Homogeneous fractional integral operators on Lebesgue and Morrey spaces, Hardy--Littlewood--Sobolev and Olsen-type inequalities

Abstract

Let $T_{\Omega,\alpha}$ be the homogeneous fractional integral operator defined as \begin{equation*} T_{\Omega,\alpha}f(x):=\int_{\mathbb R^n}\frac{\Omega(x-y)}{|x-y|^{n-\alpha}}f(y)\,dy, \end{equation*} and the related fractional maximal operator $M_{\Omega,\alpha}$ is given by \begin{equation*} M_{\Omega,\alpha}f(x):=\sup_{r>0}\frac{1}{|B(x,r)|^{1-\alpha/n}}\int_{|x-y|<r}|\Omega(x-y)f(y)|\,dy. \end{equation*} In this article, we will use the idea of Hedberg to reprove that the operators $T_{\Omega,\alpha}$ and $M_{\Omega,\alpha}$ are bounded from $L^p(\mathbb R^n)$ to $L^q(\mathbb R^n)$ provided that $\Omega\in L^s(\mathbf{S}^{n-1})$, $s'<p<n/{\alpha}$ and $1/q=1/p-{\alpha}/n$, which was obtained by Muckenhoupt and Wheeden. We also reprove that under the assumptions that $\Omega\in L^s(\mathbf{S}^{n-1})$, $s'\leq p<n/{\alpha}$ and $1/q=1/p-{\alpha}/n$, the operators $T_{\Omega,\alpha}$ and $M_{\Omega,\alpha}$ are bounded from $L^p(\mathbb R^n)$ to $L^{q,\infty}(\mathbb R^n)$, which was obtained by Chanillo, Watson and Wheeden. We will use the idea of Adams to show that $T_{\Omega,\alpha}$ and $M_{\Omega,\alpha}$ are bounded from $L^{p,\kappa}(\mathbb R^n)$ to $L^{q,\kappa}(\mathbb R^n)$ whenever $s'<p<n/{\alpha}$ and $1/q=1/p-\alpha/{n(1-\kappa)}$, and bounded from $L^{p,\kappa}(\mathbb R^n)$ to $WL^{q,\kappa}(\mathbb R^n)$ whenever $s'\leq p<n/{\alpha}$ and $1/q=1/p-\alpha/{n(1-\kappa)}$. Some new estimates in the limiting cases are also established. The results obtained are substantial improvements and extensions of some known results. Moreover, we will apply these results to several well-known inequalities such as Hardy--Littlewood--Sobolev and Olsen-type inequalities.