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Commutator estimates on contact manifolds and applications

Journal article
Authors H. Gimperlein
Magnus Goffeng
Published in Journal of Noncommutative Geometry
Volume 13
Issue 1
Pages 363-406
ISSN 1661-6952
Publication year 2019
Published at Department of Mathematical Sciences
Pages 363-406
Language en
Links dx.doi.org/10.4171/JNCG/326
Keywords Commutator estimates, Connes metrics, Hankel operators, Heisenberg calculus, Hypoelliptic operators, Weak Schatten norm estimates
Subject categories Geometry, Mathematical Analysis

Abstract

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. © European Mathematical Society.

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