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Asymptotic expansions of solutions of differential equations on complex manifolds. (English. Russian original) Zbl 0664.58037
Sov. Math., Dokl. 36, No. 1, 31-34 (1987); translation from Dokl. Akad. Nauk SSSR 295, 38-41 (1987).
The authors consider the equation $H(x,-\partial /\partial x)u(x)=\sum_{| \alpha | \leq m}a_{\alpha}(x)(-\partial /\partial x)^{\alpha}u(x)=f(x),$ where the coefficients $$a_{\alpha}$$ of the equation are holomorphic in a neighborhood of $$x_ 0\in {\mathbb{C}}^ n$$. It is assumed that f is a multiple valued analytic function which is ramified around an analytic variety X which contains $$x_ 0$$ and has defining equation $$s=0$$. For T an analytic set which does not contain X and $$q>-1$$, define $$A_ q(X,T)$$ to be the space of multiple valued analytic functions that are regular outside $$X\cup T$$ and are such that for any $$x_ 1\in X\setminus T$$ there is a neighborhood U of $$x_ 1$$ and C so that $$| f(x)| \leq C| s(x)|^ q$$ if $$x\in U$$. The authors show (under some additional assumption) that if $$u\in A_{q+m}(X,Y)$$ satisfies $$H(x,-\partial /\partial x)u=f$$, then modulo $$A_{q+m+2}(X,Z)$$, u can be represented with a residue-type integral formula in terms of f, thus establishing a quite explicit link between u and f. Here Y and Z are computed in terms of X and the characteristic set of H.
Reviewer: O.Liess
##### MSC:
 58J40 Pseudodifferential and Fourier integral operators on manifolds 35B40 Asymptotic behavior of solutions to PDEs 32A27 Residues for several complex variables
##### Keywords:
canonical operator; multiple valued analytic function