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Keywords:
multilinear fractional integral operator; multilinear fractional maximal operator; homogeneous kernel; Morrey space; Hardy-Littlewood-Sobolev inequality; Olsen-type inequality\looseness -1
multilinear fractional integral operator; multilinear fractional maximal operator; homogeneous kernel; Morrey space; Hardy-Littlewood-Sobolev inequality; Olsen-type inequality\looseness -1
Summary:
Let $m\in \mathbb {N}$ and $0<\alpha <mn$. Let $\mathcal {T}_{\Omega ,\alpha ;m}$ be the multilinear fractional integral operator with homogeneous kernels, and let $\mathcal {M}_{\Omega ,\alpha ;m}$ be the multilinear fractional maximal operator with homogeneous kernels. We will use the idea of Hedberg to reprove that the multilinear operators $\mathcal {T}_{\Omega ,\alpha ;m}$ and $\mathcal {M}_{\Omega ,\alpha ;m}$ are bounded from $L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n) \times \nobreak \cdots \times L^{p_m}(\mathbb R^n)$ into $L^q(\mathbb R^n)$ provided that $\vec {\Omega }=(\Omega _1,\Omega _2,\dots ,\Omega _m)\in [L^s({\bf S}^{n-1})]^{m}$, $s'<p_1,p_2,\dots ,p_m<\infty $, $s'/m<p<n/{\alpha }$, $$ \frac {1}{p}=\frac {1}{p_1}+\frac {1}{p_2}+\cdots +\frac {1}{p_m} \quad \text {and} \quad \frac {1}{q}=\frac {1}{p}-\frac {\alpha }{n}. $$ This result was first obtained by Chen and Xue. We also prove that under the assumptions that $\vec {\Omega }=(\Omega _1,\Omega _2,\dots ,\Omega _m) \in [L^s({\bf S}^{n-1})]^{m}$, $s'\leq p_1,p_2,\dots ,p_m<\infty $, $s'/m\leq p<n/{\alpha }$ and $(*)$, the multilinear operators $\mathcal {T}_{\Omega ,\alpha ;m}$ and $\mathcal {M}_{\Omega ,\alpha ;m}$ 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 $L^{q,\infty }(\mathbb R^n)$, which are completely new. Moreover, we will use the idea of Adams to show that $\mathcal {T}_{\Omega ,\alpha ;m}$ and $\mathcal {M}_{\Omega ,\alpha ;m}$ 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'<p_1,p_2,\dots ,p_m<\infty $, $0<\kappa <1$, $s'/m<p<{n(1-\kappa )}/{\alpha }$, $$ \frac {1}{p}=\frac {1}{p_1}+\frac {1}{p_2}+\cdots +\frac {1}{p_m} \quad \text {and} \quad \frac {1}{q}=\frac {1}{p}-\frac {\alpha }{n(1-\kappa )}, $$ 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 $WL^{q,\kappa }(\mathbb R^n)$ whenever $s'\leq p_1,p_2,\dots ,p_m<\infty $, $0<\kappa <1$, $s'/m\leq p<{n(1-\kappa )}/{\alpha }$ 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.
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