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

### Ben Krause

King’s College London, UK### Joris Roos

University of Massachusetts Lowell, USA; The University of Edinburgh, Scotland, UK

## Abstract

Consider the maximal operator

$\mathscr{C} f(x) = \sup_{\lambda\in\mathbb{R}}\,\Bigl|\sum_{\substack{y\in\mathbb{Z}^n\setminus\{0\}}} f(x-y) e(\lambda |y|^{2d}) K(y)\Bigr|\quad\ (x\in\mathbb{Z}^n),$

where $d$ is a positive integer, $K$ a Calderón–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.

## Cite this article

Ben Krause, Joris Roos, Discrete analogues of maximally modulated singular integrals of Stein–Wainger type. J. Eur. Math. Soc. 24 (2022), no. 9, pp. 3183–3213

DOI 10.4171/JEMS/1160