Multilinear fractional maximal and integral operators with homogeneous kernels, Hardy--Littlewood--Sobolev and Olsen-type inequalities
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2024
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| _version_ | 1866913591278436352 |
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| author | Chen, Cong Yang, Kaikai Wang, Hua |
| author_facet | Chen, Cong Yang, Kaikai Wang, Hua |
| contents | Let $m\in \mathbb{N}$ and $0<α<mn$.In this paper, we will use the idea of Hedberg to reprove that the multilinear operators $\mathcal{T}_{Ω,α;m}$ and $\mathcal{M}_{Ω,α;m}$ 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 $\vecΩ=(Ω_1,Ω_2,\dots,Ω_m)\in L^s(\mathbf{S}^{n-1})$, $s'<p_1,p_2,\dots,p_m<n/α$, \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 $\vecΩ=(Ω_1,Ω_2,\dots,Ω_m)\in L^s(\mathbf{S}^{n-1})$, $s'\leq p_1,p_2,\dots,p_m<n/α$ and $(*)$, the multilinear operators $\mathcal{T}_{Ω,α;m}$ and $\mathcal{M}_{Ω,α;m}$ 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 $\mathcal{T}_{Ω,α;m}$ and $\mathcal{M}_{Ω,α;m}$ are bounded from $L^{p_1,κ}(\mathbb R^n)\times L^{p_2,κ}(\mathbb R^n)\times \cdots\times L^{p_m,κ}(\mathbb R^n)$ into $L^{q,κ}(\mathbb R^n)$ whenever $s'<p_1,p_2,\dots,p_m<n/α$, $0<κ<1$, \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 $L^{p_1,κ}(\mathbb R^n)\times L^{p_2,κ}(\mathbb R^n)\times \cdots\times L^{p_m,κ}(\mathbb R^n)$ into $WL^{q,κ}(\mathbb R^n)$ whenever $s'\leq p_1,p_2,\dots,p_m<n/α$, $0<κ<1$ and $(**)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_19676 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multilinear fractional maximal and integral operators with homogeneous kernels, Hardy--Littlewood--Sobolev and Olsen-type inequalities Chen, Cong Yang, Kaikai Wang, Hua Classical Analysis and ODEs 42B20, 42B25, 42B35 Let $m\in \mathbb{N}$ and $0<α<mn$.In this paper, we will use the idea of Hedberg to reprove that the multilinear operators $\mathcal{T}_{Ω,α;m}$ and $\mathcal{M}_{Ω,α;m}$ 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 $\vecΩ=(Ω_1,Ω_2,\dots,Ω_m)\in L^s(\mathbf{S}^{n-1})$, $s'<p_1,p_2,\dots,p_m<n/α$, \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 $\vecΩ=(Ω_1,Ω_2,\dots,Ω_m)\in L^s(\mathbf{S}^{n-1})$, $s'\leq p_1,p_2,\dots,p_m<n/α$ and $(*)$, the multilinear operators $\mathcal{T}_{Ω,α;m}$ and $\mathcal{M}_{Ω,α;m}$ 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 $\mathcal{T}_{Ω,α;m}$ and $\mathcal{M}_{Ω,α;m}$ are bounded from $L^{p_1,κ}(\mathbb R^n)\times L^{p_2,κ}(\mathbb R^n)\times \cdots\times L^{p_m,κ}(\mathbb R^n)$ into $L^{q,κ}(\mathbb R^n)$ whenever $s'<p_1,p_2,\dots,p_m<n/α$, $0<κ<1$, \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 $L^{p_1,κ}(\mathbb R^n)\times L^{p_2,κ}(\mathbb R^n)\times \cdots\times L^{p_m,κ}(\mathbb R^n)$ into $WL^{q,κ}(\mathbb R^n)$ whenever $s'\leq p_1,p_2,\dots,p_m<n/α$, $0<κ<1$ and $(**)$. |
| title | Multilinear fractional maximal and integral operators with homogeneous kernels, Hardy--Littlewood--Sobolev and Olsen-type inequalities |
| topic | Classical Analysis and ODEs 42B20, 42B25, 42B35 |
| url | https://arxiv.org/abs/2411.19676 |