Analytic index formulas for elliptic corner operators
[Formules analytiques de l'indice des opérateurs elliptiques "coin"]
Fedosov, Boris ; Schulze, Bert-Wolfgang ; Tarkhanov, Nikolai
Annales de l'Institut Fourier, Tome 52 (2002), p. 899-982 / Harvested from Numdam

Les espaces à singularités de type "coins", localement modélisés par des cônes dont la base est un espace à singularités coniques, appartiennent à la catégorie des (pseudo-) variétés à géométrie lisse par morceaux. Nous étudions ici le cas typique d'une variété avec coins appelée "fuseau avec arêtes". Sur cette dernière, nous considérons l'algèbre canonique des opérateurs pseudodifférentiels dont les symboles présentent une dégénérescence particulière. Cette algèbre est appelée "algèbre coin" et il y a trois types de symboles principaux : les symboles intérieurs, les symboles à arêtes et les symboles "coins" conormaux. Un opérateur à singularités "coins", possède la propriété de Fredholm sur des espaces de Sobolev appropriés. Par ailleurs, nous établissons une formule de l'indice analytique pour ce genre d'opérateurs. Cette formule est composée de deux termes de natures différentes, oú apparaissent la contribution intérieure et la contribution des "coins".

Spaces with corner singularities, locally modelled by cones with base spaces having conical singularities, belong to the hierarchy of (pseudo-) manifolds with piecewise smooth geometry. We consider a typical case of a manifold with corners, the so-called "edged spindle", and a natural algebra of pseudodifferential operators on it with special degeneracy in the symbols, the "corner algebra". There are three levels of principal symbols in the corner algebra, namely the interior, edge and corner conormal symbols. An operator is called elliptic if all the three principal symbols are invertible. Elliptic corner operators possess the Fredholm property in appropriate Sobolev spaces. We derive an analytic index formula for such operators containing two terms of different nature: the interior and corner contributions. This is a generalization of our previous index formulas for cones and wedges and it suffers the same drawback: the contributions depend not only on the three principal symbols as one could expect but rather on the complete operator-valued symbol along the edge.

Publié le : 2002-01-01
DOI : https://doi.org/10.5802/aif.1906
Classification:  58J20,  58J05
Mots clés: varietés à singularités, opérateurs pseudodifférentiels, opérateurs elliptiques, index
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     author = {Fedosov, Boris and Schulze, Bert-Wolfgang and Tarkhanov, Nikolai},
     title = {Analytic index formulas for elliptic corner operators},
     journal = {Annales de l'Institut Fourier},
     volume = {52},
     year = {2002},
     pages = {899-982},
     doi = {10.5802/aif.1906},
     zbl = {1010.58018},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2002__52_3_899_0}
}
Fedosov, Boris; Schulze, Bert-Wolfgang; Tarkhanov, Nikolai. Analytic index formulas for elliptic corner operators. Annales de l'Institut Fourier, Tome 52 (2002) pp. 899-982. doi : 10.5802/aif.1906. http://gdmltest.u-ga.fr/item/AIF_2002__52_3_899_0/

[AB64] M. F. Atiyah; R. Bott The index problem for manifolds with boundary, Differential Analysis (papers presented at the Bombay Colloquium 1964), Oxford University Press (1964), pp. 175-186 | Zbl 0163.34603

[AD62] M. S. Agranovich; A. S. Dynin General boundary value problems for elliptic systems in a multidimensional domain, Dokl. Akad. Nauk SSSR, Tome 146 (1962), pp. 511-514 | MR 140820 | Zbl 0132.35403

[APS75] M. F. Atiyah; V. K. Patodi; I. M. Singer Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Phil. Soc, Tome 77 (1975), pp. 43-69 | Article | MR 397797 | Zbl 0297.58008

[ES97] Y. Egorov; B.- W. Schulze Pseudo-Differential Operators, Singularities, Applications, Birkhäuser Verlag, Basel (1997) | MR 1443430 | Zbl 0877.35141

[Fed74] B. V. Fedosov Analytic formulas for the index of elliptic operators, Trans. Moscow Math. Soc, Tome 30 (1974), pp. 159-241 | MR 420731 | Zbl 0349.58006

[Fed78] B. V. Fedosov A periodicity theorem in the algebra of formal symbols, Mat. Sb, Tome 105 (1978), pp. 622-637 | MR 488180 | Zbl 0412.47030

[FS96] B. Fedosov; B.-W. Schulze; M. Demuth, E. Schrohe, B.-W. Schulze, J. Sjöstrand (Eds) On the index of elliptic operators on a cone, Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Akademie-Verlag, Berlin (Advances in Partial Differential Equations) Tome Vol. 3 (1996), pp. 347-372 | Zbl 0865.35152

