# Discrete analogues of maximally modulated singular integrals of Stein-Wainger type

## TL;DR

This paper establishes -bounds for a discrete maximal operator with oscillatory phases, extending Stein-Wainger type results to the integer lattice and addressing a question by Lillian Pierce.

## Contribution

It introduces a discrete analogue of Stein-Wainger maximal operators with nonlinear phases and proves -bounds, a novel result in the discrete setting.

## Key findings

- Proved -bounds for the discrete maximal operator ()
- Extended Stein-Wainger results to the integer lattice setting
- Addressed and answered a question posed by Lillian Pierce

## Abstract

Consider the maximal operator $$\mathscr{C} f(x) = \sup_{\lambda\in\mathbb{R}}\Big|\sum_{\substack{y\in\mathbb{Z}^n\setminus\{0\}}} f(x-y) e(\lambda |y|^{2d}) K(y)\Big|,\quad (x\in\mathbb{Z}^n),$$ where $d$ is a positive integer, $K$ a Calder\'on-Zygmund kernel and $n\ge 1$. This is a discrete analogue of a real-variable operator studied by Stein and Wainger. The nonlinearity of the phase introduces a variety of new difficulties that are not present in the real-variable setting. We prove $\ell^2(\mathbb{Z}^n)$-bounds for $\mathscr{C}$, answering a question posed by Lillian Pierce.