Maximal function inequalities and a theorem of Birch

Abstract

In this paper we prove an analogue of the discrete spherical maximal theorem of Magyar, Stein, and Wainger, an analogue which concerns maximal functions associated to homogenous algebraic hypersurfaces. Let p be a homogenous polynomial in n variables with integer coefficients of degree d > 1. The maximal functions we consider are defined by A * f (y) = sup N≥1

A note on discrete spherical averages over sparse sequences

“…The proof follows the argument of [5], which is itself a blend of the arguments of Bourgain in ([2], section 7) and Magyar, Stein, and Wainger [11]. The current setup is different from that of [5], although with the sequence chosen in the introduction the differences end up being minimal and much of that work will apply to our current setup with minimal modifications.…”

“…The discussion here follows along the lines of the argument in [5] and a byproduct of this is that, in contrast to the arguments of [8] and [9], there is no reliance on estimates for Kloostermann sums. As a result one can easily extend our result to more general positive definite integral forms of higher degree like those considered in [10].…”

Sparse bounds for the discrete spherical maximal functions

“…We leave the details to the reader. (5) The main results of [10] prove sparse bounds for the Magyar Stein Wainger discrete spherical maximal function. Those inequalities can be combined with Theorem 1.1 and Theorem 1.2 to give novel sparse bounds for these operators.…”

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.…”

“…The next theorem states our asymptotic formula for the multiplier T λ (ξ , η). While we do not need this result directly in the proof of Theorem 1, such approximations are of independent interest: see [2,15,27,41,44]. We include this theorem here, since its proof requires little work beyond what is needed to prove our main results.…”