Pseudodifferential operators on manifolds with fibred corners

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 Φ-calculus of Mazzeo and Melrose. Our starting point is the observation, going back to Melrose, 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. Contents 10 3. The definition of S-pseudodifferential operators 13 4. Groupoids 17 5. Action of S-pseudodifferential operators 21 6. Suspended S-operators 24 7. Symbol Maps 28 8. Composition 31 9. Mapping properties 36 10. The semiclassical S-calculus 46 11. Poincaré duality 53 References 62

Weighted Hodge cohomology of iterated fibred cusp metrics

“…Second, we note that the iterated fibred cusp metrics of [7] constitute a special case of quasi iterated fibred cusp metrics. In fact, quasi iterated fibred cusp metrics could alternatively be defined as complete Riemannian metrics on the regular set M of X that are quasi-isometric to iterated fibred cusp metrics.…”

“…Let X be a smoothly stratified space in the sense of [1], see also [14] and [7]. There is a resolution of X by a manifold with fibred corners, q : M → X, where q is a diffeomorphism from the interior of M to the regular stratum of X, and is a fiber bundle q i : H i → S i over some singular stratum when restricted to the interior of each boundary hypersurface of M .…”

“…Furthermore, by [7, Lemma 1.4], we can assume that each other stratum defining function x j is in this neighborhood factorizes through φ U and projection onto a stratum defining function on L i (constant on V ) if S i ⊂ S j or a positive function uniformly bounded away from zero otherwise. Those x j factorizing through L i then form corresponding stratum defining functions on L i (see [7]). …”

Lie groupoids, pseudodifferential calculus, and index theory

“…Later, ([81]) Mazzeo and Melrose introduced and studied in the same situation the Φ calculus which corresponds to the algebroid of vector fields that are tangent to the fibers at the boundary but also whose derivative is tangent to ∂M : these are vector fields of the form X + tY + t 2 N -where t is a defining function of the boundary, X is a vector field along the fibration (extended near the boundary), Y is tangent to the boundary and N is normal to the boundary. Piazza and Zenobi ([97]) actually realized that the groupoid constructed in [44] integrating this algebroid can be obtained via a double blowup construction. See also [117] for topological aspects of indices in this context.…”

A Topological Approach to Indices of Geometric Operators on Manifolds with Fibered Boundaries

“…To study pseudodifferential operators with respect to such metrics, the corresponding pseudodifferential calculi, called Φ-calculus and e-calculus, were introduced by [MM98] and [Maz91]. Since then, analysis of elliptic operators in these calculi, in particular Fredholm theory and spectral theory of geometric operators, has been developed by many authors and there have been many applications to geometry of singular spaces, for example see [ALMP12], [DLR15] and [LMP06].…”