Analytic index formulas for elliptic corner operators

Fedosov, Boris; Schulze, Bert-Wolfgang; Tarkhanov, Nikolai

Annales de l'Institut Fourier, Tome 52 (2002) no. 3, pp. 899-982.

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.

Classification : 58J20, 58J05
Keywords: manifolds with singularities, pseudodifferential operators, elliptic operators, index
Mot clés : varietés à singularités, opérateurs pseudodifférentiels, opérateurs elliptiques, index

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