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

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.

Locations

• arXiv (Cornell University) - View - PDF
• Journal of the European Mathematical Society - View - PDF