Submitted Paper
Inserted: 22 jul 2018
Last Updated: 24 jul 2018
Year: 2018
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 "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.
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