Ranges and Inversion Formulas for Spherical Mean Operator and Its Dual

We consider the spherical mean operator R and its dual tR. We establish some results from harmonic analysis for the generalized Fourier transform associated with the operator R. We deduce that the operators R and tR are transmutation operators on some spaces of functions. Next, we give inversion formulas for these operators and a Plancherel theorem for the operator tR.