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 continuous and discrete setting, giving a positive answer to two questions posed recently, one of them regarding the continuity of the map $f \mapsto \big(\widetilde M_{\beta}f\big)'$from$W^{1,1}($\mathbb{R}$)$to$L^q($\mathbb{R}$)$,for$q=\frac{1}{1-\beta}$\mathrm{.}Here$\widetilde M_{\beta}$ denotes the non-centered fractional maximal operator on $\mathbb{R}$ with $\beta \in (0,1)$. The second one regarding the continuity of the discrete centered maximal operator in the space of functions of bounded variation BV$(\mathbb{Z})$, complementing some recent boundedness results.