  We prove $L^p_{comp}\to L^p_{s}$ boundedness for averaging operators
associated to a class of curves in the Heisenberg group $\mathbb{H}^1$ via
$L^2$ estimates for related oscillatory integrals and Bourgain-Demeter
decoupling inequalities on the cone. We also construct a Sobolev space adapted
to translations on the Heisenberg group to which these averaging operators map
all $L^p$ functions boundedly.
