On the continuity of maximal operators of convolution type at the derivative level

In this paper we study a question related to the continuity of maximal operators of convolution type acting on $W^{1,1}(\mathbb{R})$. We prove that the map $u\mapsto (u^{\ast }{)}^{\mathrm{\prime }}$ is continuous from $W^{1,1}(\mathbb{R})$ to $L^1(\mathbb{R})$, where $u^{\ast }$ is the maximal function associated to the Poisson kernel, the Heat kernel or a family of kernels related to the fractional Laplacian. This is the first result of this type for a centered maximal operator.