# Gradient bounds for radial maximal functions

## TL;DR

This paper proves gradient bounds for radial maximal functions of convolution type, including heat and Poisson maximal operators, showing they are weakly differentiable with controlled gradient norms, extending recent results to centered and spherical settings.

## Contribution

It establishes new gradient bounds for centered and spherical maximal operators acting on radial data, extending prior uncentered Hardy-Littlewood results.

## Key findings

- Maximal functions are weakly differentiable with bounded gradients.
- Gradient norms are controlled by initial data's gradient norms.
- Results apply to both Euclidean space and spheres.

## Abstract

In this paper we study the regularity properties of certain maximal operators of convolution type at the endpoint $p=1$, when acting on radial data. In particular, for the heat flow maximal operator and the Poisson maximal operator, when the initial datum $u_0 \in W^{1,1}( \mathbb{R}^d)$ is a radial function, we show that the associated maximal function $u^*$ is weakly differentiable and $$\|\nabla u^*\|_{L^1(\mathbb{R}^d)} \lesssim_d \|\nabla u_0\|_{L^1(\mathbb{R}^d)}.$$ This establishes the analogue of a recent result of H. Luiro for the uncentered Hardy-Littlewood maximal operator, now in a centered setting with smooth kernels. In a second part of the paper, we establish similar gradient bounds for maximal operators on the sphere $\mathbb{S}^d$, when acting on polar functions. Our study includes the uncentered Hardy-Littlewood maximal operator, the heat flow maximal operator and the Poisson maximal operator on $\mathbb{S}^d$.