Helgason-Marchaud inversion formulas for Radon transforms

Let X be either the hyperbolic space ℍⁿ or the unit sphere Sⁿ, and let Ξ be the set of all k-dimensional totally geodesic submanifolds of X, 1 ≤ k ≤ n-1. For x ∈ X and ξ ∈ Ξ, the totally geodesic Radon transform f(x) → f̂(ξ) is studied. By averaging f̂(ξ) over all ξ at a distance θ from x, and applying Riemann-Liouville fractional differentiation in θ, S. Helgason has recovered f(x). We show that in the hyperbolic case this method blows up if f does not decrease sufficiently fast. The situation can be saved if one employs Marchaud's fractional derivatives instead of the Riemann-Liouville ones. New inversion formulas for f̂(ξ), f ∈ L^P(X), are obtained.