On the variation of maximal operators of convolution type

TL;DR

This paper investigates how certain convolution-based maximal operators, like heat flow and Poisson maximal operators, affect the regularity of functions across different dimensions, showing they do not increase variation in specific cases.

Contribution

The paper establishes that these maximal operators do not increase the $L^p$-variation in one dimension and the $L^2$-variation in higher dimensions, including discrete versions.

Findings

Maximal operators do not increase $L^p$-variation for $p \,\geq\, 1$ in 1D.

In higher dimensions, the operators do not increase $L^2$-variation.

Results extend to discrete convolution-based maximal operators.

Abstract

In this paper we study the regularity properties of two maximal operators of convolution type: the heat flow maximal operator (associated to the Gauss kernel) and the Poisson maximal operator (associated to the Poisson kernel). In dimension $d=1$ we prove that these maximal operators do not increase the $L^p$-variation of a function for any $p \geq 1$, while in dimensions $d>1$ we obtain the corresponding results for the $L^2$-variation. Similar results are proved for the discrete versions of these operators.