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 ∈ W 1,1 (R d ) is a radial function, we show that the associated maximal function u * is weakly differentiable andThis 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 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 S d .
We study the Sobolev regularity on the sphere S d of the uncentered fractional Hardy-Littlewood maximal operator M β at the endpoint p = 1, when acting on polar data. We firstWe then prove that the mapwhen restricted to polar data. Our methods allow us to give a new proof of the continuity of the map f → |∇ M β f | from W 1,1 rad (R d ) to L q (R d ). Moreover, we prove that a conjectural local boundedness for the centered fractional Hardy-Littlewood maximal operator M β implies the continuity of the map f → |∇M β f | from W 1,1 to L q , in the context of polar functions on S d and radial functions on R d .
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