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 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 ofT ^αƒ are obtained for ƒ belonging to the spacesC ^∞,C, L^p and for the case when ƒ is replaced by a finite Borel measure. All admissiblen ≥ 2,α ε ℂ, andp are considered. As a tool we use spherical wavelet transforms associated withT ^α. Wavelet type representations are obtained forT ^α ƒ, ƒ εL ^p, in the case Reα ≤ 0, provided thatT ^α is a linear bounded operator inL ^p.