A remark on analytic pseudodifferential operators with singularities
type: Item
instance of: scholarly article
title: A remark on analytic pseudodifferential operators with singularities
zbMATH Open document ID: 0928.35216
author: Nguyen Si Anh Tuan
published in: Vietnam Journal of Mathematics
publication date: 2000-01-12T00:00:00Z
review text: In the book by \textit{Yu. A. Dubinskij}: [``Analytic pseudodifferential operators and their applications''. Dordrecht: Kluwer (1991; Zbl 0743.35090)], an analytic pseudodifferential operator (an APO for short) \(A(D)\) with holomorphic symbol \(a(\zeta)\) was defined, and the algebra of APOs on \(\Omega\) was constructed. It was proved that if an APO \(A(D)\in A(\Omega)\) has the inverse \(A^{-1}(D)\in A(\Omega)\), then the analytic pseudodifferential equation \(A(D)u(z)= \nu(z)\), \(\nu(z)\in \text{Exp}_\Omega (C_z)\) has a unique solution \(u(z)=A^{-1} (D) \nu(z)\in \text{Exp}_\Omega (C_z)\). However, the existence condition for \(A^{-1}(D)\) in \(A(\Omega)\) is very strong, which leads to a loss of solutions. The main purpose of this paper is to introduce a class of APOs with pole-singularities and to define the operators with meromorphic symbol \(a(\zeta)\), \(A(D)\nu(z)= {1\over 2\pi i}\int_\gamma a(\zeta) B\nu(\zeta) e^{\zeta z} d\zeta\), where \(\gamma\) is a closed simple, oriented anticlockwise contour enclosing the spectrum of \(\nu(z)\) and \(B\nu(z)\) is the Borel transform of \(\nu(z)\). By this formula, the author proves that the operators, an APO and its value, can be represented as a sum of regular and singular parts. For a holomorphic symbol \(a(\zeta)\), the inverse of \(A(D)\), is also an APO with symbol \(a^{-1}(\zeta)\), and can be represented in this way.
reviewed by: Zeng Yuesheng
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 32F99
zbMATH DE Number: 1247958
zbMATH Keywords: holomorphic function
zbMATH Keywords: Borel transform
zbMATH Keywords: pole singularities
zbMATH Keywords: holomorphic symbol
zbMATH Keywords: meromorphic symbol
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1247958