Removable singularities of holomorphic solution of linear partial differential equations
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.
Fuchsian linear partial differential operators
scientific article; zbMATH DE number 2092009