Pseudo-differential calculus and applications to non-smooth configurations (Q2728736)

In these rather substantial lecture notes, the author presents some of the basic tools of the calculus of pseudodifferential operators. The aim of the author is to provide the tools necessary to study partial differential equations on manifolds with singularities of a certain kind.NEWLINENEWLINENEWLINEThe presented material, which is not self-contained, is organized in three chapters. The first chapter covers the basic notions and definitions relevant to the classical theory of pseudodifferential operators, where the Fourier transform plays a fundamental role. Roughly speaking, the classes considered here correspond to operators with smooth symbols in the Hörmander class \(S^m_{1,0}\). The author extends these notions to symbols with values on a Hilbert space and to operators acting on functions defined on smooth compact manifolds. He analyzes the correspondence between the operations on symbols and the operations on operators. The author also considers the case of operators that depend on a parameter. He introduces appropriate Sobolev-type spaces in these settings.NEWLINENEWLINENEWLINEIn the second chapter the author studies the main properties of so-called Fuchs-type operators, for which the relevant transform is the Mellin transform. Accordingly he introduces appropriate modifications of the Sobolev-type spaces introduced in Chapter 1. This is a suitable framework to consider manifolds that are no longer smooth.NEWLINENEWLINENEWLINEChapter 3 is dedicated to the analysis of manifolds with edges. Roughly speaking, a manifold \(X\) will have edges when one can single out a subspace \(Y\) so that \(Y\) and \(X\setminus Y\) are both \(C^\infty\) manifolds of appropriate dimensions and each point \(y\in Y\) has a neighborhood in \(X\) that is homeomorphic to a model non-smooth manifold. The rest of the chapter is dedicated to translate into this framework the construction of Sobolev-type spaces, appropriate differential and pseudodifferential operators, symbols, asymptotics.NEWLINENEWLINENEWLINEThe lecture notes end with an extensive list of references.