On the continuity of maximal operators of convolution type at the derivative level

TL;DR

This paper proves the continuity of the derivative of maximal convolution operators on Sobolev spaces, specifically for kernels like Poisson, Heat, and fractional Laplacians, advancing understanding of their regularity properties.

Contribution

It establishes the first known continuity result for the derivative of centered maximal operators of convolution type on Sobolev spaces.

Findings

Proves continuity of the map u ↦ (u*)' from W^{1,1}(R) to L^1(R).

Applies to maximal functions associated with Poisson, Heat, and fractional Laplacian kernels.

First result of its kind for centered maximal operators.

Abstract

In this paper we study a question related to the continuity of maximal operators of convolution type acting on $W^{1,1}(\mathbb{R})$. We prove that the map $u\mapsto (u^{*})'$ is continuous from $W^{1,1}(\mathbb{R})$ to $L^{1}(\mathbb{R})$, where $u^{*}$ is the maximal function associated to the Poisson kernel, the Heat kernel or a family of kernels related to the fractional Laplacian. This is the first result of this type for a centered maximal operator.