Geodesic
Reduction of the Calculus of Pseudodifferential Operators on a Noncompact Manifold to the Calculus on a Compact Manifold of Doubled Dimension
Matematičeskie zametki, Tome 94 (2013) no. 4, pp. 488-505.

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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, elliptic operator, Fredholm operator, Atya–Singer formula.
Mots-clés : Sobolev norm 
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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. Matematičeskie zametki, Tome 94 (2013) no. 4, pp. 488-505. http://geodesic.mathdoc.fr/item/MZM_2013_94_4_a1/

[1] A. A. Arutyunov, A. S. Mischenko, “Reduktsiya PDO ischisleniya na nekompaktnom mnogoobrazii”, Dokl. RAN, 451:4 (2013), 1–5

[2] U. Rudin, Funktsionalnyi analiz, Mir, M., 1975 | MR | Zbl

[3] M. A. Shubin, Psevdodifferentsialnye operatory i spektralnaya teoriya, Dobrosvet, 2003 | MR | Zbl

[4] M. S. Agranovich, “Ellipticheskie operatory na zamknutykh mnogoobraziyakh”, Differentsialnye uravneniya s chastnymi proizvodnymi – 6, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 63, VINITI, M., 1990, 5–129 | MR | Zbl

[5] V. S. Rabinovich, “Apriornye otsenki i fredgolmovost odnogo klassa psevdodiffentsialnykh operatorov”, Matem. sb., 92(134):2(10) (1973), 195–208 | MR | Zbl

[6] A. S. Mischenko, Vektornye rassloeniya i ikh primeneniya, Nauka, M., 1984 | MR | Zbl

[7] V. E. Nazaikinskii, A. Yu. Savin, B. Yu. Sternin, Elliptic Theory and Noncommutative Geometry. Nonlocal Elliptic Operators, Oper. Theory Adv. Appl., 183, Birkhäuser Verlag, Basel, 2008 | MR | Zbl

[8] V. V. Grushin, “Psevdodifferentsialnye operatory v $\mathbf R^n$ s ogranichennymi simvolami”, Funkts. analiz i ego pril., 4:3 (1970), 37–50 | MR | Zbl