Refined Sobolev inequalities on manifolds with ends

TL;DR

This paper develops refined Sobolev inequalities on manifolds with ends using a Besov norm, ensuring stability under oscillations and spectral localization, with a general abstract version for selfadjoint operators.

Contribution

It introduces a new Besov space-based approach to refined Sobolev inequalities on manifolds with ends, stable under oscillations and spectral localization, and extends the results to abstract selfadjoint operators.

Findings

Refined Sobolev inequalities on manifolds with ends using Besov norms.

Stability of inequalities under oscillating functions and spectral localization.

General abstract inequalities for selfadjoint operators on measure spaces.

Abstract

By considering a suitable Besov type norm, we obtain refined Sobolev inequalities on a family of Riemannian manifolds with (possibly exponentially large) ends. The interest is twofold: on one hand, these inequalities are stable by multiplication by rapidly oscillating functions, much as the original ones \cite{GMO}, and on the other hand our Besov space is stable by spectral localization associated to the Laplace-Beltrami operator (while $ L^p $ spaces, with $ p \ne 2 $, are in general not preserved by such localizations on manifolds with exponentially large ends). We also prove an abstract version of refined Sobolev inequalities for any selfadjoint operator on a measure space (Proposition \ref{general}).