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,
P(z, ∂)u(z)=f(z),
where u(z) and f(z) admit singularities on the surface K={z₀=0}. Our main result is the following:
For the operator P we define an exponent γ^* called the minimal irregularity of K and show that if u(z) grows at most exponentially with exponent γ^* as z₀ tends to 0 and if f(z) has a Gevrey type expansion of exponent γ^* with respect to z₀, then u(z) also has the same one.