Divergent formal solutions to Fuchsian partial differential equations (Q1089498)

The author studies the solvability of certain type of Fuchsian partial differential equations which do not fall into the framework of a so- called Fuchsian partial differential equations of Baouendi-Goulaouic. More precisely it is of the form \[ (\#)\quad P(x;x\cdot \partial)u\equiv (\sum _{| \alpha | =m}a_{\alpha}(x\cdot \partial)^{\alpha}+\sum _{| \beta | \leq m- \sigma}b_{\beta}(x)(x\cdot \partial)^{\beta})u(x)=f(x) \] where \(m\geq 1\), \(\sigma\) \(\geq 1\) are integers, \(a_{\alpha}'s\) are complex constants and \(b_{\beta}(x)\) is analytic at the origin. Here f(x) is a given analytic function and where \((x\cdot \partial)^{\alpha}=(x_ 1\cdot \partial _ 1)^{\alpha _ 1}... (x_ d\cdot \partial _ d)^{\alpha _ d}\), \(\partial _ j=\partial /\partial x_ j\), \(j=1,...,d.\) The author assumes that ({\#}) is hyperbolic and it satisfies a Levi- condition. Moreover he assumes that the equation ({\#}) has a formal solution û. Then, there exists a smooth solution u of ({\#}) in some neighbourhood of the origin such that u is asymptotically equal to û.