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_{Ω,α}f(x):=\int_{\mathbb R^n}\frac{Ω(x-y)}{|x-y|^{n-α}}f(y)\,dy, \end{equation*} where $\Omega$ is homogeneous of degree zero in ${\mathbb{R}}^n$ for $n\ge 2$, 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 ${\mathcal{C}}_{\omega }^{\gamma ,\ell }$(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 $p/q<\kappa <1$.

20 pages