Endpoint Sobolev and BV continuity for maximal operators, II

Abstract

In this paper we study some questions about the continuity of classical and fractional maximal operators in the Sobolev space $W^{1,1}$, in both the continuous and discrete setting, giving a positive answer to two questions posed recently, one of them regarding the continuity of the map $f\mapsto ({\tilde{M}}_{\beta }f{)}^{\prime }$ from $W^{1,1}(\mathbb{R})$ to $L^q(\mathbb{R})$, for $q=1/(1-\beta )$. Here ${\tilde{M}}_{\beta }$ denotes the non-centered fractional maximal operator on $\mathbb{R}$, with $\beta \in (0,1)$. The second one is related to the continuity of the discrete centered maximal operator in the space of functions of bounded variation $\mathrm{BV}(\mathbb{Z})$, complementing some recent boundedness results.