Fractional integrals and wavelet transforms associated with Blaschke-Levy representations on the sphere

A family of the spherical fractional integrals T^α f = γ_n,α ∫_∑n |cursive Greek chiy|^α-1 f(y)dy on the unit sphere ∑_n in ℝⁿ⁺¹ is investigated. This family includes the spherical Radon transform (α = 0) and the Blaschke-Levy representation (α > 1). Explicit inversion formulas and a characterization of T^α f are obtained for f belonging to the spaces C^∞, C, L^p and for the case when f is replaced by a finite Borel measure. All admissible n ≥ 2, α ∈ ℂ, and p are considered. As a tool we use spherical wavelet transforms associated with T^α. Wavelet type representations are obtained for T^α f, f ∈ L^p, in the case Re α ≤ 0, provided that T^α is a linear bounded operator in L^p.