ABSTRACT
Pseudodifferential operators (ψDO) are one of the main analytic tools of elliptic theory on smooth as well as singular manifolds. Although it is the latter that are of main interest to us, we start from ψDO on smooth manifolds. On the one hand, this permits us to avoid additional technical complications in describing the main ideas, which will help us motivate the constructions of ψDO on singular manifolds. On the other hand, the calculus of ψDO on smooth manifolds serves as building material for more complicated calculi; here the most important role is played by ψDO acting in spaces of sections of infinite-dimensional bundles.
