Solvability of systems of invariant differential equations on H2 and beyond

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 H2 and partial results for products H^2 × · · · × H^2 and the hyperbolic 3-space H^3.