Pseudodifferential operators on manifolds with fibred corners (Q5962683)

The purpose of the article is to study a calculus of pseudodifferential operators on stratified pseudomanifolds endowed with an iterated fibred cusp metric, named \(\mathsf S\)-calculus. The article contains eleven sections distributed as follows. In the first section, the authors introduce the following definitions: a manifold with fibred corners \(X\), a family of tube system, a diffeomorphism map between two manifolds with fibred corners, a stratified pseudomanifold, and the depth of a stratum. The second section deals with vector fields on \(X\). The third (fifth) section copes with the definition and properties of \(\mathsf{S}\)-pseudodifferential operators. In the fourth section, the authors make a survey on (alge)groupoids and justify their usefulness in the present article by giving a meaningful example. The seventh (sixth) section states several symbol maps for \(\mathsf{S}\)-operators. The eighteen section concentrates on the stability of the set of \(\mathsf{S}\)-operators under a law of composition. The ninth section focuses on criteria for some mapping properties as the boundedness, compactness, and Fredholmness of a \(\mathsf{S}\)-operator; these notions are based on the creation of a parametrix for elliptic operators. In the tenth section, the authors announce the semiclassical \(\mathsf{S}\)-double space and the corresponding semiclassical \(\mathsf{S}\)-calculus. The last section, deals with the Poincaré duality between \(T^{\text{FC}}X\), a continuous family groupoid (Definition 4.2) and the stratified pseudomanifold associated to \(X\), this duality is described through the Kasparov bivariant K-theory.