{"title":"Marcinkiewicz–Zygmund inequalities in variable Lebesgue spaces","authors":"Marcos Bonich, Daniel Carando, Martin Mazzitelli","doi":"10.1007/s43037-024-00344-y","DOIUrl":null,"url":null,"abstract":"<p>We study <span>\\(\\ell ^r\\)</span>-valued extensions of linear operators defined on Lebesgue spaces with variable exponent. Under some natural (and usual) conditions on the exponents, we characterize <span>\\(1\\le r\\le \\infty \\)</span> such that every bounded linear operator <span>\\(T:L^{q(\\cdot )}(\\Omega _2, \\mu )\\rightarrow L^{p(\\cdot )}(\\Omega _1, \\nu )\\)</span> has a bounded <span>\\(\\ell ^r\\)</span>-valued extension. We consider both non-atomic measures and measures with atoms and show the differences that can arise. We present some applications of our results to weighted norm inequalities of linear operators and vector-valued extensions of fractional operators with rough kernel.</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"115 1","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Banach Journal of Mathematical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s43037-024-00344-y","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We study \(\ell ^r\)-valued extensions of linear operators defined on Lebesgue spaces with variable exponent. Under some natural (and usual) conditions on the exponents, we characterize \(1\le r\le \infty \) such that every bounded linear operator \(T:L^{q(\cdot )}(\Omega _2, \mu )\rightarrow L^{p(\cdot )}(\Omega _1, \nu )\) has a bounded \(\ell ^r\)-valued extension. We consider both non-atomic measures and measures with atoms and show the differences that can arise. We present some applications of our results to weighted norm inequalities of linear operators and vector-valued extensions of fractional operators with rough kernel.
期刊介绍:
The Banach Journal of Mathematical Analysis (Banach J. Math. Anal.) is published by Birkhäuser on behalf of the Tusi Mathematical Research Group.
Banach J. Math. Anal. is a peer-reviewed electronic journal publishing papers of high standards with deep results, new ideas, profound impact, and significant implications in all areas of functional analysis and operator theory and all modern related topics. Banach J. Math. Anal. normally publishes survey articles and original research papers numbering 15 pages or more in the journal’s style. Shorter papers may be submitted to the Annals of Functional Analysis or Advances in Operator Theory.