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Complex-analytic Cauchy problem in a bounded domain. (English. Russian original) Zbl 0704.35026
Differ. Equations 26, No. 1, 121-131 (1990); translation from Differ. Uravn. 26, No. 1, 136-147 (1990).
This article deals with the global solvability of linear partial differential equation $$H\hat u=f$$ of order m with constant coefficients in the space of (multi-valued) holomorphic functions on a domain $$D\subset {\mathbb{C}}^ n$$ with (branching) singularity along a complex hypersurface $$X=\{s(x)=0\}$$. Let $$A_ q(X)$$ denote the space of such functions satisfying the estimate $$| f| \leq C| s(x)|^ q$$. D is called $$(H,X)$$-convex if every bicharacteristic line starting from a point of $$X\setminus \partial D$$ with the holomorphic conormal direction to $$\partial D$$, or from a point of $$X\cap \partial D$$ with a holomorphic conormal direction to $$X\cap \partial D$$. Then if X has no characteristic points in Int(D), $$\hat Hu=f\in A_ q(X)$$ is solvable in $$u\in A_{q+m}(X)$$. If there are some, it is solvable in the space of functions having singularity further along the subvariety Y formed by the bicharacteristic lines emanating from these characteristic points. The proof is made on examination of the singularity of the solution given by the Radon integral.
Reviewer: A.Kaneko

##### MSC:
 35E20 General theory of PDEs and systems of PDEs with constant coefficients 35C15 Integral representations of solutions to PDEs 35A20 Analyticity in context of PDEs