  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.
