{"title":"Characterization of Lipschitz Functions via Commutators of Multilinear Singular Integral Operators in Variable Lebesgue Spaces","authors":"Jiang Long Wu, Pu Zhang","doi":"10.1007/s10114-023-2164-0","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\(\\overrightarrow{b}=(b_{1},b_{2},\\ldots,b_{m})\\)</span> be a collection of locally integrable functions and <span>\\(T_{\\Sigma\\overrightarrow{b}}\\)</span> the commutator of multilinear singular integral operator <i>T</i>. Denote by <span>\\(\\mathbb{L}(\\delta)\\)</span> and <span>\\(\\mathbb{L}(\\delta(\\cdot))\\)</span> the Lipschitz spaces and the variable Lipschitz spaces, respectively. The main purpose of this paper is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of multilinear commutator <span>\\(T_{\\Sigma\\overrightarrow{b}}\\)</span> in the context of the variable exponent Lebesgue spaces, that is, the authors give the necessary and sufficient conditions for <i>b</i><sub><i>j</i></sub> (<i>j</i> = 1, 2, …, <i>m</i>) to be <span>\\(\\mathbb{L}(\\delta)\\)</span> or <span>\\(\\mathbb{L}(\\delta(\\cdot))\\)</span> via the boundedness of multilinear commutator from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces. The authors do so by applying the Fourier series technique and some pointwise estimate for the commutators. The key tool in obtaining such pointwise estimate is a certain generalization of the classical sharp maximal operator.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-023-2164-0","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: 1
Abstract
Let \(\overrightarrow{b}=(b_{1},b_{2},\ldots,b_{m})\) be a collection of locally integrable functions and \(T_{\Sigma\overrightarrow{b}}\) the commutator of multilinear singular integral operator T. Denote by \(\mathbb{L}(\delta)\) and \(\mathbb{L}(\delta(\cdot))\) the Lipschitz spaces and the variable Lipschitz spaces, respectively. The main purpose of this paper is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of multilinear commutator \(T_{\Sigma\overrightarrow{b}}\) in the context of the variable exponent Lebesgue spaces, that is, the authors give the necessary and sufficient conditions for bj (j = 1, 2, …, m) to be \(\mathbb{L}(\delta)\) or \(\mathbb{L}(\delta(\cdot))\) via the boundedness of multilinear commutator from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces. The authors do so by applying the Fourier series technique and some pointwise estimate for the commutators. The key tool in obtaining such pointwise estimate is a certain generalization of the classical sharp maximal operator.
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.