GRADIENT BOUNDS FOR RADIAL MAXIMAL FUNCTIONS

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 u0 is an element of W1,1(Rd) is a radial function, we show that the associated maximal function u* is weakly differentiable and 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 Sd, 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 Sd.