It is well known that uniform resolvent estimates imply spectral cluster
estimates. We show that the converse is also true in some cases. In particular,
the universal spectral cluster estimates of Sogge \cite{MR930395} for the
Laplace--Beltrami operator on compact Riemannian manifolds without boundary
directly imply the uniform Sobolev inequality of Dos Santos Ferreira, Kenig and
Salo \cite{MR3200351}, without any reference to parametrices. This observation
also yields new resolvent estimates for manifolds with boundary or with
nonsmooth metrics, based on spectral cluster bounds of Smith--Sogge
\cite{MR2316270} and Smith, Koch and Tataru~\cite{MR2443996}, respectively. We
also convert the recent spectral cluster bounds of Canzani and Galkowski
\cite{Canzani--Galkowski} to improved resolvent bounds. Moreover, we show that
the resolvent estimates are stable under perturbations and use this to
establish uniform Sobolev and spectral cluster inequalities for Schr\"odinger
operators with singular potentials.Comment: Theorem 4.1 now also converts improved spectral cluster bounds to
improved resolvent bounds. A new application of the recent spectral cluster
bounds of Canzani and Galkowski is included. Some minor typos correcte
