Regularity of the Hardy-Littlewood maximal operator on block decreasing functions

TL;DR

This paper investigates how the Hardy-Littlewood maximal operator affects the regularity of block decreasing functions, showing it can improve their differentiability properties, especially when defined via an unconditional norm.

Contribution

It demonstrates that the uncentered maximal operator enhances the regularity of block decreasing functions of special bounded variation, extending results to the cube-based case.

Findings

Uncentered maximal operator maps block decreasing functions of special bounded variation to functions with integrable derivatives.

For the cube-based maximal operator, the regularity improvement extends to all block decreasing functions of bounded variation.

Results highlight the regularizing effect of the Hardy-Littlewood maximal operator on specific function classes.

Abstract

We study the Hardy-Littlewood maximal operator defined via an unconditional norm, acting on block decreasing functions. We show that the uncentered maximal operator maps block decreasing functions of special bounded variation to functions with integrable distributional derivatives, thus improving their regularity. In the special case of the maximal operator defined by the l_infty-norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation, not necessarily special.