双线性向量值奇异积分算子交换子的加权紧性及其应用

IF 0.8 3区 数学 Q2 MATHEMATICS
Zhengyang Li, Liu Lu, Fanghui Liao, Qingying Xue
{"title":"双线性向量值奇异积分算子交换子的加权紧性及其应用","authors":"Zhengyang Li,&nbsp;Liu Lu,&nbsp;Fanghui Liao,&nbsp;Qingying Xue","doi":"10.1007/s10114-025-3465-2","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>T</i> be a bilinear vector-valued singular integral operator satisfies some mild regularity conditions, which may not fall under the scope of the theory of standard Calderón–Zygmund classes. For any <span>\\(\\vec{b}=(b_{1},b_{2})\\in (\\text{CMO}(\\mathbb{R}^{n}))^{2}\\)</span>, let <span>\\([T,b_{j}]_{e_{j}}\\ (j=1,2),\\ [T,\\vec{b}]_{\\alpha}\\)</span> be the commutators in the <i>j</i>-th entry and the iterated commutators of <i>T</i>, respectively. In this paper, for all <i>p</i><sub>0</sub> &gt; 1, <span>\\({p_{0}\\over 2} &lt; p &lt; \\infty\\)</span>, and <i>p</i><sub>0</sub> ≤ <i>p</i><sub>1</sub>, <i>p</i><sub>2</sub> &lt; ∞ with 1/<i>p</i> = 1/<i>p</i><sub>1</sub> + 1/<i>p</i><sub>2</sub>, we prove that <span>\\([T,b_{j}]_{e_{j}}\\)</span> and <span>\\([T,\\vec{b}]_{\\alpha}\\)</span> are weighted compact operators from <span>\\(L^{p_{1}}(w_{1})\\times L^{p_{2}}(w_{2})\\)</span> to <span>\\(L^{p}(\\nu_{\\vec{w}})\\)</span>, where <span>\\(\\vec{w}=(w_{1},w_{2})\\in A_{\\vec{p}/p_{0}}\\)</span> and <span>\\(\\nu_{\\vec{w}}=w_{1}^{p/p_{1}}w_{2}^{p/p_{2}}\\)</span>. As applications, we obtain the weighted compactness of commutators in the <i>j</i>-th entry and the iterated commutators of several kinds of bilinear Littlewood–Paley square operators with some mild kernel regularity, including bilinear <i>g</i> function, bilinear <i>g</i>*<sub><i>λ</i></sub> function and bilinear Lusin’s area integral. In addition, we also get the weighted compactness of commutators in the <i>j</i>-th entry and the iterated commutators of bilinear Fourier multiplier operators, and bilinear square Fourier multiplier operators associated with bilinear <i>g</i> function, bilinear <i>g</i>*<sub><i>λ</i></sub> function and bilinear Lusin’s area integral, respectively.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"41 1","pages":"169 - 190"},"PeriodicalIF":0.8000,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Weighted Compactness of Commutators of Bilinear Vector-valued Singular Integral Operators and Applications\",\"authors\":\"Zhengyang Li,&nbsp;Liu Lu,&nbsp;Fanghui Liao,&nbsp;Qingying Xue\",\"doi\":\"10.1007/s10114-025-3465-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>T</i> be a bilinear vector-valued singular integral operator satisfies some mild regularity conditions, which may not fall under the scope of the theory of standard Calderón–Zygmund classes. For any <span>\\\\(\\\\vec{b}=(b_{1},b_{2})\\\\in (\\\\text{CMO}(\\\\mathbb{R}^{n}))^{2}\\\\)</span>, let <span>\\\\([T,b_{j}]_{e_{j}}\\\\ (j=1,2),\\\\ [T,\\\\vec{b}]_{\\\\alpha}\\\\)</span> be the commutators in the <i>j</i>-th entry and the iterated commutators of <i>T</i>, respectively. In this paper, for all <i>p</i><sub>0</sub> &gt; 1, <span>\\\\({p_{0}\\\\over 2} &lt; p &lt; \\\\infty\\\\)</span>, and <i>p</i><sub>0</sub> ≤ <i>p</i><sub>1</sub>, <i>p</i><sub>2</sub> &lt; ∞ with 1/<i>p</i> = 1/<i>p</i><sub>1</sub> + 1/<i>p</i><sub>2</sub>, we prove that <span>\\\\([T,b_{j}]_{e_{j}}\\\\)</span> and <span>\\\\([T,\\\\vec{b}]_{\\\\alpha}\\\\)</span> are weighted compact operators from <span>\\\\(L^{p_{1}}(w_{1})\\\\times L^{p_{2}}(w_{2})\\\\)</span> to <span>\\\\(L^{p}(\\\\nu_{\\\\vec{w}})\\\\)</span>, where <span>\\\\(\\\\vec{w}=(w_{1},w_{2})\\\\in A_{\\\\vec{p}/p_{0}}\\\\)</span> and <span>\\\\(\\\\nu_{\\\\vec{w}}=w_{1}^{p/p_{1}}w_{2}^{p/p_{2}}\\\\)</span>. As applications, we obtain the weighted compactness of commutators in the <i>j</i>-th entry and the iterated commutators of several kinds of bilinear Littlewood–Paley square operators with some mild kernel regularity, including bilinear <i>g</i> function, bilinear <i>g</i>*<sub><i>λ</i></sub> function and bilinear Lusin’s area integral. In addition, we also get the weighted compactness of commutators in the <i>j</i>-th entry and the iterated commutators of bilinear Fourier multiplier operators, and bilinear square Fourier multiplier operators associated with bilinear <i>g</i> function, bilinear <i>g</i>*<sub><i>λ</i></sub> function and bilinear Lusin’s area integral, respectively.</p></div>\",\"PeriodicalId\":50893,\"journal\":{\"name\":\"Acta Mathematica Sinica-English Series\",\"volume\":\"41 1\",\"pages\":\"169 - 190\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2025-01-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Sinica-English Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10114-025-3465-2\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-025-3465-2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

