Let m ∈ ℕ and 0 < α < mn. Let \(\cal{T}_{\Omega,\alpha;m}\) be the multilinear fractional integral operator with homogeneous kernels, and let \(\cal{M}_{\Omega,\alpha;m}\) be the multilinear fractional maximal operator with homogeneous kernels. We will use the idea of Hedberg to reprove that the multilinear operators \(\cal{T}_{\Omega,\alpha;m}\) and \(\cal{M}_{\Omega,\alpha;m}\) are bounded from \(L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n) \times\ldots\times L^{p_m}(\mathbb R^n)\) into Lq (ℝn) provided that \(\vec{\Omega}=(\Omega_1,\Omega_2,\dots,\Omega_m)\in[L^s({{\mathbf{S}}}^{n-1})]^{m}, s^{\prime}< p_1,p_2,\dots,p_m <\infty, s^{\prime} /m< p < n /{\alpha}\)
(*) \({1 \over p} = {1 \over {{p_1}}} + {1 \over {{p_2}}} + \cdots + {1 \over {{p_m}}}\quad {\text{and}}\quad {1 \over q} = {1 \over p} - {\alpha \over n}.\)
This result was first obtained by Chen and Xue. We also prove that under the assumptions that \(\vec{\Omega}=(\Omega_1,\Omega_2,\ldots,\Omega_m) \in[L^s({{\mathbf S}}^{n-1})]^{m}, s^{\prime}\leqslant p_1,p_2,\ldots,p_m<\infty, s^{\prime} /m\leqslant p< n /{\alpha}\) and (∗), the multilinear operators \(\cal{T}_{\Omega,\alpha;m}\) and \(\cal{M}_{\Omega,\alpha;m}\) are bounded from \(L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n) \times\ldots\times L^{p_m}(\mathbb R^n)\) into Lq,∞(ℝn), which are completely new. Moreover, we will use the idea of Adams to show that \(\cal{T}_{\Omega,\alpha;m}\) and \(\cal{M}_{\Omega,\alpha;m}\) are bounded from \(L^{p_1,\kappa}(\mathbb R^n)\times L^{p_2,\kappa}(\mathbb R^n) \times\ldots\times L^{p_m,\kappa}(\mathbb R^n)\) into Lq,κ(ℝn) whenever \(s^{\prime}<p_1,p_2,\ldots,p_m <\infty, 0 <\kappa <1, s^{\prime} /m<p <{n(1-\kappa)} /{\alpha}\)
(**) \({1 \over p} = {1 \over {{p_1}}} + {1 \over {{p_2}}} + \cdots + {1 \over {{p_m}}}\quad {\text{and}}\quad {1 \over q} = {1 \over p} - {\alpha \over {n(1 - \kappa )}},\)
and also bounded from \(L^{p_1,\kappa}(\mathbb R^n)\times L^{p_2,\kappa}(\mathbb R^n) \times\ldots\times L^{p_m,\kappa}(\mathbb R^n)\) into WLq,κ(ℝn) whenever \(s^{\prime}\leqslant p_1,p_2,\ldots,p_m<\infty, 0 <\kappa <1, s^{\prime} /m\leqslant p<{n(1-\kappa)} /{\alpha}\) and (∗∗). These results mentioned above are also completely new. In addition, some new estimates in the limiting cases are also established. Applications to the Hardy-Littlewood-Sobolev and Olsen-type inequalities are discussed as well.
