On Discrete Hardy–Littlewood Maximal Functions over the Balls in $${\boldsymbol {\mathbb {Z}^d}}$$ : Dimension-Free Estimates

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“…The second part is based on the dimension-free estimates for short variations (a square function corresponding to two-variations that are taken over all dyadic blocks); this is the place where the constraint for p ∈ (3/2, 4) comes from. Our approach is based on certain bootstrap arguments, which were recently adjusted to abstract settings in [230] At the same time as [227], [232], [233], in collaboration with Bourgain, Stein, and Wróbel we originated systematic studies of dimension-free phenomena in the discrete setting. For every x ∈ Z d and t > 0 and for every function f ∈ 1 (Z d ), we define the discrete Hardy-Littlewood averaging operator over G t ∩ Z d by…”

“…On the other hand, we proved in [227] some results with positive conclusions. In [227] we were also able to obtain analogues of r-variational estimates (II.3.10) and (II.3.12) for M B ∞ t with the same ranges for p and r. In [230] we extended these r-variational estimates to the endpoint r = 2, and we obtained analogues of jump inequalities (II.3.9) and (II.3.11) for M B ∞ t with the same ranges for p. The case of discrete Euclidean balls was considered in [232], [233]. Finally, it is worth mentioning that the operators from (II.3.7) and (II.3.13) have ergodic interpretations and some pointwise ergodic theorems hold.…”

Discrete Maximal Operators Over Surfaces of Higher Codimension

“…Initiated by work of Bourgain [9] in ergodic theory, research in this direction has continued to evolve into a standalone subfield of harmonic analysis following the pivotal work of Magyar, Stein and Wainger [44], where they considered the discrete analog of the spherical maximal function. Several authors have proved maximal and/or improving inequalities for discrete operators over lattice points on surfaces of arithmetic interest; see [1][2][3]12,15,24,27,28,32,36,40,41,46] for some such results. A distinctive feature of such work is the interplay between analysis and number theory, as the arithmetic properties of the underlying discrete set play a central role when the analogous continuous operator involves curvature.…”

Discrete maximal operators over surfaces of higher codimension

“…Initiated by work of Bourgain [7] in ergodic theory, research in this direction has continued to evolve into a standalone subfield of harmonic analysis following the pivotal work of Magyar, Stein and Wainger [39], where they considered the discrete analogue of the spherical maximal function. Several authors have proved maximal and/or improving inequalities for discrete operators over lattice points on surfaces of arithmetic interest; see [1,2,10,13,21,24,25,29,31,35,36,41] for some such results. A distinctive feature of such work is the interplay between analysis and number theory, as the arithmetic properties of the underlying discrete set play a central role when the analogous continuous operator involves curvature.…”