Boundedness of homogeneous fractional integrals on $L^p$ for $N/\alpha \leq \infty$

Abstract

In this paper we study the map properties of the homogeneous fractional integral operator $T_{\Omega,\alpha}$ on $L^p({\Bbb R}^n)$ for $n/\alpha\le p\le\infty.$ We prove that if $\Omega$ satisfies some smoothness conditions on $S^{n-1},$ then $T_{\Omega,\alpha}$ is bounded from $L^{n/\alpha}({\Bbb R}^n)$ to $BMO({\Bbb R}^n),$ and from $L^p({\Bbb R}^n)\, (n/\alpha<p\le\infty)$ to a class of the Campanato spaces ${\cal L}_{l,\lambda} ({\Bbb R}^n),$ respectively. As the corollary of the results above, we show that when $\Omega$ satisfies some smoothness conditions on $S^{n-1},$ the homogeneous fractional integral operator $T_{\Omega,\alpha}$ is also bounded from $H^p({\Bbb R}^n)\, (n/(n+\alpha)\le p\le1)$ to $L^q({\Bbb R}^n)$ for $1/q=1/p-\alpha/n.$ The results are the extensions of Stein-Weiss (for $p=1$) and Taibleson-Weiss's (for $n/(n+\alpha)\le p<1$) results on the boundedness of the Riesz potential operator $I_\alpha$ on the Hardy spaces $H^p({\Bbb R}^n).$