K-Homology classes of elliptic uniform pseudodifferential operators

K-Homology classes of elliptic uniform pseudodifferential operators Ann Glob Anal Geom https://doi.org/10.1007/s10455-018-9614-4 K-Homology classes of elliptic uniform pseudodifferential operators Alexander Engel Received: 21 February 2018 / Accepted: 7 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract We show that an elliptic uniform pseudodifferential operator over a manifold of bounded geometry defines a class in uniform K -homology, and that this class only depends on the principal symbol of the operator. Keywords Pseudodifferential operators · K -Homology · Index theory 1 Introduction Pseudodifferential operators are an indispensable tool in the study of elliptic differential operators (like Dirac operators) and their index theory. The calculus of pseudodifferential operators on compact manifolds encompasses parametrices of elliptic differential operators, i.e., their inverses up to smoothing operators, which enables one to deduce the usual important results about elliptic operators like elliptic regularity. Also, the first published proof of the Atiyah–Singer index theorem [2] goes via pseudodifferential operators. The first goal of the present paper is to set up a suitable calculus of pseudodifferential operators on non-compact manifolds. It turns out that for us the only useful definition of such a calculus is the uniform one, and that such a definition is only possible on manifolds of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Global Analysis and Geometry Springer Journals

K-Homology classes of elliptic uniform pseudodifferential operators

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Mathematics; Global Analysis and Analysis on Manifolds; Differential Geometry; Analysis; Geometry; Mathematical Physics
ISSN
0232-704X
eISSN
1572-9060
D.O.I.
10.1007/s10455-018-9614-4
Publisher site
See Article on Publisher Site

Abstract

Ann Glob Anal Geom https://doi.org/10.1007/s10455-018-9614-4 K-Homology classes of elliptic uniform pseudodifferential operators Alexander Engel Received: 21 February 2018 / Accepted: 7 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract We show that an elliptic uniform pseudodifferential operator over a manifold of bounded geometry defines a class in uniform K -homology, and that this class only depends on the principal symbol of the operator. Keywords Pseudodifferential operators · K -Homology · Index theory 1 Introduction Pseudodifferential operators are an indispensable tool in the study of elliptic differential operators (like Dirac operators) and their index theory. The calculus of pseudodifferential operators on compact manifolds encompasses parametrices of elliptic differential operators, i.e., their inverses up to smoothing operators, which enables one to deduce the usual important results about elliptic operators like elliptic regularity. Also, the first published proof of the Atiyah–Singer index theorem [2] goes via pseudodifferential operators. The first goal of the present paper is to set up a suitable calculus of pseudodifferential operators on non-compact manifolds. It turns out that for us the only useful definition of such a calculus is the uniform one, and that such a definition is only possible on manifolds of

Journal

Annals of Global Analysis and GeometrySpringer Journals

Published: Jun 5, 2018

References

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