Regularity of maximal operators: recent progress and some open problems

TL;DR

This paper reviews recent advances and open problems in the regularity theory of maximal operators acting on Sobolev and BV functions, highlighting key research challenges in harmonic analysis and geometric measure theory.

Contribution

It provides a comprehensive overview of current progress and presents fifteen open research problems in the regularity of maximal operators.

Findings

Summarizes recent progress in regularity theory.

Identifies key open problems in the field.

Provides a list of fifteen research challenges.

Abstract

This is an expository paper on the regularity theory of maximal operators, when these act on Sobolev and BV functions, with a special focus on some of the current open problems in the topic. Overall, a list of fifteen research problems is presented. It summarizes the contents of a talk delivered by the author at the CIMPA 2017 Research School - Harmonic Analysis, Geometric Measure Theory and Applications, in Buenos Aires, Argentina.