Sur l’existence d’une solution ramifiée pour des équations de Fuchs à caractéristique simple

Abstract:

The aim of this paper is to construct a holomorphic solution, ramified around a simple characteristic hypersurface, for some linear Fuchsian equation of order $m\geq 1$. We consider an operator $L$, holomorphic in a neighborhood of the origin in ${\mathbb {C}}_t\times {\mathbb {C}}_x^n$, of the form $L=tA+B$ where $A$ and $B$ are linear partial differential operators of order $m$ and $m-1$, and where $A$ has a simple characteristic hypersurface transverse to $S:t=0$. Under an assumption linking the principal symbols of $A$ and $B$, the question is reduced to the study of an integro-differential Fuchsian equation with an additional variable $z$ that describes the universal covering of a pointed disk. It is an equation where terms like $t^lD_t^hD_x^\alpha (tD_t+1)^{-1}D_z^{-q}, l,h,q\in \mathbb {N}, \alpha \in \mathbb {N}^n$ with $l\leq 1$ and $h+|\alpha |\leq l+q$ appear. The problem is solved by the fixed-point theorem with appropriate estimations in a Banach space.