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Journal of Noncommutative Geometry

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Volume 13, Issue 1, 2019, pp. 363–406
DOI: 10.4171/JNCG/326

Published online: 2019-03-11

Commutator estimates on contact manifolds and applications

Heiko Gimperlein[1] and Magnus Goffeng[2]

(1) Heriot-Watt University, Edinburgh, UK and University of Paderborn, Germany
(2) Chalmers University of Technology and University of Gothenburg, Sweden

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 Œ$[D, f]$ extends to an $L^2$-bounded operator. Using interpolation, it implies sharpweak-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.

Keywords: Commutator estimates, Heisenberg calculus, hypoelliptic operators, weak Schatten norm estimates, Hankel operators, Connes metrics

Gimperlein Heiko, Goffeng Magnus: Commutator estimates on contact manifolds and applications. J. Noncommut. Geom. 13 (2019), 363-406. doi: 10.4171/JNCG/326