Commutator estimates on contact manifolds and applications

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.