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

A maximal function characterization of absolutely continuous measures and Sobolev functions