This book is devoted to the study of pseudo-di?erential operators, with special emphasis on non-selfadjoint operators, a priori estimates and localization in the phase space. We have tried here to expose the most recent developments of the theory with its applications to local solvability and semi-classical estimates for non-selfadjoint operators. The?rstchapter,Basic Notions of Phase Space Analysis,isintroductoryand gives a presentation of very classical classes of pseudo-di?erential operators, along with some basic properties. As an illustration of the power of these methods, we give a proof of propagation of singularities for real-principal type operators (using aprioriestimates,andnotFourierintegraloperators),andweintroducethereader to local solvability problems. That chapter should be useful for a reader, say at the graduate level in analysis, eager to learn some basics on pseudo-di?erential operators. The second chapter, Metrics on the Phase Space begins with a review of symplectic algebra, Wigner functions, quantization formulas, metaplectic group and is intended to set the basic study of the phase space. We move forward to the more general setting of metrics on the phase space, following essentially the basic assumptions of L. H¨ ormander (Chapter 18 in the book [73]) on this topic.

Hersteller:
Springer Basel AG in Springer Science + Business Media
juergen.hartmann@springer.com
Heidelberger Platz 3
DE 14197 Berlin

Preface.- 1 Basic Notions of Phase Space Analysis.- 1.1 Introduction to pseudodifferential operators.- 1.2 Pseudodifferential operators on an open subset of Rn.- 1.3 Pseudodifferential operators in harmonic .- 2 Metrics on the Phase Space.- 2.1 The structure of the phase space.- 2.2 Admissible metrics.- 2.3 General principles of pseudodifferential calculus.- 2.4 The Wick calculus of pseudodifferential operators.- 2.5 Basic estimates for pseudodifferential operators.- 2.6 Sobolev spaces attached to a pseudodifferential calculus.- 3 Estimates for Non-selfadjoint Operators.- 3.1 Introduction.- 3.2 First bracket analysis.- 3.3 The geometry of condition (Y).- 3.4 The necessity of condition (Y).- 3.5 Estimates with loss of k/k + 1 derivative.- 3.6 Estimates with loss of one derivative.- 3.7 (Y) does not imply solvability with loss of one derivative.- 3.8 (Y) implies solvability with loss of 3/2 derivatives.- 3.9 Open problems.- 4 Appendix.- 4.1 Some elements of Fourier analysis.- 4.2 Some remarks of algebra.- 4.3 Lemmas of classical analysis.- 4.4 On the symplectic and metaplectic groups.- 4.5 Composing a large number of symbols.- 4.6 A few elements of operator theory.- 4.7 On Sjöstrand algebra.- 4.8 On preparation theorems.- 4.9 On the pseudospectrum.- 4.10 More on symbolic calculus.