Calculus of Variations and Geometric Measure Theory

E. Bruè - Q. H. Nguyen - G. Stefani

A maximal function characterization of absolutely continuous measures and Sobolev functions

created by stefani on 22 Jul 2018
modified on 03 Oct 2021


Published Paper

Inserted: 22 jul 2018
Last Updated: 3 oct 2021

Journal: Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.
Volume: 30
Number: 3
Pages: 599--614
Year: 2019
Doi: 10.4171/RLM/862

ArXiv: 1807.08266 PDF


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 "Estimates and regularity results for the DiPerna-Lions flow" and "Differential equations with singular fields" 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.

Keywords: Singular measures, Maximal functions, Regular Lagrangian flows, BV and Sobolev functions