Dubinskij, Yu. A. Analytic pseudo-differential operators and their applications. (English) Zbl 0743.35090 Mathematics and Its Applications: Soviet Series. 68. Dordrecht etc.: Kluwer Academic Publishers. xii, 252 p. (1991). 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. Reviewer: V.S.Rabinovich (Rostov-na-Donu) Cited in 1 ReviewCited in 4 Documents 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 Keywords:pseudo-differential operators; symbols; analytic functions; algebras; analytic symbols; Cauchy problem; local analytic solvability; global exponential solvability; well-posedness; real pseudodifferential operators PDFBibTeX XMLCite \textit{Yu. A. Dubinskij}, Analytic pseudo-differential operators and their applications. Dordrecht etc.: Kluwer Academic Publishers (1991; Zbl 0743.35090)