Bertrand Monthubert

Pseudodifferential calculus on manifolds with corners and groupoids

We build a longitudinally smooth, differentiable groupoid associated to any manifold with corners. The pseudodifferential calculus on this groupoid coincides with the pseudodifferential calculus of Melrose (also called b-calculus). We also define an algebra of rapidly decreasing functions on this groupoid; it contains the kernels of the smoothing operators of the (small)

Groupoids and pseudodifferential calculus on manifolds with corners

We associate to any manifold with corners (even with non-embedded hyperfaces) a (non-Hausdorff) longitudinally smooth Lie groupoid, on which we define a pseudodifferential calculus. This calculus generalizes the b-calculus of Melrose, defined for manifolds with embedded corners. The groupoid of a manifold with corners is shown to be unique up to equivalence for manifolds with corners of same codimension. Using tools from the theory of C∗-algebras of groupoids, we also obtain new proofs for the study of b-calculus.

SPECTRAL INVARIANCE FOR CERTAIN ALGEBRAS OF PSEUDODIFFERENTIAL OPERATORS

We construct algebras of pseudodifferential operators on a continuous family groupoid G that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on G as a dense subalgebra, and reflect the smooth structure of the groupoid G, when G is smooth. As an application, we get a better understanding on the structure of inverses of elliptic pseudodifferential operators on classes of non-compact manifolds. For the construction of these algebras closed under holomorphic functional calculus, we develop three methods: one using two-sided semi-ideals, one using commutators, and one based on Schwartz spaces on the groupoid.

A topological index theorem for manifolds with corners

We define an analytic index and prove a topological index theorem for a non-compact manifold M 0 with poly-cylindrical ends. Our topological index theorem depends only on the principal symbol, and establishes the equality of the topological and analytical index in the group K 0 ( C * ( M )), where C * ( M ) is a canonical C * -algebra associated to the canonical compactification M of M 0 . Our topological index is thus, in general, not an integer, unlike the usual Fredholm index appearing in the Atiyah–Singer theorem, which is an integer. This will lead, as an application in a subsequent paper, to the determination of the K -theory groups K 0 ( C * ( M )) of the groupoid C * -algebra of the manifolds with corners M . We also prove that an elliptic operator P on M 0 has an invertible perturbation P + R by a lower-order operator if and only if its analytic index vanishes.

Boutet de Monvel’s calculus and groupoids I

Can Boutet de Monvels algebra on a compact manifold with boundary be obtained as the algebra

Invariance spectrale des algèbres d'opérateurs pseudodifférentiels

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Errata to “Pseudodifferential calculus on manifolds with corners and groupoids”

of pseudodifferential operators on some Lie groupoid

Pseudodifferential Analysis on Continuous Family Groupoids

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Indice analytique et groupoïdes de Lie

? If it could, the kernel

PSEUDODIFFERENTIAL OPERATORS ON NON-COMPACT MANIFOLDS AND ANALYSIS ON POLYHEDRAL DOMAINS

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