Pseudodifferential operators on manifolds with singularities and localization.

*(English. Russian original)*Zbl 1272.58012
Dokl. Math. 71, No. 3, 457-460 (2005); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 402, No. 6, 743-747 (2005).

Summary: At present, there is extensive literature on pseudodifferential operators (PDOs) on manifolds with singularities. In this paper, we develop an approach related to the definition of PDOs in terms of localization and show that this approach makes it possible to give a simple and uniform description of PDO calculus and prove its fundamental theorems (the composition formula and the finiteness theorem) for a very large class of manifolds with singularities. The description is closely related to the well-known localization principle in \(C^*\)-algebras, which has been developed by many authors (we mention only Antonevich, Vasilevskii, Gokhberg, Dauns, Douglas, Krupnik, Lebedev, Plamenevskii, Senichkin, Simonenko, and Hofmann; see, e.g., and the references therein). Although this principle has been extensively applied to studying PDOs on manifolds with singularities, it has never been used as a basis for the definition of such PDOs as far as we know. We consider only the case of PDOs of order zero in spaces of type \(L^2\); the general case will be considered elsewhere.