Consider the maximal operator
where is a positive integer, a Calderón–Zygmund kernel and . 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 -bounds for , 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. 3183–3213DOI 10.4171/JEMS/1160