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

Consider the maximal operator

Cf(x)=supλRyZn{0}f(xy)e(λy2d)K(y) (xZn),\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 dd is a positive integer, KK a Calderón–Zygmund kernel and n1n\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 2(Zn)\ell^2(\mathbb{Z}^n)-bounds for C\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