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 Mat. Zametki, 2013, Volume 94, Issue 4, Pages 488–505 (Mi mz10318)

Reduction of the Calculus of Pseudodifferential Operators on a Noncompact Manifold to the Calculus on a Compact Manifold of Doubled Dimension

A. A. Arutyunova, A. S. Mishchenkob

a Steklov Mathematical Institute of the Russian Academy of Sciences
b M. V. Lomonosov Moscow State University

Abstract: The paper is devoted to the exposition of results announced in [1] We construct a reduction (following an idea of S. P. Novikov) of the calculus of pseudodifferential operators on Euclidean space $\mathbb{R}^{n}$ to a similar calculus in the space of sections of a one-dimensional fiber bundle $\xi$ on the $2n$-dimensional torus $\mathbb{T}^{2n}$. This reduction enables us to identify the Schwartz space on $\mathbb{R}^{n}$ with the space of smooth sections $\Gamma^{\infty}(T^{2n},\xi)$, compare the Sobolev norms on the corresponding spaces and pseudodifferential operators in them, and describe the class of elliptic operators that reduce to Fredholm operators in Sobolev norms. Thus, for a natural class of elliptic pseudodifferential operators on a noncompact manifold of $\mathbb{R}^n$, we construct an index formula in accordance with the classical Atya–Singer formula.

Keywords: pseudodifferential operator, Euclidean space $\mathbb{R}^{n}$, fiber bundle, space of sections, $2n$-dimensional torus $\mathbb{T}^{2n}$, Schwartz space, Sobolev norm, elliptic operator, Fredholm operator, Atya–Singer formula.

 Funding Agency Grant Number Russian Foundation for Basic Research 11-01-00057-à

DOI: https://doi.org/10.4213/mzm10318

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English version:
Mathematical Notes, 2013, 94:4, 455–469

Bibliographic databases:

UDC: 515.168.5+517.983.37

Citation: A. A. Arutyunov, A. S. Mishchenko, “Reduction of the Calculus of Pseudodifferential Operators on a Noncompact Manifold to the Calculus on a Compact Manifold of Doubled Dimension”, Mat. Zametki, 94:4 (2013), 488–505; Math. Notes, 94:4 (2013), 455–469

Citation in format AMSBIB
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\jour Mat. Zametki
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\pages 488--505
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\jour Math. Notes
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• http://mi.mathnet.ru/eng/mz10318
• https://doi.org/10.4213/mzm10318
• http://mi.mathnet.ru/eng/mz/v94/i4/p488

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This publication is cited in the following articles:
1. A. A. Arutyunov, “Reduction of Nonlocal Pseudodifferential Operators on a Noncompact Manifold to Classical Pseudodifferential Operators on a Double-Dimensional Compact Manifold”, Math. Notes, 97:4 (2015), 502–509
2. Arutyunov A.A., Liou Y.-Ch., “On a Method of Reducing Partial Differential Equations To Solving the ODEs”, J. Nonlinear Convex Anal., 17:4 (2016), 787–790
3. Mishchenko A.S., “Correlation Between the Hochschild Cohomology and the Eilenberg-Maclane Cohomology of Group Algebras From a Geometric Point of View”, Russ. J. Math. Phys., 27:2 (2020), 236–250
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