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  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
Annals of Global Analysis and Geometry – Springer Journals
Published: Jun 5, 2018
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