Discrete analogues of maximally modulated singular integrals of Stein-Wainger type: $\ell^p$ bounds for $p>1$

TL;DR

This paper establishes $ ext{ell}^p$ bounds for discrete analogues of Stein-Wainger type singular integrals for all $p$ in $(1, olinebreak \infty)$, extending previous results from the case $p=2$ and opening avenues for future research.

Contribution

The authors prove $ ext{ell}^p$ boundedness of discrete maximally modulated singular integrals of Stein-Wainger type for all $p$ in $(1, olinebreak \infty)$, generalizing earlier $p=2$ results.

Findings

Boundedness of discrete Stein-Wainger type integrals on $ ext{ell}^p$ for all $p$ in $(1, olinebreak \infty)$.

Extension of previous $p=2$ results to a full range of $p$ values.

Discussion of open problems for further research.

Abstract

It is proved that certain discrete analogues of maximally modulated singular integrals of Stein-Wainger type are bounded on $\ell^p(\mathbb{Z}^n)$ for all $p\in (1,\infty)$. This extends earlier work of the authors concerning the case $p=2$. Some open problems for further investigation are briefly discussed.