Singular solutions with asymptotic expansion of linear partial differential equations in the complex domain (Q1974495)

Summary: We consider a linear partial differential equation with holomorphic coefficients in a neighbourhood of \(z= 0\) in \(\mathbb{C}^{d+1}\), \(P(z,\partial)u(z)= f(z)\), 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 \(\gamma^*\) called the minimal irregularity of \(K\) and show that if \(u(z)\) grows at most exponentially with exponent \(\gamma^*\) as \(z_0\) tends to \(0\) and if \(f(z)\) has a Gevrey type expansion of exponent \(\gamma^*\) with respect to \(z_0\), then \(u(z)\) also has the same one.