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

Dec 2022

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

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