Uniform Sobolev Estimates on compact manifolds involving singular potentials

TL;DR

This paper extends uniform Sobolev inequalities and quasimode estimates to compact Riemannian manifolds with critically singular potentials, providing sharper results and optimal exponent ranges.

Contribution

It generalizes existing Sobolev and quasimode estimates to include critically singular potentials on compact manifolds, improving previous bounds and establishing sharp inequalities.

Findings

Extended uniform Sobolev inequalities to singular potentials on manifolds.

Obtained sharp inequalities on spheres for optimal exponents.

Improved quasimode estimates for broader exponent ranges.

Abstract

We obtain generalizations of the uniform Sobolev inequalities of Kenig, Ruiz and the fourth author \cite{KRS} for Euclidean spaces and Dos Santos Ferreira, Kenig and Salo \cite{DKS} for compact Riemannian manifolds involving critically singular potentials $V\in L^{n/2}$. We also obtain the analogous improved quasimode estimates of the the first, third and fourth authors \cite{BSS} , Hassell and Tacy \cite{HassellTacy}, the first and fourth author \cite{SBLog}, and Hickman \cite{Hickman} as well as analogues of the improved uniform Sobolev estimates of \cite{BSSY} and \cite{Hickman} involving such potentials. Additionally, on $S^n$, we obtain sharp uniform Sobolev inequalities involving such potentials for the optimal range of exponents, which extend the results of S. Huang and the fourth author \cite{SHSo}. For general Riemannian manifolds we improve the earlier results in \cite{BSS} by obtaining quasimode estimates for a larger (and optimal) range of exponents under the weaker assumption that $V\in L^{n/2}$.