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

Abstract

For any \(0<\alpha<n\), the homogeneous fractional integral operator \(T_{\Omega,\alpha}\) is defined by \begin{equation*} T_{\Omega,\alpha}f(x)=\int_{\mathbb R^n}\frac{\Omega(x-y)}{|x-y|^{n-\alpha}}\mkern1mu f(y)\,dy. \end{equation*} We prove that if \(\Omega\) satisfies certain Dini smoothness conditions on \(\mathbf{S}^{n-1}\), then \(T_{\Omega,\alpha}\) is bounded from \(L^{p,\kappa}(w^p\!,w^q)\) (weighted Morrey space) to \(\mathrm{BMO}(\mathbb R^n)\) whenever \(p={(1-\kappa)\mkern1mu n}/{\alpha}\). It is also shown that the fractional maximal operator \(M_{\Omega,\alpha}\) is bounded from the weighted Lebesgue space \(L^p(w^p)\) to \(L^{\infty}(w)\) when \(p=n/{\alpha}\), and bounded from \(L^{p,\kappa}(w^p\!,w^q)\) to \(L^{\infty}(\mathbb R^n)\) when \(p={(1-\kappa)\mkern1mu n}/{\alpha}\).