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 Calderón commutator estimate: If D is a first-order operator in the Heisenberg calculus and f is Lipschitz in the Carnot–Carathéodory metric, then ŒD; 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 Hölder 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.