  This article studies sharp norm estimates for the commutator of
pseudo-differential operators with multiplication operators on closed
Heisenberg manifolds. In particular, we obtain a Calderon commutator estimate:
If $D$ is a first-order operator in the Heisenberg calculus and $f$ is
Lipschitz in the Carnot-Caratheodory metric, then $[D,f]$ extends to an
$L^2$-bounded operator. Using interpolation, it implies sharp weak--Schatten
class properties for the commutator between zeroth order operators and H\"older
continuous functions. We present applications to sub-Riemannian spectral
triples on Heisenberg manifolds as well as to the regularization of a
functional studied by Englis-Guo-Zhang.
