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

Abstract

Let $0<\alpha<n$ 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 $\mathbb R^n$ for $n\geq2$, and is integrable on the unit sphere $\mathbb{S}^{n-1}$. 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 $L^{p}(\omega^p)$ to $\mathcal{C}^{\gamma,\ell}_{\omega}$(a class of Campanato spaces) for appropriate indices, when $n/{\alpha}<p<\infty$. Moreover, we prove that if $\Omega$ satisfies certain Dini-type smoothness condition on $\mathbb{S}^{n-1}$, then $T_{\Omega,\alpha}$ is bounded from $\mathcal{M}^{p,\kappa}(\omega^p,\omega^q)$ to $\mathcal{C}^{\gamma,\ell}(\omega^q)$(weighted Campanato spaces) for appropriate indices, when $p/q<\kappa<1$.