Sharp higher order regularity of discrete maximal functions

We derive sharp ${\mathrm{\ell }}^p(\mathbb{Z})$ bounds for the $k$th derivative of the discrete uncentered maximal operator applied to characteristic functions $f:\mathbb{Z}\to \{0,1\}$ in the cases $k=0,1,2$. When $k=1,2$ these are the first sharp bounds for derivatives of a Hardy-Littlewood maximal function in continuous or discrete settings when $1<p<\mathrm{\infty }$. We also establish several lower bounds for $k\ge 3$ and for general functions $f:\mathbb{Z}\to \mathbb{R}$.