Multilinear fractional maximal and integral operators with homogeneous kernels, Hardy--Littlewood--Sobolev and Olsen-type inequalities

Let $m\in \mathbb{N}$ and $0<\alpha <mn$.In this paper, we will use the idea of Hedberg to reprove that the multilinear operators ${\mathcal{T}}_{\mathrm{\Omega },\alpha ;m}$ and ${\mathcal{M}}_{\mathrm{\Omega },\alpha ;m}$ are bounded from $L^{p_1}({\mathbb{R}}^n)\times L^{p_2}({\mathbb{R}}^n)\times \cdots \times L^{p_m}({\mathbb{R}}^n)$ into $L^q({\mathbb{R}}^n)$ provided that $\vec{\mathrm{\Omega }}=({\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2,\dots ,{\mathrm{\Omega }}_m)\in L^s({\mathbf{S}}^{n-1})$, $s^{\mathrm{\prime }}<p_1,p_2,\dots ,p_m<n\mathrm{/}\alpha$, \begin{equation*} \frac{,1,}{p}=\frac{1}{p_1}+\frac{1}{p_2}+\cdots+\frac{1}{p_m} \quad \mbox{and} \quad \frac{,1,}{q}=\frac{,1,}{p}-\fracα{n}. \qquad () \end{equation} We also prove that under the assumptions that $\vec{\mathrm{\Omega }}=({\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2,\dots ,{\mathrm{\Omega }}_m)\in L^s({\mathbf{S}}^{n-1})$, $s^{\mathrm{\prime }}\le p_1,p_2,\dots ,p_m<n\mathrm{/}\alpha$ and $(\ast )$, the multilinear operators ${\mathcal{T}}_{\mathrm{\Omega },\alpha ;m}$ and ${\mathcal{M}}_{\mathrm{\Omega },\alpha ;m}$ are bounded from $L^{p_1}({\mathbb{R}}^n)\times L^{p_2}({\mathbb{R}}^n)\times \cdots \times L^{p_m}({\mathbb{R}}^n)$ into $L^{q,\mathrm{\infty }}({\mathbb{R}}^n)$, which are completely new. Moreover, we will use the idea of Adams to show that ${\mathcal{T}}_{\mathrm{\Omega },\alpha ;m}$ and ${\mathcal{M}}_{\mathrm{\Omega },\alpha ;m}$ are bounded from $L^{p_1,\kappa }({\mathbb{R}}^n)\times L^{p_2,\kappa }({\mathbb{R}}^n)\times \cdots \times L^{p_m,\kappa }({\mathbb{R}}^n)$ into $L^{q,\kappa }({\mathbb{R}}^n)$ whenever $s^{\mathrm{\prime }}<p_1,p_2,\dots ,p_m<n\mathrm{/}\alpha$, $0<\kappa <1$, \begin{equation*} \frac{,1,}{p}=\frac{1}{p_1}+\frac{1}{p_2}+\cdots+\frac{1}{p_m} \quad \mbox{and} \quad \frac{,1,}{q}=\frac{,1,}{p}-\fracα{n(1-κ)},\qquad (**) \end{equation*} and also bounded from $L^{p_1,\kappa }({\mathbb{R}}^n)\times L^{p_2,\kappa }({\mathbb{R}}^n)\times \cdots \times L^{p_m,\kappa }({\mathbb{R}}^n)$ into $W{L}^{q,\kappa }({\mathbb{R}}^n)$ whenever $s^{\mathrm{\prime }}\le p_1,p_2,\dots ,p_m<n\mathrm{/}\alpha$, $0<\kappa <1$ and $(\ast \ast )$.