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

Abstract

In this paper we study a question related to the continuity of maximal operators of convolution type acting on W1,1(ℝ). We prove that the map u ↦ (u*)′ is continuous from W1,1(ℝ) to L1(ℝ), where u* 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.