Ranges and inversion formulas for spherical mean operator and its dual (Q1910099)

The spherical mean operator is defined for a function \(f\) on \(\mathbb{R}^{n + 1}\) by \[ R(f) (r,x) = \int_{S^n} f(r \eta, x + r \xi) d \sigma_n (\eta, \xi); \quad (r,x) \in \mathbb{R} \times \mathbb{R}^n, \] where \(S^n\) is the unit sphere in \(\mathbb{R}^{n + 1}\) and \(\sigma_n\) is the normalized surface measure on \(S^n\). Some results from harmonic analysis are established for the generalized Fourier transform associated with the operator \(R\). The operator \(R\) and its dual \(^tR\) are transmutation operators on some function spaces. The authors prove inversion formulas for \(R\), \(^tR\) and a Plancherel theorem for \(^tR\).