Gradient bounds for radial maximal functions

In this paper we study the regularity properties of certain maximal operators convolution type at endpoint \(p=1\), when acting on radial data. particular, for heat flow operator and Poisson operator, initial datum \(u_0 \in W^{1,1}(\mathbf{R}^d)\) is a function, show that associated function \(u^*\) weakly differentiable \[\|\nabla u^*\|_{L^1(\mathbf{R}^d)} \lesssim_d \|\nabla u_0\|_{L^1(\mathbf{R}^d)}.\] This establishes analogue recent result Luiro uncentered Hardy-Littlewood now in centered setting with smooth kernels. second part paper, establish similar gradient bounds sphere \(\mathbf{S}^d\), polar functions. Our includes \(\mathbf{S}^d\).