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 $u_0\in W^{1,1}({\mathbb{R}}^d)$ is a radial function, we show that the associated maximal function $u^{\ast }$ is weakly differentiable and $$\Vert \nabla u^{\ast }{\Vert }_{L^1({\mathbb{R}}^d)}{\lesssim }_d\Vert \nabla u_0{\Vert }_{L^1({\mathbb{R}}^d)}.$$ 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 ${\mathbb{S}}^d$.

28 pages. V2 with minor updates and typos corrected

References in corpus (2)

• Regularity of fractional maximal functions through Fourier multipliers
• Regularity and continuity of local Multilinear Maximal type operator

Cited by in corpus (1)

• Weak differentiability for fractional maximal functions of general $L^p$ functions on domains