Homogeneous fractional integral operators on weighted Lebesgue, Morrey and Campanato spaces

Abstract: Let $0&lt;\alpha&lt;n$ and $T_{\Omega,\alpha}$ be the homogeneous fractional integral operator which 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*} where $\Omega$ is homogeneous of degree zero in $\mathbb R<sup>n$ for $n\geq2$, and is integrable on the unit sphere $\mathbb{S}<sup>{n-1}$. In this paper we study boundedness properties of the homogeneous fractional integral operator $T_{\Omega,\alpha}$ acting on weighted Lebesgue and Morrey spaces. Under certain Dini-type smoothness condition on $\Omega$, we prove that $T_{\Omega,\alpha}$ is bounded from $L<sup>{p}(\omega<sup>p)$ to $\mathcal{C}<sup>{\gamma,\ell}_{\omega}$(a class of Campanato spaces) for appropriate indices, when $n/{\alpha}&lt;p&lt;\infty$. Moreover, we prove that if $\Omega$ satisfies certain Dini-type smoothness condition on $\mathbb{S}<sup>{n-1}$, then $T_{\Omega,\alpha}$ is bounded from $$\mathcal{M}<sup>{p,\kappa}(\omega<sup>p,\omega<sup>q)$$ to $\mathcal{C}<sup>{\gamma,\ell}(\omega<sup>q)$(weighted Campanato spaces) for appropriate indices, when $p/q&lt;\kappa&lt;1$.