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