From spectral cluster to uniform resolvent estimates on compact manifolds
Abstract
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 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 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 MR2316270 and Smith, Koch and Tataru MR2443996, respectively. We also convert the recent spectral cluster bounds of Canzani and Galkowski 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.