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

Abstract

A family of the spherical fractional integrals $$T^\alpha f = \gamma _{n,\alpha } \int {_{\Sigma _n } } \left| {xy} \right|^{\alpha - 1} f(y)dy$$ on the unit sphere Σ n in ℝ n+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 ofT αƒ are obtained for ƒ belonging to the spacesC ∞,C, Lp and for the case when ƒ is replaced by a finite Borel measure. All admissiblen ≥ 2,α e ℂ, andp are considered. As a tool we use spherical wavelet transforms associated withT α. Wavelet type representations are obtained forT α ƒ, ƒ eL p, in the case Reα ≤ 0, provided thatT α is a linear bounded operator inL p.