Almost sharp variational estimates for discrete truncated operators of Stein-Wainger type

TL;DR

This paper proves sharp r-variational estimates for discrete Stein-Wainger operators on ^p, improving prior results by reducing logarithmic losses and extending to quadratic phases in higher dimensions.

Contribution

It introduces nearly sharp variational bounds for discrete Stein-Wainger operators, refining previous estimates and removing logarithmic losses in specific cases.

Findings

Established sharp r-variational estimates for Stein-Wainger operators.

Improved previous bounds by reducing logarithmic scale losses.

Extended results to quadratic phases in higher dimensions.

Abstract

We establish $r$-variational estimates for discrete truncated Stein-Wainger type operators on $\ell^p$ for $1<p<\infty$. Notably, these estimates are sharp and enhance the results obtained by Krause and Roos (J. Eur. Math. Soc. 2022, J. Funct. Anal. 2023), up to a logarithmic loss related to the scale. On the other hand, as $r$ approaches infinity, the consequences align with the estimates proved by Krause and Roos. Moreover, for the case of quadratic phases, we remove this logarithmic loss with respect to the scale in two and higher dimensions, at the cost of increasing $p$ slightly.