Commutator estimates on contact manifolds and applications

Journal article, 2019

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 Holder 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.