[FST98] B. Fedosov; B.-W. Schulze; N. Tarkhanov On the index of elliptic operators on a wedge, J. Funct. Anal, Tome 156 (1998), pp. 164-208 | Article | MR 1637925 | Zbl 0961.58010

[FST99] B. Fedosov; B.-W. Schulzel; N. Tarkhanov The index of elliptic operators on manifolds with conical points, Sel. Math., New ser., Tome 5 (1999), pp. 467-506 | Article | MR 1740679 | Zbl 0951.58026

[GSS00] J.B. Gil; B.-W. Schulze; J. Seiler Cone Pseudodifferential Operators in the Edge Symbolic Calculus, Osaka J. Math, Tome 37 (2000), pp. 221-260 | MR 1750278 | Zbl 1005.58010

[Hir90] T. Hirschmann Functional analysis in cone and wedge Sobolev spaces, Ann. Global Anal. Geom, Tome 8 (1990), pp. 167-192 | Article | MR 1088510 | Zbl 0739.46023

[Luk72] G. Luke Pseudodifferential operators on Hilbert bundles, J. Diff. Equ., Tome 12 (1972), pp. 566-589 | Article | MR 346856 | Zbl 0238.35077

[Maz91] R. Mazzeo Elliptic theory of differential edge operators. I, Comm. Part. Diff. Equ., Tome 16 (1991), pp. 1615-1664 | Article | MR 1133743 | Zbl 0745.58045

[Mel87] R. B. Melrose Pseudodifferential Operators on Manifolds with Corners, Manuscript MIT, Boston (1987)

[MM98] R. Mazzeo; R. B. Melrose Pseudodifferential Operators on Manifolds with Fibred Boundary, Asian J. Math, Tome 2 (1998), pp. 833-866 | MR 1734130 | Zbl 01531011

[MN98] R.B. Melrose; V. Nistor K-theory of C * -algebras of b-pseudodifferential operators, Geom. and Funct. Anal, Tome 8 (1998), pp. 99-122 | MR 1601850 | Zbl 0898.46060

[MP77] V. G. Maz'Ya; B. A. Plamenevskii Elliptic boundary value problems on manifolds with singularities, Univ. of Leningrad (Problems of Mathematical Analysis) Tome Vol. 6 (1977), pp. 85-142 | Zbl 0453.58022

[Roz00] G. Rozenblum On Some Analytical Index Formulas Related to Operator-Valued Symbols (2000) (Preprint 16, Univ. of Postdam, 35pp)

[Sch00] B.-W. Schulze Pseudo-Differential Calculus and Applications to Non-Smooth Configurations, Lecture Notes of TICMI, Tbilisi University Press, Tome Vol. 1 (2000), pp. 129 pp. | Zbl 0981.35109

[Sch01] B.-W. Schulze Operators with Symbol Hierarchies and Iterated Asymptotics (2001) (Preprint 10, Univ. of Postdam, 54 pp.) | MR 1917163 | Zbl 1051.58011

[Sch89] B.-W. Schulze Corner Mellin operators and reductions of orders with parameters, Ann. Scuola Norm. Super. Pisa, Tome 16 (1989) no. 1, pp. 1-81 | Numdam | MR 1056128 | Zbl 0711.58030

[Sch91] B.-W. Schulze Pseudo-Differential Operators on Manifolds with Singularities, North-Holland, Amsterdam (1991) | MR 1142574 | Zbl 0747.58003

[Sch92] B.-W. Schulze The Mellin pseudodifferential calculus on manifolds with corners, Symposium "Analysis on Manifolds with Singularities", Breitenbrunn, 1990, Teubner-Verlag, Leipzig (Teubner-Texte zur Mathematik) Tome 131 (1992), pp. 208-289 | Zbl 0810.58041

[ST00] B.-W. Schulze; N. Tarkhanov Pseudodifferential Operators on Manifolds with Corners (2000) (Preprint 13, Univ. of Postdam, 95pp.)

[ST98] B.-W. Schulze; N. Tarkhanov Green pseudodifferential operators on manifolds with edges, Comm. Part. Diff. Equ., Tome 23 (1998) no. 1-2, pp. 171-201 | MR 1608512 | Zbl 0901.58062

[ST99] B.-W. Schulze; N. Tarkhanov; M. Demuth, E. Schrohe, B.-W. Schulze, J. Sjöstrand (Eds) Elliptic complexes of pseudodifferential operators on manifolds with edges, Evolution Equations, Feshbach Resonances, Singular Hodge Theory, Wiley-VCH, Berlin et al. (Advances in Partial Differential Equations) Tome Vol. 16 (1999), pp. 287-431 | Zbl 0945.58018