Weighted estimates for Multilinear Singular Integrals with Rough Kernels
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911693122043904 |
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| author | Park, Bae Jun |
| author_facet | Park, Bae Jun |
| contents | We establish weighted norm inequalities for multilinear singular integral operators with rough kernels. Specifically, we consider the multilinear singular integral operator $\mathcal{L}_Ω$ associated with an integrable function $Ω$ on the unit sphere $\mathbb{S}^{mn-1}$ satisfying the vanishing mean condition. Extending the classical results of Watson and Duoandikoetxea to the multilinear setting, we prove that $\mathcal{L}_Ω$ is bounded from $L^{p_1}(w_1)\times\cdots\times L^{p_m}(w_m)$ to $L^p(v_{\vec{\boldsymbol{w}}})$ under the assumption that $Ω\in L^q(\mathbb{S}^{mn-1})$ and that the $m$ tuple of weights $\vec{\boldsymbol{w}}= (w_1,\ldots,w_m)$ lies in the multiple weight class $A_{\vec{\boldsymbol{p}}/q'}((\mathbb{R}^n)^m)$. Here, $q'$ denotes the Hölder conjugate of $q$, and we assume $q'\le p_1,\dots,p_m<\infty$ with $1/p = 1/p_1 + \cdots + 1/p_m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12119 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted estimates for Multilinear Singular Integrals with Rough Kernels Park, Bae Jun Classical Analysis and ODEs 42B20, 42B25, 42B35, 47H60 We establish weighted norm inequalities for multilinear singular integral operators with rough kernels. Specifically, we consider the multilinear singular integral operator $\mathcal{L}_Ω$ associated with an integrable function $Ω$ on the unit sphere $\mathbb{S}^{mn-1}$ satisfying the vanishing mean condition. Extending the classical results of Watson and Duoandikoetxea to the multilinear setting, we prove that $\mathcal{L}_Ω$ is bounded from $L^{p_1}(w_1)\times\cdots\times L^{p_m}(w_m)$ to $L^p(v_{\vec{\boldsymbol{w}}})$ under the assumption that $Ω\in L^q(\mathbb{S}^{mn-1})$ and that the $m$ tuple of weights $\vec{\boldsymbol{w}}= (w_1,\ldots,w_m)$ lies in the multiple weight class $A_{\vec{\boldsymbol{p}}/q'}((\mathbb{R}^n)^m)$. Here, $q'$ denotes the Hölder conjugate of $q$, and we assume $q'\le p_1,\dots,p_m<\infty$ with $1/p = 1/p_1 + \cdots + 1/p_m$. |
| title | Weighted estimates for Multilinear Singular Integrals with Rough Kernels |
| topic | Classical Analysis and ODEs 42B20, 42B25, 42B35, 47H60 |
| url | https://arxiv.org/abs/2504.12119 |