An analogue of the Fuglede formula in integral geometry on matrix spaces

The well known formula of B. Fuglede expresses the mean value of the Radon k-plane transform on $R^n$ as a Riesz potential. We extend this formula to the space of $n\times m$ real matrices and show that the corresponding matrix k-plane transform $f\to \widehat{f}$ is injective if and only if $n-k\ge m$. Different inversion formulas for this transform are obtained. We assume that $f \in L^p$or$f$ is a continuous function satisfying certain “minimal” conditions at infinity.