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

A family of the spherical fractional integrals T-alpha f = gamma(n, alpha) integral(Sigma n) \xy\(alpha-1) f(y)dy on the unit sphere Sigma(n) in Rn+1 is investigated. This family includes the spherical Radon transform (alpha = 0) and the Blaschke-Levy representation (alpha > 1). Explicit inversion formulas and a characterization of T(alpha)f are obtained for f belonging to the spaces C-infinity, C, L-p and for the case when f is replaced by a finite Borel measure. All admissible n greater than or equal to 2, alpha is an element of C, and p are considered. As a tool we use spherical wavelet transforms associated with T alpha. Wavelet type representations are obtained for T-alpha f, f is an element of L-p, in the case Re alpha less than or equal to 0, provided that T alpha is a linear bounded operator in L-p.