A new estimate for homogeneous fractional integral operators on the weighted Morrey space $L^{p,\kappa }$ when $\alpha p=(1-\kappa )n$

For any $0<\alpha <n$, the homogeneous fractional integral operator $T_{\mathrm{\Omega },\alpha }$ is defined by \begin{equation*} T_{Ω,α}f(x)=\int_{\mathbb R^n}\frac{Ω(x-y)}{|x-y|^{n-α}}f(y),dy. \end{equation*} In this paper, we prove that if $\mathrm{\Omega }$ satisfies certain Dini smoothness conditions on ${\mathbf{S}}^{n-1}$, then $T_{\mathrm{\Omega },\alpha }$ is bounded from $L^{p,\kappa }(w^p,w^q)$ (weighted Morrey space) to $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$.