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
We associate to a pseudomanifold X with a conical singularity a differentiable groupoid G which plays the role of the tangent space of X : We construct a Dirac element and a dual Dirac element which induce a K-duality between the C Ã -algebras C Ã ðGÞ and CðX Þ: This is a first step toward an index theory for pseudomanifolds. r 2004 Elsevier Inc. All rights reserved.A basic point in the Atiyah-Singer index theory for closed manifolds lies in the isomorphism:induced by the map which assigns to the class of an elliptic pseudodifferential operator on a closed manifold V ; the class of its principal symbol [2]. To prove this isomorphism, Kasparov and Connes and Skandalis [6,15], define two elements D V AKKðCðV Þ#C 0 ðT Ã V Þ; CÞ and l V AKKðC; CðV Þ#C 0 ðT Ã V ÞÞ
We review the properties of transversality of distributions with respect to submersions. This allows us to construct a convolution product for a large class of distributions on Lie groupoids. We get a unital involutive algebra E ′ r,s (G, Ω 1/2 ) enlarging the convolution algebra C ∞ c (G, Ω 1/2 ) associated with any Lie groupoid G. We prove that G-operators are convolution operators by transversal distributions. We also investigate the microlocal aspects of the convolution product. We give conditions on wave front sets sufficient to compute the convolution product and we show that the wave front set of the convolution product of two distributions is essentially the product of their wave front sets in the symplectic groupoid T * G of Coste-Dazord-Weinstein. This also leads to a subalgebra E ′ a (G, Ω 1/2 ) of E ′ r,s (G, Ω 1/2 ) which contains for instance the algebra of pseudodifferential G-operators and a class of Fourier integral G-operators which will be the central theme of a forthcoming paper.
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