Discrete analogues in harmonic analysis: A theorem of Stein-Wainger

Abstract

For d≥2,D≥1, let Pd,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,supP∈Pd,D,N⁡|∑0<|m|≤Nf(x−m)K(m)e2πiP(m)|, is bounded on ℓp(ZD), 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:supP∈Pd,D⁡|∫RDf(x−t)K(t)e2πiP(t)dt|.