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Analytic pseudo-differential operators and their applications. (English) Zbl 0743.35090

The main purpose of the present book is to give an introduction to the foundations of the theory of pseudo-differential operators, the symbols of which are arbitrary analytic functions of complex arguments or, as the author says, the foundation of the theory of analytic PD-operators. For the author the starting point for setting up this theory is the desire to solve differential equations of the type \(A(D)u(x)=f(x)\), \(x\in\mathbb{R}^ n\), by some natural operator method, that is in the form \(u(x)=(1/A(D))f(x)\).
The book consists of three parts. The first part is devoted to the construction of the algebras of PD-operators with constant analytic symbols. The author introduces the spaces of entire functions on \(\mathbb{C}^ n\), \(\text{ Exp}_{\Omega}(\mathbb{C}^ n_ z)\) as the domain of definition of a PD-operator and defines the action of a PD-operator on \(\text{Exp}_ \Omega(\mathbb{C}^ n)\).
Part II is devoted to the Cauchy problem for differential equations in a complex domain of \(\mathbb{C}^ n\). More precisely, the author is concerned with the three classical problems: (1) local analytic solvability, (2) global exponential solvability, (3) the connections between these theories. The well-posedness of the Cauchy problem is proved not only for differential equations but also for analytic PD-equations with variable analytic symbols.
In part III the author gives a version of the theory of real pseudodifferential operators wose symbols are arbitrary analytic functions in \(G\subset\mathbb{R}^ n\). He gives interesting applications of this theory to the investigation of some problem of mathematical physics by the mentioned operator method.

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
35S10 Initial value problems for PDEs with pseudodifferential operators
47G30 Pseudodifferential operators
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
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