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Commutator estimates on contact manifolds and applications

Goffeng, Carl Henrik Tryggve Magnus LU orcid and Gimperlein, Heiko (2019) In Journal of Noncommutative Geometry 13(1). p.363-406
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 [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.
Please use this url to cite or link to this publication:
author
and
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Commutator estimates, Heisenberg calculus, hypoelliptic operators, weak Schatten norm estimates, Hankel operators, Connes metrics
in
Journal of Noncommutative Geometry
volume
13
issue
1
pages
363 - 406
publisher
European Mathematical Society Publishing House
external identifiers
  • scopus:85064332790
ISSN
1661-6960
DOI
10.4171/JNCG/326
language
English
LU publication?
no
id
9aa4a902-38c7-4ebc-88ed-7d74b201620b
date added to LUP
2021-03-12 11:53:44
date last changed
2025-10-14 11:55:20
@article{9aa4a902-38c7-4ebc-88ed-7d74b201620b,
  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 [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.}},
  author       = {{Goffeng, Carl Henrik Tryggve Magnus and Gimperlein, Heiko}},
  issn         = {{1661-6960}},
  keywords     = {{Commutator estimates; Heisenberg calculus; hypoelliptic operators; weak Schatten norm estimates; Hankel operators; Connes metrics}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{363--406}},
  publisher    = {{European Mathematical Society Publishing House}},
  series       = {{Journal of Noncommutative Geometry}},
  title        = {{Commutator estimates on contact manifolds and applications}},
  url          = {{http://dx.doi.org/10.4171/JNCG/326}},
  doi          = {{10.4171/JNCG/326}},
  volume       = {{13}},
  year         = {{2019}},
}