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

TL;DR

This paper establishes a new estimate for homogeneous fractional integral operators on weighted Morrey spaces, demonstrating boundedness under certain smoothness and weight conditions when a specific relation between parameters holds.

Contribution

The paper introduces a novel boundedness result for $T_{oldsymbol{ ext{Omega}},oldsymbol{ ext{alpha}}}$ on weighted Morrey spaces under Dini smoothness assumptions, specifically when $oldsymbol{ ext{alpha}}oldsymbol{ ext{p}}=(1-oldsymbol{ ext{kappa}})oldsymbol{ ext{n}}$.

Findings

Boundedness of $T_{oldsymbol{ ext{Omega}},oldsymbol{ ext{alpha}}}$ from weighted Morrey spaces to BMO.

Establishment of conditions on $oldsymbol{ ext{Omega}}$ for boundedness.

Extension of fractional integral operator estimates to 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}}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 $\mathrm{BMO}(\mathbb R^n)$.