Solvability of invariant systems of differential equations on ${\mathbb{H}}^2$ and beyond

Abstract

We show how the Fourier transform for distributional sections of vector bundles over symmetric spaces of non-compact type $G/K$ can be used for questions of solvability of systems of invariant differential equations in analogy to Hörmander's proof of the Ehrenpreis-Malgrange theorem. We get complete solvability for the hyperbolic plane ${\mathbb{H}}^2$ and partial results for products ${\mathbb{H}}^2\times \cdots \times {\mathbb{H}}^2$ and the hyperbolic 3-space ${\mathbb{H}}^3$.

Corrected misprints and improved the presentation. Lemma 1 is new