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

TL;DR

This paper explores the solvability of invariant differential systems on hyperbolic spaces using Fourier analysis, achieving complete results for the hyperbolic plane and partial results for higher-dimensional hyperbolic spaces.

Contribution

It extends Fourier transform techniques to distributional sections over symmetric spaces, providing new solvability results for invariant differential equations.

Findings

Complete solvability for the hyperbolic plane $\

Partial solvability results for products of hyperbolic planes and hyperbolic 3-space.

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 H\"ormander'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 $\mathbb{H}^3$.