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Linear partial differential operators in Gevrey spaces. (English) Zbl 0869.35005

Singapore: World Scientific. ix, 251 p. (1993).
This book deals with the study of linear partial differential operators in Gevrey spaces. It is a very good introduction to microlocal theory of ultradistributions and there are many applications for linear partial differential operators. The proofs are precise, clear, and short.
The book is divided into four chapters. Chapter I deals with Gevrey ultradistributions and Gevrey wave front sets. It contains all necessary facts used further on. Chapter II is devoted mainly to the investigation of operators with constant coefficients in Gevrey spaces. Special attention is paid to the problems of hypoellipticity and local solvability. The microlocal point of view (Gevrey and analytic microhypoellipticity) is dominating everywhere. The pseudodifferential calculus is developed in Chapter III. Pseudodifferential operators \((\psi\)do) of infinite order are studied in detail. The study of \(\psi\)do of finite order enables the author to propose many results on Gevrey hypoellipticity. Canonical transformations are also introduced and the classical theorem for propagation of singularities along the zero bicharacteristics of operators of real principal type is proved in the framework of Gevrey spaces. Chapter IV deals with operators with multiple characteristics of constant multiplicity and precise microlocal analysis of their properties is proposed there. It is interesting to note that operators of the type \(D^m_1+q(x,D)\) are not \(G_s\) hypoelliptic for \(1<s<{m\over m-1}\). Under the assumption: \(\xi^m_1+ q_{m-1}\neq 0\) they turn out to be \(s\)-hypoelliptic for \(s\geq{m\over m-1}\). The problems of \(s\)-local solvability are also studied here. It is very interesting to point out that nonsolvability in the \(C^\infty\) category can be easily proved by using Gevrey type arguments and tools.
The book is based mainly on the investigations of the author as well as on the investigations of Hörmander, Liess, Zanghirati, and others. The book is a good introduction to the Gevrey microlocal analysis for students and post-graduate students, but it is also useful for all specialists working in the domain of the general theory of linear partial differential operators.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
65H10 Numerical computation of solutions to systems of equations
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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