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 $u^{\ast }$ is weakly differentiable and $$\Vert \mathrm{\nabla }{u}^{\ast }{\Vert }_{L^1({\mathbf{R}}^d)}{\lesssim }_d\Vert \mathrm{\nabla }{u}_0{\Vert }_{L^1({\mathbf{R}}^d)}.$$ 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 ${\mathbf{S}}^d$