A maximal function characterization of absolutely continuous measures and Sobolev functions

TL;DR

This paper introduces new characterizations of Sobolev functions and absolutely continuous measures using Hardy-Littlewood maximal functions, and discusses limitations of existing approaches for BV vector fields.

Contribution

It provides novel maximal function characterizations for Sobolev $W^{1,1}$ functions and absolutely continuous measures, and analyzes the extension limits of certain flow theories to BV vector fields.

Findings

Characterization of Sobolev $W^{1,1}$ functions among $BV$ functions.

New characterization of absolutely continuous measures.

Limitations of existing flow theories for BV vector fields.

Abstract

In this note, we give a new characterisation of Sobolev $W^{1,1}$ functions among $BV$ functions via Hardy-Littlewood maximal function. Exploiting some ideas coming from the proof of this result, we are also able to give a new characterisation of absolutely continuous measures via a weakened version of Hardy-Littlewood maximal function. Finally, we show that the approach adopted in [Crippa and De Lellis, J. Reine Angew. Math. (2008)] and [Jabin, J. Math. Pures Appl. (2010)] to establish existence and uniqueness of regular Lagrangian flows associated to Sobolev vector fields cannot be further extended to the case of $BV$ vector fields.