\(L^{p}\) Theory of Pseudo-Differential Operators

In this chapter we present a brief description of the basic concepts and results of the \(L^{p}\) theory of pseudo-differential operators, which may be considered as a modern version of the classical potential approach. First, we define Hölder spaces and various Sobolev spaces and Besov spaces on a smooth domain of Euclidean space \({\mathbf {R}}^{n}\) . It is the imbedding characteristics of \(L^{p}\) Sobolev spaces that render these spaces so useful in the study of partial differential equations. It should be emphasized that Besov spaces on the boundary enter naturally in connection with boundary value problems in the framework of \(L^{p}\) Sobolev spaces. Secondly, we present a brief description of basic concepts and results of the theory of Fourier integral operators and pseudo-differential operators. The \(L^{p}\) theory of pseudo-differential operators may be considered as a modern version of the classical potential theory, and provides a constructive tool to deal with existence and smoothness of solutions of partial differential equations. Moreover, we formulate the Besov space boundedness theorem and hypoellipticity in the framework of \(L^{p}\) Sobolev spaces, which plays an essential role in the proof of our main results. Finally, by using the Riesz–Schauder theory we prove some of the most important results about elliptic pseudo-differential operators on a manifold and their indices.