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

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}}f(y)\,dy. \end{equation*} In this paper, 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 $\mathrm{BMO}(\mathbb R^n)$.