设T为双线性向量值奇异积分算子,满足一些温和的正则性条件,这些条件可能不属于标准Calderón-Zygmund类的理论范围。对于任意\(\vec{b}=(b_{1},b_{2})\in (\text{CMO}(\mathbb{R}^{n}))^{2}\),设\([T,b_{j}]_{e_{j}}\ (j=1,2),\ [T,\vec{b}]_{\alpha}\)分别为第j项的换向子和T的迭代换向子。在本文中,对于所有p0 &gt;1, \({p_{0}\over 2} < p < \infty\), p0≤p1, p2 &lt;∞时1/p = 1/p1 + 1/p2,证明\([T,b_{j}]_{e_{j}}\)和\([T,\vec{b}]_{\alpha}\)是\(L^{p_{1}}(w_{1})\times L^{p_{2}}(w_{2})\)到\(L^{p}(\nu_{\vec{w}})\)的加权紧算子,其中\(\vec{w}=(w_{1},w_{2})\in A_{\vec{p}/p_{0}}\)和\(\nu_{\vec{w}}=w_{1}^{p/p_{1}}w_{2}^{p/p_{2}}\)。作为应用,我们得到了第j项上对易子的加权紧性,以及几种具有温和核正则性的双线性Littlewood-Paley平方算子的迭代对易子,包括双线性g函数、双线性g*λ函数和双线性Lusin面积积分。此外,我们还分别得到了与双线性g函数、双线性g*λ函数和双线性Lusin面积积分相关的双线性傅里叶乘子算子、双线性平方傅里叶乘子算子在第j项上的加权紧性和对易子的迭代紧性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On Weighted Compactness of Commutators of Bilinear Vector-valued Singular Integral Operators and Applications

Let T be a bilinear vector-valued singular integral operator satisfies some mild regularity conditions, which may not fall under the scope of the theory of standard Calderón–Zygmund classes. For any \(\vec{b}=(b_{1},b_{2})\in (\text{CMO}(\mathbb{R}^{n}))^{2}\), let \([T,b_{j}]_{e_{j}}\ (j=1,2),\ [T,\vec{b}]_{\alpha}\) be the commutators in the j-th entry and the iterated commutators of T, respectively. In this paper, for all p0 > 1, \({p_{0}\over 2} < p < \infty\), and p0p1, p2 < ∞ with 1/p = 1/p1 + 1/p2, we prove that \([T,b_{j}]_{e_{j}}\) and \([T,\vec{b}]_{\alpha}\) are weighted compact operators from \(L^{p_{1}}(w_{1})\times L^{p_{2}}(w_{2})\) to \(L^{p}(\nu_{\vec{w}})\), where \(\vec{w}=(w_{1},w_{2})\in A_{\vec{p}/p_{0}}\) and \(\nu_{\vec{w}}=w_{1}^{p/p_{1}}w_{2}^{p/p_{2}}\). As applications, we obtain the weighted compactness of commutators in the j-th entry and the iterated commutators of several kinds of bilinear Littlewood–Paley square operators with some mild kernel regularity, including bilinear g function, bilinear g*λ function and bilinear Lusin’s area integral. In addition, we also get the weighted compactness of commutators in the j-th entry and the iterated commutators of bilinear Fourier multiplier operators, and bilinear square Fourier multiplier operators associated with bilinear g function, bilinear g*λ function and bilinear Lusin’s area integral, respectively.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
1.00
自引率
0.00%
发文量
138
审稿时长
14.5 months
期刊介绍: Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信