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Metrics on the phase space and non-selfadjoint pseudo-differential operators. (English) Zbl 1186.47001
Pseudo-Differential Operators. Theory and Applications 3. Basel: Birkhäuser (ISBN 978-3-7643-8509-5/pbk). xi, 397 p. (2010).
This very interesting book by 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 due to the author.
The book is structured in three chapters plus a lengthy appendix. The first chapter, “Basic Notions of Phase Space Analysis”, is introductory and gives a presentation of very classical classes of pseudo-differential operators, along with some basic properties (classical and semi-classical calculus). As an application of these methods, the author gives a proof of propagation of singularities for operators of principal type with complex symbols with a nonnegative imaginary part (using a priori estimates, and not using Fourier integral operators) and also some consequences to local solvability problems.
The second chapter, “Metrics on the Phase Space”, deals with the general notion of metrics on the phase space, following essentially the basic assumptions of L. Hörmander, but using the notion of confinement introduced by J.-M. Bony and the author. The chapter begins with a review of symplectic algebra, Wigner functions, quantization formulas, metaplectic group. Then the admissible metrics and the general principles of the Weyl pseudo-differential calculus are presented. The author also exposes some elements of the Wick calculus and some key examples related to the Calderón-Zygmund decompositions such as the Fefferman-Phong inequality and proves that the analytic functional calculus works for admissible metrics. Finally, a description of the construction of Sobolev spaces attached to a pseudo-differential calculus, following a paper by J.-M. Bony and J.-Y. Chemin, is given.
The third chapter, “Estimates for Non-Selfadjoint Operators”, is devoted to the discussion of the details of the various types of estimates that can be proved or disproved, depending on the geometry of the symbols. The author starts with some examples and various classical models such as H. Lewy’s example and a discussion of the first Poisson bracket of the imaginary and real part of the symbol. Then some facts concerning the geometry of condition $$(\Psi)$$ of L. Nirenberg and F. Trèves, including the contribution of N. Dencker, are presented. The necessity of condition $$(\Psi)$$ for local solvability and some subelliptic estimates are formulated, but not proved. Instead, the author gives a detailed proof of the Beals-Fefferman results on local solvability with loss of one derivative under condition (P). He shows, following his counterexample, that an estimate with loss of one derivative is not a consequence of condition $$(\Psi)$$. Finally, he proves an estimate with loss of $$3/2$$ derivatives under condition $$(\Psi)$$, following some articles of N. Dencker (were the loss was of $$z$$ derivatives) and the author’s own. The chapter ends with some interesting open problems, such as the following:
Let $$P$$ be a principal type pseudo-differential operator whose principal symbol $$p$$ satisfies condition $$(\Psi)$$. Then:
What is the minimal loss of derivatives for the local solvability of $$P$$?
Let $$p$$ be real analytic. Is $$P$$ locally solvable with loss of one derivative?
Is there a geometric condition, stronger than $$(\Psi)$$, on $$p$$ which is equivalent to local solvability with loss of one derivative?
Is it true that, for every $$x_0$$ and every $$f\in C^\infty$$, there is some $$u\in C^\infty$$ such that $$Pu= f$$ in a neighborhood of $$x_0$$?
Some topics of the appendix are very classical and concern classical and Fourier analysis, symplectic and metaplectic groups, and symplectic geometry. Other parts are devoted to technical questions.
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.

##### MSC:
 47-02 Research exposition (monographs, survey articles) pertaining to operator theory 47G30 Pseudodifferential operators 35S05 Pseudodifferential operators as generalizations of partial differential operators
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