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

Abstract

Consider the maximal operator $$\mathcal{C}f(x)=\underset{\lambda \in \mathbb{R}}{\sup }|\sum _{\begin{array}{c}y\in {\mathbb{Z}}^n\setminus \{0\}\end{array}}f(x-y)e(\lambda |y{|}^{2d})K(y)|,(x\in {\mathbb{Z}}^n),$$ 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 $\mathcal{C}$, answering a question posed by Lillian Pierce.

24 pages. Additional details inserted; final version to appear in JEMS

Cited by in corpus (1)

• Discrete analogues of maximally modulated singular integrals of Stein-Wainger type: ${\ell }^p$ bounds for $p>1$