Radon inversion on Grassmannians via Gårding-Gindikin fractional integrals

We study the Radon transform Rf of functions on Stiefel and Grassmann manifolds. We establish a connection between Rf and Gårding-Gindikin fractional integrals associated to the cone of positive definite matrices. By using this connection, we obtain Abel-type representations and explicit inversion formulae for Rf and the corresponding dual Radon transform. We work with the space of continuous functions and also with L^p spaces.