Homogeneous fractional integral operators on weighted Lebesgue, Morrey and Campanato spaces

Let 0<α<n and T⏨, ⏐ be the homogeneous fractional integral operator which is defined by equation* T⏨, ⏐f (x): = ₑ䂞Ω (x-y) |x-y|^{n-α}f (y) \, dy, equation* where Ω is homogeneous of degree zero in Rⁿ for n2, and is integrable on the unit sphere S^n-1. In this paper we study boundedness properties of the homogeneous fractional integral operator T⏨, ⏐ acting on weighted Lebesgue and Morrey spaces. Under certain Dini-type smoothness condition on Ω, we prove that T⏨, ⏐ is bounded from L^p (ωᵖ) to C^γ, _ω (a class of Campanato spaces) for appropriate indices, when n/α<p<. Moreover, we prove that if Ω satisfies certain Dini-type smoothness condition on S^n-1, then T⏨, ⏐ is bounded from M^p, κ (ωᵖ, ωq) to C^γ, (ωq) (weighted Campanato spaces) for appropriate indices, when p/q<κ<1.