Sharp resolvent estimates outside of the uniform boundedness range

In this paper we are concerned with resolvent estimates for the Laplacian $\mathrm{\Delta }$ in Euclidean spaces. Uniform resolvent estimates for $\mathrm{\Delta }$ 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.