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={_z_0=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 _z_0 tends to 0 and if f(z) has a Gevrey type expansion of exponent γ* with respect to _z_0, then u(z) also has the same one.