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Let X be either the hyperbolic space H n or the unit sphere S n , 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) → f(ξ) is studied. By averaging f(ξ) over all ξ at a distance θ from x, and applying Riemann-Liouville fractional differentiation in 0, 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 f(ξ); f ∈ L p (X), are obtained.

Helgason-Marchaud inversion formulas for Radon transforms