--Riesz operators of negative index.

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.