A New Estimate for Homogeneous Fractional Integral Operators on Weighted Morrey Spaces

For any \(0<α<n\), the homogeneous fractional integral operator \(TΩ,α\) is defined by {equation*} TΩ,αf(x)=∫R^n{Ω(x-y)}{|x-y|n-α}1mu f(y)\,dy. {equation*} We prove that if \(Ω\) satisfies certain Dini smoothness conditions on \(Sⁿ⁻¹\), then \(TΩ,α\) is bounded from \(Lp,κ(w^p\!,w^q)\) (weighted Morrey space) to \(BMO( R^n)\) whenever \(p={(1-κ)1mu n}/{α}\). It is also shown that the fractional maximal operator \(MΩ,α\) is bounded from the weighted Lebesgue space \(L^p(w^p)\) to \(L∞(w)\) when \(p=n/{α}\), and bounded from \(Lp,κ(w^p\!,w^q)\) to \(L∞( R^n)\) when \(p={(1-κ)1mu n}/{α}\).