Bounds for anisotropic Carleson operators

TL;DR

This paper establishes weak (2,2) bounds for anisotropic Carleson operators, extending classical results to higher dimensions and anisotropic settings using advanced time-frequency analysis techniques.

Contribution

It generalizes the classical Carleson operator bounds to anisotropic multipliers in higher dimensions and adapts the time-frequency method accordingly.

Findings

Proved weak (2,2) bounds for anisotropic Carleson operators.

Extended time-frequency analysis techniques to anisotropic settings.

Discussed open problems related to Carleson operators along monomial curves.

Abstract

We prove weak $(2,2)$ bounds for maximally modulated anisotropically homogeneous smooth multipliers on $\mathbb{R}^n$. These can be understood as generalizing the classical one-dimensional Carleson operator. For the proof we extend the time-frequency method by Lacey and Thiele to the anistropic setting. We also discuss a related open problem concerning Carleson operators along monomial curves.