Singular Solutions with Asymptotic Expansion of Linear Partial Differential Equations in the Complex Domain

Abstract

We consider a linear partial differential equation with holomorphic coefficients in a neighbourhood of $z=0$ in $C^{d+1}$,

where $u(z)$ and $f(z)$ admit singularities on the surface $K=\{z_0=0\}$. Our main result is the following:

For the operator $P$ we define an exponent ${\gamma }^{\ast }$ called the minimal irregularity of $K$ and show that if $u(z)$ grows at most exponentially with exponent ${\gamma }^{\ast }$ as $z_0$ tends to $0$ and if $f(z)$ has a Gevrey type expansion of exponent ${\gamma }^{\ast }$ with respect to $z_0$, then $u(z)$ also has the same one.