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

Abstract

For d ≥ 2, D ≥ 1, let 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 Calderón-Zygmund kernel, K, the maximally modulated and maximally truncated discrete singular integral operator, sup m) , is bounded on ℓ p (Z D ), for each 1 < p < ∞. 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:) dt . Contents 1. Introduction 1 2. A Reveiw of Stein-Wainger 6 3. Exponential Sums and Sublevel Estimates 10 4. The Discrete Stein-Wainger Operator 13 5. Approximations 18 6. Analytic Estimates 23 Appendix A. The Proof of Theorem 3.2 30 References 36