Gradient bounds for radial maximal functions (1906.01487v2)
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 $$|\nabla u*|_{L1(\mathbb{R}d)} \lesssim_d |\nabla u_0|_{L1(\mathbb{R}d)}.$$ 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 $\mathbb{S}d$.