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_{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 $\Omega$ 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 $\mathrm{BMO}({\mathbb{R}}^n)$.

13 pages