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

Abstract

Let $m\in \mathbb{N}$ and $0<\hat{I}\pm <mn$.In this paper, we will use the idea of Hedberg to reprove that the multilinear operators ${\mathcal{T}}_{\hat{I}\text{c◯},\hat{I}\pm ;m}$ and ${\mathcal{M}}_{\hat{I}\text{c◯},\hat{I}\pm ;m}$ 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 $\vec{\hat{I}}\text{c◯}=(\hat{I}{\text{c◯}}_1,\hat{I}{\text{c◯}}_2,\dots ,\hat{I}{\text{c◯}}_m)\in L^s({\mathbf{S}}^{n-1})$, $s^{\prime }<p_1,p_2,\dots ,p_m<n/\hat{I}\pm$, \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 $\vec{\hat{I}}\text{c◯}=(\hat{I}{\text{c◯}}_1,\hat{I}{\text{c◯}}_2,\dots ,\hat{I}{\text{c◯}}_m)\in L^s({\mathbf{S}}^{n-1})$, $s^{\prime }\le p_1,p_2,\dots ,p_m<n/\hat{I}\pm$ and $(\ast )$, the multilinear operators ${\mathcal{T}}_{\hat{I}\text{c◯},\hat{I}\pm ;m}$ and ${\mathcal{M}}_{\hat{I}\text{c◯},\hat{I}\pm ;m}$ 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 ${\mathcal{T}}_{\hat{I}\text{c◯},\hat{I}\pm ;m}$ and ${\mathcal{M}}_{\hat{I}\text{c◯},\hat{I}\pm ;m}$ are bounded from $L^{p_1,\hat{I}º}({\mathbb{R}}^n)\times L^{p_2,\hat{I}º}({\mathbb{R}}^n)\times \cdots \times L^{p_m,\hat{I}º}({\mathbb{R}}^n)$ into $L^{q,\hat{I}º}({\mathbb{R}}^n)$ whenever $s^{\prime }<p_1,p_2,\dots ,p_m<n/\hat{I}\pm$, $0<\hat{I}º<1$, \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 $L^{p_1,\hat{I}º}({\mathbb{R}}^n)\times L^{p_2,\hat{I}º}({\mathbb{R}}^n)\times \cdots \times L^{p_m,\hat{I}º}({\mathbb{R}}^n)$ into $W{L}^{q,\hat{I}º}({\mathbb{R}}^n)$ whenever $s^{\prime }\le p_1,p_2,\dots ,p_m<n/\hat{I}\pm$, $0<\hat{I}º<1$ and $(\ast \ast )$.

41 pages. arXiv admin note: substantial text overlap with arXiv:2212.14774