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- W2955752220 abstract "Consider the maximal operator $$mathscr{C} f(x) = sup_{lambdainmathbb{R}}Big|sum_{substack{yinmathbb{Z}^nsetminus{0}}} f(x-y) e^{i lambda |y|^{2d}} K(y)Big|,quad (xinmathbb{Z}^n),$$ where $d$ is a positive integer, $K$ an appropriate Calderon-Zygmund kernel and $nge 1$. 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 $mathscr{C}$. Our arguments are inspired by the recent seminal paper of Krause covering the case $n=1$, earlier key partial progress by Krause and Lacey, and Bourgain's classical maximal multi-frequency lemma in combination with variation-norm estimates from recent joint work of the author with S. Guo and P.-L. Yung." @default.
- W2955752220 created "2019-07-12" @default.
- W2955752220 creator A5069161484 @default.
- W2955752220 date "2019-06-30" @default.
- W2955752220 modified "2023-09-23" @default.
- W2955752220 title "Discrete analogues of maximally modulated singular integrals of Stein-Wainger type: $ell^2(mathbb{Z}^n)$ bounds" @default.
- W2955752220 hasPublicationYear "2019" @default.
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