On the regularity of the maximal function of a BV function

TL;DR

This paper proves that the non-centered maximal function of a BV function is quasicontinuous and explores conditions under which it becomes a Sobolev function, with implications for sets of finite perimeter.

Contribution

It establishes the quasicontinuity of the maximal function of BV functions and links maximal functions of SBV functions to Sobolev functions, extending previous results.

Findings

Non-centered maximal function of BV functions is quasicontinuous.

If maximal function of SBV is BV, then it is Sobolev.

Maximal function of a set of finite perimeter is Sobolev.

Abstract

We show that the non-centered maximal function of a BV function is quasicontinuous. We also show that \emph{if} the non-centered maximal functions of an SBV function is a BV function, then it is in fact a Sobolev function. Using a recent result of Weigt, we are in particular able to show that the non-centered maximal function of a set of finite perimeter is a Sobolev function.