Radon transform of L^p-functions on the Lobachevsky space and hyperbolic wavelet transforms

The Radon transform Rf on the n-dimensional Lobachevsky space ℋ assigns to each sufficiently nice function f on ℋ the collection of integrals of f over all (n - 1)-dimensional totally geodesic submanifolds. The space ℋ is identified with the "upper" sheet of the two-sheeted hyperboloid in the pseudo-Euclidean space E^{n, 1} and represents a noncompact symmetric Riemannian space of constant negative curvature. Explicit inversion formulae for the Radon transform Rf, f ∈ L^p(ℋ), are obtained in terms of the relevant continuous wavelet transforms for all admissible values of the parameter p and all n ≥ 2. The case of continuous functions f is also considered. Our inverting construction is reminiscent of the Calderón reproducing formula and converges in the L^p-norm (or in the sup-norm) and in the a.e. sense.