Removable singularities of holomorphic solution of linear partial differential equations (Q1876305)

The author considers Fuchsian linear partial differential operators in the complex domain, of the form: \[ P(z,D)=\left( \sum^r_{s=0} a_s(z) (z_1D_1)^{r-s} \right)D_n^{m-r}+ Q(z,D) \] where the lower order part \(Q(z,D)\) satisfies suitable Fuchs conditions. The coefficients \(a_s(z)\), \(z= (z_1,\dots,z_n) \in \mathbb{C}^n\), and the coefficients of \(Q(z,D)\) are assumed holomorphic in a neighborhood \(V\) of the origin. The problem under discussion is whether a solution \(u(z)\) of \(P(z,D)u=0\), holomorphic outside the characteristic hypersurface \(S\), has a holomorphic extension to the whole neighborhood \(V\), cf. \textit{K. Igari} [Funkc. Ekvacioj, Ser. Int. 37, 537--547 (1994; Zbl 0822.35003)]. In the present paper, new precise conditions for holomorphic extensions are given on the expression of \(P(z,D)\), with applications to the propagation of singularities for equations with noninvolutive double characteristics.