Eigenfunctions of invariant differential operators on a symmetric space

TLDR: In this article, it was shown that the sum of the boundary values of K-finite functions converges in the sense of analytic functionals in the case that the rank of the symmetric space is equal to one.

Abstract: In his paper [14], S. Helgason conjectured that any joint-eigenfunction of all invariant differential operators on a symmetric space can be given by the Poisson integral. The purpose of this paper is to prove this conjecture (see corollary to the theorem in the Section 5). There have appeared several papers dealing with the conjecture in the case that the rank of the symmetric space is equal to one ([12], [13], [15], [17], [30], [34], [35], [36]). The proofs given in these papers follow an idea due to S. Helgason [14] and may be explained as follows. In the rank one case the algebra of all invariant differential operators is generated by the Laplace-Beltrami operator. First, one expands any eigenfunction of the laplacian into K-finite functions. Then these K-finite functions are also eigenfunctions of the laplacian and have boundary values in the natural way. The radial component of the laplacian gives rise to hypergeometric differential equations. Thanks to the classical results on hypergeometric functions, one can estimate the asymptotic behavior of the solution near the boundary. This enables us to prove that the sum of the boundary values of K-finite functions converges in the sense of analytic functionals. In the higher rank case, however, the radial components of invariant differential operators are not ordinary differential operators anymore, so that one is unable to apply the classical results. In the meanwhile, some of the present authors have recently generalized the notion of "regular singularity" for the ordinary differential equation to that for the system of partial differential equations. The essential point is that the system of invariant differential equations has regular singularity along the Martin boundary which assures the existence of the boundary values of a solution as "hyperfunctions." In the method mentioned above, one encounters the crucial difficulty in proving the exist-