The second derivative of the discrete Hardy-Littlewood maximal function

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.