Pseudodifferential operators and linear PDE

Abstract

In this preliminary chapter we give an outline of the theory of pseudodifferential operators as it has been developed to treat problems in linear PDE, and which will provide a basis for further developments to be discussed in the following chapters. Many results will be proved in detail, but some proofs are only sketched, with references to more details in the literature. We define pseudodifferential operators with symbols in Hörmander’s classes S ^m _p,δ , derive some useful properties of their Schwartz kernels, discuss their algebraic properties, then show how they can be used to establish regularity of solutions to elliptic PDE with smooth coefficients. We proceed to a discussion of mapping properties on L² and on the Sobolev spaces H^s, then discuss Gårding’s inequality, and some of its refinements, known as sharp Gårding inequalities. In §0.8 we apply some of the previous material to establish existence of solutions to hyperbolic equations. We introduce the notion of wave front set in §0.10 and discuss microlocal regularity of solutions to elliptic equations. We also discuss how solution operators to a class of hyperbolic equations propagate wave front sets. In §0.11 we discuss L^p estimates, particularly some fundamental results of Calderon and Zygmund, and applications to Littlewood-Paley Theory, which will be an important technical tool for basic estimates established in Chapter 2.We end this introduction with a brief discussion of pseudodifferential operators on manifolds.