Inversion of Fractional Integrals Related to the Spherical Radon Transform

Explicit inversion formulas are obtained for the analytic family of fractional integrals (T^αf)(x)=γ_n,α∫_S ⁿxy^α-1f(y)dyon the unit sphere in Rⁿ⁺¹. Arbitrary complexαandn≥2 are considered. In the easeα=0 the integralT^αfcoincides with the spherical Radon transform. Forα1(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 integralT^αfis defined by continuity as the spherical convolution with the power-logarithmic kernel. Different inversion methods are discussed.