Inversion of Fractional Integrals Related to the Spherical Radon Transform

Explicit inversion formulas are obtained for the analytic family of fractional integrals (Tαf)(x)=γn, α ∫Sn |xy|α−1 f(y) dyon the unit sphere in Rn+1. Arbitrary complexαandn⩾2 are considered. In the easeα=0 the integralTαfcoincides with the spherical Radon transform. Forα>1(α≠1, 3, 5, …) such integrals are known as the Blaschke–Levy representations and arise in convex geometry, probability, and the Banach space theory. Forα=1, 3, 5, … the integralTαfis defined by continuity as the spherical convolution with the power–logarithmic kernel. Different inversion methods are discussed.