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

Ben Krause; Joris Roos

doi:10.4171/jems/1160

JournalsjemsVol. 24, No. 9pp. 3183–3213

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

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Abstract

Consider the maximal operator

C f (x) = λ \in R sup y \in Z^{n} ∖ {0} \sum f (x - y) e (λ ∣ y ∣^{2 d}) K (y) (x \in Z^{n}),

where $d$ is a positive integer, $K$ a Calderón–Zygmund kernel and $n \geq 1$ . 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 $ℓ^{2} (Z^{n})$ -bounds for $C$ , 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