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

Let $0<α<n$ and TΩ,α be the homogeneous fractional integral operator which is defined by {equation*} TΩ,αf(x):=∫R^n{Ω(x-y)}{|x-y|n-α}f(y)\,dy, {equation*} where $Ω$ is homogeneous of degree zero in Rⁿ for n≥2, and is integrable on the unit sphere Sⁿ⁻¹. In this paper we study boundedness properties of the homogeneous fractional integral operator TΩ,α acting on weighted Lebesgue and Morrey spaces. Under certain Dini-type smoothness condition on $Ω$, we prove that TΩ,α is bounded from Lᵖ(ωᵖ) to Cγ,_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 Sⁿ⁻¹, then TΩ,α is bounded from Mp,κ(ωᵖ,ωq) to Cγ,(ωq)(weighted Campanato spaces) for appropriate indices, when $p/q<κ<1$.