This is a four-hundred-page book on the topic of pseudodifferential operators, with special emphasis on non-selfadjoint operators, a priori estimates and localization in the phase space. The first chapter, Basic Notions of Phase Space Analysis, is introductory and gives a presentation of very classical classes of pseudodifferential operators, along with some basic properties. The second chapter, Metrics on the Phase Space, begins with a review of classical structures. We expose as well some elements of the so-called Wick calculus. We present some key examples related to the Calderón-Zygmund decompositions such that the Fefferman-Phong inequality. We give a description of the construction of Sobolev spaces attached to a pseudodifferential calculus. The third and last chapter is entitled Estimates for Non-Selfadjoint Operators. We discuss the details of the various types of estimates that can be proved or disproved, depending on the geometry of the symbols. We start with a rather elementary section containing examples and various classical models such as the Hans Lewy example. The following sections are more involved; in particular we start a discussion on the geometry of condition (Ψ) with some known facts on flow-invariant sets, but we expose also the contribution of N. Dencker in the understanding of that geometric condition, with various inequalities satisfied by symbols. Then we enter into the discussion of estimates with loss of one derivative; we start with a detailed proof of the Beals-Fefferman result on local solvability with loss of one derivative under condition (P). Following the author's counterexample, we show that an estimate with loss of one derivative is not a consequence of (Ψ). Finally, we give a proof of an estimate with loss of 3/2 derivatives under condition (Ψ), following the articles of N. Dencker and the author's. It is our hope that the first two parts of the book are accessible to graduate students with a decent background in Analysis. The third chapter is directed more to researchers but should also be accessible to the readers able to get some good familiarity with the first two chapters, in which the main tools for the proofs of Chapter 3 are provided. |