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Index formulas for stratified manifolds. (English. Russian original) Zbl 1205.35351
Differ. Equ. 46, No. 8, 1145-1156 (2010); translation from Differ. Uravn. 46, No. 8, 1135-1146 (2010).
Summary: We consider elliptic operators on stratified manifolds with a stratification of arbitrary length. Under some (symmetry-like) conditions imposed on the symbols of these operators, we obtain index formulas in which the index of an operator is expressed as the sum of indices of some (explicitly written out) elliptic operators on the strata.
MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
58J40 Pseudodifferential and Fourier integral operators on manifolds
47G30 Pseudodifferential operators
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[1] Plamenevskii, B.A. and Senichkin, V.N., Representations of C*-Algebras of Pseudodifferential Operators on Piecewise Smooth Manifolds, Algebra i Analiz, 2001, vol. 13, no. 6, pp. 124–174.
[2] Nazaikinskii, V.E., Savin, A.Yu., and Sternin, B.Yu., Pseudodifferential Operators on Stratified Manifolds. I, Differ. Uravn., 2007, vol. 43, no. 4, pp. 519–532.
[3] Nazaikinskii, V.E., Savin, A.Yu., and Sternin, B.Yu., Pseudodifferential Operators on Stratified Manifolds. II, Differ. Uravn., 2007, vol. 43, no. 5, pp. 685–696. · Zbl 1140.58008
[4] Schulze, B.-W., Pseudo-Differential Calculus on Manifolds with Geometric Singularities, Pseudo-Differential Operators. Partial Differential Equations and Time-Frequency Analysis: Selected Papers of the ISAAC Workshop, Toronto, Canada, December 11–15, 2006, vol. 52 of The Fields Institute for Research in Mathematical Sciences, Fields Institute Communications, 2007, pp. 37–83.
[5] Nistor, V., Pseudodifferential Operators on Non-Compact Manifolds and Analysis on Polyhedral Domains, in Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds, vol. 366 of Contemp. Math.: Amer. Math. Soc., Providence, RI, 2005, pp. 307–328. · Zbl 1091.58017
[6] Schulze, B.-W., Sternin, B., and Shatalov, V., On the Index of Differential Operators on Manifolds with Conical Singularities, Ann. Global Anal. Geom., 1998, vol. 16, no. 2, pp. 141–172. · Zbl 0914.58030 · doi:10.1023/A:1006521714633
[7] Nazaikinskii, V.E., Savin, A.Yu., Sternin, B.Yu., and Schulze, B.-W., On the Index of Elliptic Operators on Manifolds with Edges, Mat. Sb., 2005, vol. 196, no. 9, pp. 23–58. · doi:10.4213/sm1419
[8] Nazaikinskii, V., Savin, A., Schulze, B.-W., and Sternin, B., Elliptic Theory on Singular Manifolds, Boca Raton, 2005. · Zbl 1138.58310
[9] Nazaikinskii, V.E., Savin, A.Yu., and Sternin, B.Yu., On the Homotopy Classification of Elliptic Operators on Stratified Manifolds, Izv. Ross. Akad. Nauk Ser. Mat., 2007, vol. 71, no. 6, pp. 91–118. · Zbl 1154.58013 · doi:10.4213/im781
[10] Atiyah, M.F. and Bott, R., The Index Problem for Manifolds with Boundary, in Bombay Colloquium on Differential Analysis, Oxford, 1964, pp. 175–186. · Zbl 0163.34603
[11] Savin, A.Yu. and Sternin, B.Yu., Index for a Class of Nonlocal Elliptic Operators, in Spectral and Evolution Problems: Proceedings of the Fourteenth Crimean Autumn Mathematical School-Symposium, 2004, vol. 14, pp. 35–41.
[12] Savin, A.Yu. and Sternin, B.Yu., Defect of Index in the Theory of Non-Local Problems and the \(\eta\)-Invariant, Mat. Sb., 2004, vol. 195, no. 9, pp. 85–126. · Zbl 1083.58022 · doi:10.4213/sm847
[13] Luke, G., Pseudodifferential Operators on Hilbert Bundles, J. Differential Equations, 1972, vol. 12, pp. 566–589. · Zbl 0238.35077 · doi:10.1016/0022-0396(72)90026-5
[14] Rozenblum, G., Regularization of Secondary Characteristic Classes and Unusual Index Formulas for Operator-Valued Symbols, in Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations, vol. 145 of Oper. Theory Adv. Appl., Basel, 2003, pp. 419–437. · Zbl 1161.58311
[15] Nazaikinskii, V.E., Savin, A.Yu., and Sternin, B.Yu., On the Homotopy Classification of Elliptic Operators on Stratified Manifolds, Dokl. Akad. Nauk, 2006, vol. 408, no. 5, pp. 591–595. · Zbl 1327.58025
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