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.
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".
Keywords: manifolds with singularities, pseudodifferential operators, elliptic operators, index
Mot clés : varietés à singularités, opérateurs pseudodifférentiels, opérateurs elliptiques, index
Fedosov, Boris 1; Schulze, Bert-Wolfgang 1; Tarkhanov, Nikolai 1
@article{AIF_2002__52_3_899_0, author = {Fedosov, Boris and Schulze, Bert-Wolfgang and Tarkhanov, Nikolai}, title = {Analytic index formulas for elliptic corner operators}, journal = {Annales de l'Institut Fourier}, pages = {899--982}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {52}, number = {3}, year = {2002}, doi = {10.5802/aif.1906}, zbl = {1010.58018}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1906/} }
TY - JOUR AU - Fedosov, Boris AU - Schulze, Bert-Wolfgang AU - Tarkhanov, Nikolai TI - Analytic index formulas for elliptic corner operators JO - Annales de l'Institut Fourier PY - 2002 SP - 899 EP - 982 VL - 52 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1906/ DO - 10.5802/aif.1906 LA - en ID - AIF_2002__52_3_899_0 ER -
%0 Journal Article %A Fedosov, Boris %A Schulze, Bert-Wolfgang %A Tarkhanov, Nikolai %T Analytic index formulas for elliptic corner operators %J Annales de l'Institut Fourier %D 2002 %P 899-982 %V 52 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1906/ %R 10.5802/aif.1906 %G en %F 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, Volume 52 (2002) no. 3, pp. 899-982. doi : 10.5802/aif.1906. https://aif.centre-mersenne.org/articles/10.5802/aif.1906/
[AB64] The index problem for manifolds with boundary, Differential Analysis (papers presented at the Bombay Colloquium 1964) (1964), pp. 175-186 | Zbl
[AD62] General boundary value problems for elliptic systems in a multidimensional domain, Dokl. Akad. Nauk SSSR, Volume 146 (1962), pp. 511-514 | MR | Zbl
[APS75] Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Phil. Soc, Volume 77 (1975), pp. 43-69 | DOI | MR | Zbl
[ES97] Pseudo-Differential Operators, Singularities, Applications, Birkhäuser Verlag, Basel, 1997 | MR | Zbl
[Fed74] Analytic formulas for the index of elliptic operators, Trans. Moscow Math. Soc, Volume 30 (1974), pp. 159-241 | MR | Zbl
[Fed78] A periodicity theorem in the algebra of formal symbols, Mat. Sb, Volume 105 (1978), pp. 622-637 | MR | Zbl
[FS96] On the index of elliptic operators on a cone, Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras (Advances in Partial Differential Equations), Volume Vol. 3 (1996), pp. 347-372 | Zbl
[FST98] On the index of elliptic operators on a wedge, J. Funct. Anal, Volume 156 (1998), pp. 164-208 | DOI | MR | Zbl
[FST99] The index of elliptic operators on manifolds with conical points, Sel. Math., New ser., Volume 5 (1999), pp. 467-506 | DOI | MR | Zbl
[GSS00] Cone Pseudodifferential Operators in the Edge Symbolic Calculus, Osaka J. Math, Volume 37 (2000), pp. 221-260 | MR | Zbl
[Hir90] Functional analysis in cone and wedge Sobolev spaces, Ann. Global Anal. Geom, Volume 8 (1990), pp. 167-192 | DOI | MR | Zbl
[Luk72] Pseudodifferential operators on Hilbert bundles, J. Diff. Equ., Volume 12 (1972), pp. 566-589 | DOI | MR | Zbl
[Maz91] Elliptic theory of differential edge operators. I, Comm. Part. Diff. Equ., Volume 16 (1991), pp. 1615-1664 | DOI | MR | Zbl
[Mel87] Pseudodifferential Operators on Manifolds with Corners, Manuscript MIT (1987)
[MM98] Pseudodifferential Operators on Manifolds with Fibred Boundary, Asian J. Math, Volume 2 (1998), pp. 833-866 | MR | Zbl
[MN98] -theory of -algebras of -pseudodifferential operators, Geom. and Funct. Anal, Volume 8 (1998), pp. 99-122 | MR | Zbl
[MP77] Elliptic boundary value problems on manifolds with singularities (Problems of Mathematical Analysis), Volume Vol. 6 (1977), pp. 85-142 | Zbl
[Roz00] On Some Analytical Index Formulas Related to Operator-Valued Symbols (2000) (Preprint 16, Univ. of Postdam, 35pp)
[Sch00] Pseudo-Differential Calculus and Applications to Non-Smooth Configurations, Lecture Notes of TICMI, Volume Vol. 1 (2000), pp. 129 pp. | Zbl
[Sch01] Operators with Symbol Hierarchies and Iterated Asymptotics (2001) (Preprint 10, Univ. of Postdam, 54 pp.) | MR | Zbl
[Sch89] Corner Mellin operators and reductions of orders with parameters, Ann. Scuola Norm. Super. Pisa, Volume 16 (1989) no. 1, pp. 1-81 | Numdam | MR | Zbl
[Sch91] Pseudo-Differential Operators on Manifolds with Singularities, North-Holland, Amsterdam, 1991 | MR | Zbl
[Sch92] The Mellin pseudodifferential calculus on manifolds with corners, Symposium "Analysis on Manifolds with Singularities", Breitenbrunn, 1990 (Teubner-Texte zur Mathematik), Volume 131 (1992), pp. 208-289 | Zbl
[ST00] Pseudodifferential Operators on Manifolds with Corners (2000) (Preprint 13, Univ. of Postdam, 95pp.)
[ST98] Green pseudodifferential operators on manifolds with edges, Comm. Part. Diff. Equ., Volume 23 (1998) no. 1-2, pp. 171-201 | MR | Zbl
[ST99] Elliptic complexes of pseudodifferential operators on manifolds with edges, Evolution Equations, Feshbach Resonances, Singular Hodge Theory (Advances in Partial Differential Equations), Volume Vol. 16 (1999), pp. 287-431 | Zbl
Cited by Sources: