Endpoint Sobolev and BV continuity for maximal operators, II

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 $f \mapsto (\widetilde M_{\beta}f)'$ from $W^{1,1}(\mathbb{R})$ to $L^q(\mathbb{R})$, for $q={1}/{(1-\beta)}$. Here $\widetilde M_{\beta}$ 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 ${\rm BV}(\mathbb{Z})$, complementing some recent boundedness results.