  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.
