Gradient bounds for radial maximal functions

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 \(u_0 \in W^{1,1}(\mathbf{R}^d)\) is a radial function, we show that the associated maximal function \(u^*\) is weakly differentiable and \[\|\nabla u^*\|_{L^1(\mathbf{R}^d)} \lesssim_d \|\nabla u_0\|_{L^1(\mathbf{R}^d)}.\] This establishes the analogue of a recent result of 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 \(\mathbf{S}^d\), 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 \(\mathbf{S}^d\).