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 ove r all (n - 1)-dimensional totally geodesic submanifolds. The space H is ide ntified with the "upper" sheet of the two-sheeted hyperboloid in the pseudo -Euclidean space E-n,E-1 and represents a noncompact symmetric Riemannian s pace of constant negative curvature. Explicit inversion formulae for the Ra don transform Rf,f is an element of L-p(H), are obtained in terms of the re levant continuous wavelet transforms for all admissible values of the param eter p and all n greater than or equal to 2. The case of continuous functio ns f is also considered. Our inverting construction is reminiscent of the C alderon reproducing formula and converges in the L-p-norm (or in the sup-no rm) and in the a.e. sense.