A remark on analytic pseudodifferential operators with singularities (Q1277224)

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.