Discrete Analogues in Harmonic Analysis: A Theorem of Stein-Wainger

TL;DR

This paper proves the boundedness of a class of discrete polynomial-modulated singular integral operators on al^p spaces, extending continuous harmonic analysis results to the discrete setting using equidistribution theory.

Contribution

It introduces a novel approach using equidistribution of polynomial orbits to relate discrete operators to their continuous analogues, establishing boundedness results.

Findings

Boundedness of discrete polynomial-modulated singular integrals on al^p spaces.

Extension of Stein-Wainger continuous results to the discrete setting.

Use of equidistribution theory to connect discrete and continuous harmonic analysis.

Abstract

For $d \geq 2, \ D \geq 1$, let $\mathscr{P}_{d,D}$ denote the set of all degree $d$ polynomials in $D$ dimensions with real coefficients without linear terms. We prove that for any Calder\'{o}n-Zygmund kernel, $K$, the maximally modulated and maximally truncated discrete singular integral operator, \begin{align*} \sup_{P \in \mathscr{P}_{d,D}, \ N} \Big| \sum_{0 < |m| \leq N} f(x-m) K(m) e^{2\pi i P(m)} \Big|, \end{align*} is bounded on $\ell^p(\mathbb{Z}^D)$, for each $1 < p < \infty$. Our proof introduces a stopping time based off of equidistribution theory of polynomial orbits to relate the analysis to its continuous analogue, introduced and studied by Stein-Wainger: \begin{align*} \sup_{P \in \mathscr{P}_{d,D}} \Big| \int_{\mathbb{R}^D} f(x-t) K(t) e^{2\pi i P(t)} \ dt \Big|. \end{align*}