Bounds for anisotropic Carleson operators

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.