The Poisson transform on a compact real analytic Riemannian manifold

We study the transform mapping an function f on a compact, real analytic Riemannian manifold X to the analytic continuation of to the interior of a Grauert tube tube about X. We show that after precomposing with an elliptic pseudodifferential operator this becomes a unitary map from onto the holomorphic functions on . If a compact Lie group of isometries acts transitively on X then the inverse of this unitarized map can be constructed by the restrict ion principle.