Analytic index formulas for elliptic corner operators

Boris Fedosov[1]; Bert-Wolfgang Schulze[1]; Nikolai Tarkhanov[1]

  • [1] Universität Potsdam, Institut für Mathematik, Postfach 60 15 53, 14415 Potsdam (Allemagne)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 3, page 899-982
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Fedosov, Boris, Schulze, Bert-Wolfgang, and Tarkhanov, Nikolai. "Analytic index formulas for elliptic corner operators." Annales de l’institut Fourier 52.3 (2002): 899-982. <http://eudml.org/doc/115999>.

@article{Fedosov2002,
abstract = {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.},
affiliation = {Universität Potsdam, Institut für Mathematik, Postfach 60 15 53, 14415 Potsdam (Allemagne); Universität Potsdam, Institut für Mathematik, Postfach 60 15 53, 14415 Potsdam (Allemagne); Universität Potsdam, Institut für Mathematik, Postfach 60 15 53, 14415 Potsdam (Allemagne)},
author = {Fedosov, Boris, Schulze, Bert-Wolfgang, Tarkhanov, Nikolai},
journal = {Annales de l’institut Fourier},
keywords = {manifolds with singularities; pseudodifferential operators; elliptic operators; index; elliptic pseudodifferential operators; Fredholm index; index formula},
language = {eng},
number = {3},
pages = {899-982},
publisher = {Association des Annales de l'Institut Fourier},
title = {Analytic index formulas for elliptic corner operators},
url = {http://eudml.org/doc/115999},
volume = {52},
year = {2002},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 3
SP - 899
EP - 982
AB - 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.
LA - eng
KW - manifolds with singularities; pseudodifferential operators; elliptic operators; index; elliptic pseudodifferential operators; Fredholm index; index formula
UR - http://eudml.org/doc/115999
ER -

References

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