A maximal function characterisation of absolutely continuous measures and Sobolev functions

In this note, we give a new characterisation of Sobolev W1; 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 [3, 8] 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.