Boundary Values of Eigenfunctions on Riemannian Symmetric Spaces

TL;DR

This paper provides a new, self-contained proof of a generic version of the Helgason conjecture, establishing the Poisson transform as a topological isomorphism with boundary values on Riemannian symmetric spaces.

Contribution

It offers a novel proof of the Helgason conjecture for generic spectral parameters, extending previous work to higher rank symmetric spaces and different functional settings.

Findings

Poisson transform is a topological isomorphism for generic spectral parameters

Boundary value maps serve as inverses to the Poisson transform

The proof applies to both hyperfunction and distribution frameworks

Abstract

We give a new and self-contained proof of a generic version of the (former) Helgason conjecture. It says that for generic spectral parameters the Poisson transform is a topological isomorphism, with the inverse given by a boundary value map. Following Oshima's approach to a simplified definition of boundary values, and using the earlier work of Baouendi and Goulaouic on Fuchsian type equations, our proof is along the lines of our earlier work in the rank one distributional case, and works for both the hyperfunction and the distribution setting.