Sharp resolvent estimates outside of the uniform boundedness range

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.

arXiv comment39 pages, v2 minor corrections, to appear in Commun. Math. Phys

References in corpus (2)

• Eigenvalue bounds for Schrödinger operators with complex potentials
• Improved bounds for Stein's square functions

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• Counterexample to the Laptev--Safronov conjecture
• Keller-type bounds for Dirac operators perturbed by rigid potentials
• Resolvent estimates for time-harmonic Maxwell's equations in the partially anisotropic case
• Bochner-Riesz means at the critical index: Weighted and sparse bounds