Invariant hyperfunction solutions to invariant differential equations on the space of real symmetric matrices
The author wants to find a \(\mathbb{G}\)-invariant hyperfunction solution for the linear differential equation \(P(x,\partial)u(x)=v(x)\) by using the Laurent expansion coefficient of the complex power of the determinant function, where \(P(x,\partial)\) is a \(\mathbb{G}\)-invariant homogeneous differential operator and \(v(x)\) is a \(\mathbb{G}\)-invariant hyperfunction. If \(v(x) = 0\) and the degree of \(b_p(s)\) equals the degree of \(P(x, \partial)\) then the solution space is finite dimensional, and solutions \(u(x)\) are given as finite linear combinations of quasi-homogeneous \(\mathbb{G}\)-invariant hyperfunctions. If \(v(x)\) is a quasi-homogeneous \( \mathbb{G}\)-invariant hyperfunction and \(b_p(s) \neq 0\) for all \(s\), then we can construct a \( \mathbb{G}\)-invariant hyperfunction solution \(u(x)\) which is given as a sum of Laurent expansion, and any \( \mathbb{G}\)-invariant hyperfunction solution \(u(x)\) is given as finite linear combinations of quasi-homogeneous \(\mathbb{G}\)-invariant hyperfunctions. For the proof he applies the method using the holonomic system theory of \(D\)-modules, since Methée's method cannot be applied in this more general case.
scientific article; zbMATH DE number 1836682
scientific article