  The regularity of the Hardy-Littlewood maximal function, in both discrete and
continuous contexts, and for both centered and noncentered variants, has been
subjected to intense study for the last two decades. But efforts so far have
concentrated on first order differentiability and variation, as it is known
that in the continuous context higher order regularity is impossible. This
short note gives the first positive result on the higher order regularity of
the discrete noncentered maximal function.
