{"title":"Compactness of averaging operators on Banach function spaces","authors":"Katsuhisa Koshino","doi":"10.1007/s43037-024-00383-5","DOIUrl":null,"url":null,"abstract":"<p>Let <i>X</i> be a Borel metric measure space such that every closed ball is of positive and finite measure. In this paper, we give a sufficient condition and a necessary condition for averaging operators on a Banach function space <i>E</i>(<i>X</i>) on <i>X</i> to be compact. As a corollary, we show that the averaging operators on the Lorentz space <span>\\(L^{p,q}(X)\\)</span> of <i>X</i> are compact if and only if <i>X</i> is bounded, in the case where <i>X</i> is a doubling and Borel-regular metric measure space with some continuity between metric and measure.</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2024-09-11","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/s43037-024-00383-5","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
Let X be a Borel metric measure space such that every closed ball is of positive and finite measure. In this paper, we give a sufficient condition and a necessary condition for averaging operators on a Banach function space E(X) on X to be compact. As a corollary, we show that the averaging operators on the Lorentz space \(L^{p,q}(X)\) of X are compact if and only if X is bounded, in the case where X is a doubling and Borel-regular metric measure space with some continuity between metric and measure.
设 X 是一个 Borel 度量空间，其中每个闭球都是正有限度量。本文给出了 X 上巴拿赫函数空间 E(X) 的平均算子紧凑的充分条件和必要条件。作为推论，我们证明了在 X 是倍增和博尔规则度量空间且度量与度量之间具有某种连续性的情况下，X 的洛伦兹空间 \(L^{p,q}(X)\)上的平均算子是紧凑的，当且仅当 X 是有界的。
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.
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