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EQUIVALENCE OF ELLIPTICITY AND THE FREDHOLM PROPERTY IN THE WEYL-HÖRMANDER CALCULUS

Part of: Miscellaneous topics - Partial differential equations Linear function spaces and their duals Integral, integro-differential, and pseudodifferential operators

Published online by Cambridge University Press:  20 January 2021

Stevan Pilipović  and
Bojan Prangoski Open the ORCID record for Bojan Prangoski [Opens in a new window]
Stevan Pilipović
Affiliation:
Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000Novi Sad, Serbia ([email protected])
Bojan Prangoski
Affiliation:
Department of Mathematics, Faculty of Mechanical Engineering-Skopje, University “Ss. Cyril and Methodius”, Karposh 2 b.b., 1000Skopje, Macedonia ([email protected])
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Abstract

The main result is that the ellipticity and the Fredholm property of a $\Psi $ DO acting on Sobolev spaces in the Weyl-Hörmander calculus are equivalent when the Hörmander metric is geodesically temperate and its associated Planck function vanishes at infinity. The proof is essentially related to the following result that we prove for geodesically temperate Hörmander metrics: If $\lambda \mapsto a_{\lambda }\in S(1,g)$ is a $\mathcal {C}^N$ , $0\leq N\leq \infty $ , map such that each $a_{\lambda }^w$ is invertible on $L^2$ , then the mapping $\lambda \mapsto b_{\lambda }\in S(1,g)$ , where $b_{\lambda }^w$ is the inverse of $a_{\lambda }^w$ , is again of class $\mathcal {C}^N$ . Additionally, assuming also the strong uncertainty principle for the metric, we obtain a Fedosov-Hörmander formula for the index of an elliptic operator. At the very end, we give an example to illustrate our main result.

Keywords

Hörmander metricgeodesic temperanceSobolev spaces H(M,g)

MSC classification

Primary: 35S05: Pseudodifferential operators
Secondary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 47G30: Pseudodifferential operators
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 4 , July 2022 , pp. 1363 - 1389
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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