Pseudodifferential operators on manifolds with fibred corners

[Opérateurs pseudodifférentiels sur les variétés à coins fibrés]

Debord, Claire ; Lescure, Jean-Marie ; Rochon, Frédéric

Annales de l'Institut Fourier, Tome 65 (2015), p. 1799-1880 / Harvested from Numdam

Résumé

Un moyen géométrique d’encoder les singularités d’une pseudovariété stratifiée est de munir son intérieur d’une métrique cuspidale fibrée itérée. Pour une telle métrique, nous développons et étudions un calcul pseudodifférentiel généralisant le $\Phi $-calcul de Mazzeo et Melrose. Notre point de départ est l’observation bien connue qu’une pseudovariété stratifiée peut être « désingularisée » en variété à coins fibrés. Cela nous permet de définir les opérateurs pseudodifférentiels comme des distributions conormales sur un espace double éclaté approprié. Des applications symboles sont introduites, conduisant à la notion d’ellipticité pleine. Nous utilisons cela pour construire des paramétrix fins et pour caractériser les propriétés de nos opérateurs pseudodifférentiels, comme le fait d’être de Fredholm ou compacts. Nous introduisons aussi une version semi-classique du calcul que nous utilisons pour établir une dualité de Poincaré entre la $K$-homologie de la pseudovariété stratifiée et le $K$-groupe des opérateurs pleinement elliptiques.

Abstract

One way to geometrically encode the singularities of a stratified pseudomanifold is to endow its interior with an iterated fibred cusp metric. For such a metric, we develop and study a pseudodifferential calculus generalizing the $\Phi $-calculus of Mazzeo and Melrose. Our starting point is the well-known observation that a stratified pseudomanifold can be ‘resolved’ into a manifold with fibred corners. This allows us to define pseudodifferential operators as conormal distributions on a suitably blown-up double space. Various symbol maps are introduced, leading to the notion of full ellipticity. This is used to construct refined parametrices and to provide criteria for the mapping properties of operators such as Fredholmness or compactness. We also introduce a semiclassical version of the calculus and use it to establish a Poincaré duality between the $K$-homology of the stratified pseudomanifold and the $K$-group of fully elliptic operators.

Détail

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2974
Classification:  58J40,  58J05,  19K35
Mots clés: Géométrie différentielle, Analyse des équations aux dérivées partielles, $K$-théorie, $K$-homologie.

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