  We establish the $L^p$ boundedness of Hilbert transforms and maximal
functions along flat curves in the Heisenberg group. This generalizes the
$\mathbb{R}^n$ result by Carbery, Christ, Vance, Wainger, and Watson. What is
new about our result compared to the Heisenberg group generalization by
Carbery, Wainger, and Wright is that we allow all three components of the
curves to vary independently, we keep the original form of the conditions
required in the $\mathbb{R}^n$ case, and our method is likely to be generalized
to other stratified nilpotent groups.
