Maximal inequalities for dual Sobolev spaces $W^{-1,p}$ and applications to interpolation

TL;DR

This paper establishes a novel maximal inequality for dual Sobolev spaces W^{-1,p}, extending Hardy-Littlewood maximal operator properties to Sobolev contexts on Riemannian manifolds, and applies it to interpolation of Sobolev spaces.

Contribution

It introduces a new maximal inequality for W^{-1,p} spaces and applies it to derive interpolation results, extending classical harmonic analysis tools to Sobolev spaces on manifolds.

Findings

New maximal inequality for dual Sobolev spaces W^{-1,p}

Extension of Hardy-Littlewood maximal properties to Sobolev spaces

Interpolation results for Sobolev spaces on Riemannian manifolds

Abstract

We firstly describe a maximal inequality for dual Sobolev spaces W^{-1,p}. This one corresponds to a "Sobolev version" of usual properties of the Hardy-Littlewood maximal operator in Lebesgue spaces. Even in the euclidean space, this one seems to be new and we develop arguments in the general framework of Riemannian manifold. Then we present an application to obtain interpolation results for Sobolev spaces.