Let \(0<\alpha <n\) and \(\Omega \) be homogeneous of degree zero in \(\mathbb R^n\) for \(n\ge 2\) and integrable on the unit sphere \(\textbf{S}^{n-1}\) . Let \(T_{\Omega ,\alpha }\) be the homogeneous fractional integral operator and \(M_{\Omega ,\alpha }\) be the related fractional maximal operator. 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 \(\Omega \in L^s(\textbf{S}^{n-1})\) , \(s'<p<n/{\alpha }\) and \(1/q=1/p-{\alpha }/n\) , which was obtained by Muckenhoupt–Wheeden. We also reprove that under the assumptions that \(\Omega \in L^s(\textbf{S}^{n-1})\) , \(s'=p<n/{\alpha }\) 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–Wheeden. We will use the idea of Adams to show that both \(T_{\Omega ,\alpha }\) 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 \(s'=p<n/{\alpha }\) and \(1/q=1/p-\alpha /{n(1-\kappa )}\) . Some new estimates in the limiting cases are also established, under certain smoothness condition on \(\Omega \) . The results obtained are substantial improvements and extensions of some known results. Moreover, we will apply these results to several well-known inequalities on \(\mathbb R^n\) such as Hardy–Littlewood–Sobolev inequalities and Olsen-type inequalities.
