Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators

ISBN-13: 9783764385095 / Angielski / Miękka / 2010 / 397 str.

Nicolas Lerner

Opis

This book contains a thorough exposition of pseudodifferential calculus defined by metrics on the phase space. It contains a proof of the Nirenberg-Treves conjecture and a construction of counterexamples to "optimal" solvability under condition (psi).

Recenzje

From the reviews:

"The present book is devoted to the theory of pseudodifferential operators and some applications. The presentation starts from basic results and continues up to the most advanced and recent developments of the local solvability theory ... . the present monograph will become a reference book for researchers in microlocal analysis. ... Ph. D. students will greatly appreciate this up-to-date overview of such a deep subject ... ." (Fabio Nicola, Mathematical Reviews, Issue 2011 b)

"This very interesting book of a well-known specialist in partial differential equations is devoted to the study of pseudo-differential operators and describes the most recent developments of the theory with its applications to local solvability and semi-classical estimates for non-selfadjoint operators, most of them belonging to the author. ... To sum up, the present book is really excellent. The first two parts are accessible to graduate students in analysis. The third chapter is highly recommended to researchers, providing an up-to-date overview of the subject." (Viorel Iftimie, Zentralblatt MATH, Vol. 1186, 2010)

Spis treści

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.