{"title":"Grand Lebesgue空间中的复合算子","authors":"A. Karapetyants, M. Lanza de Cristoforis","doi":"10.1007/s10476-023-0201-y","DOIUrl":null,"url":null,"abstract":"<div><p>Let Ω be an open subset of ℝ<sup><i>n</i></sup> of finite measure. Let <i>f</i> be a Borel measurable function from ℝ to ℝ. We prove necessary and sufficient conditions on <i>f</i> in order that the composite function <i>T</i><sub><i>f</i></sub>[<i>g</i>] = <i>f</i> o <i>g</i> belongs to the Grand Lebesgue space <i>L</i><sub><i>p</i>),<i>θ</i></sub>(Ω) whenever <i>g</i> belongs to <i>L</i><sub><i>p</i>),<i>θ</i></sub>(Ω).</p><p>We also study continuity, uniform continuity, Hölder and Lipschitz continuity of the composition operator <i>T</i><sub><i>f</i></sub>[·] in <i>L</i><sub><i>p</i>),<i>θ</i></sub>(Ω).</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Composition Operators in Grand Lebesgue Spaces\",\"authors\":\"A. Karapetyants, M. Lanza de Cristoforis\",\"doi\":\"10.1007/s10476-023-0201-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let Ω be an open subset of ℝ<sup><i>n</i></sup> of finite measure. Let <i>f</i> be a Borel measurable function from ℝ to ℝ. We prove necessary and sufficient conditions on <i>f</i> in order that the composite function <i>T</i><sub><i>f</i></sub>[<i>g</i>] = <i>f</i> o <i>g</i> belongs to the Grand Lebesgue space <i>L</i><sub><i>p</i>),<i>θ</i></sub>(Ω) whenever <i>g</i> belongs to <i>L</i><sub><i>p</i>),<i>θ</i></sub>(Ω).</p><p>We also study continuity, uniform continuity, Hölder and Lipschitz continuity of the composition operator <i>T</i><sub><i>f</i></sub>[·] in <i>L</i><sub><i>p</i>),<i>θ</i></sub>(Ω).</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-02-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10476-023-0201-y\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10476-023-0201-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
设Ω是ℝ有限测度的n。设f是Borel可测函数ℝ 到ℝ. 我们证明了f上复合函数Tf[g]=f o g属于Grand Lebesgue空间Lp），θ（Ω）（当g属于Lp）时）的充要条件。我们还研究了Lp）中复合算子Tf[·]的连续性、一致连续性、Hölder和Lipschitz连续性。
Let Ω be an open subset of ℝn of finite measure. Let f be a Borel measurable function from ℝ to ℝ. We prove necessary and sufficient conditions on f in order that the composite function Tf[g] = f o g belongs to the Grand Lebesgue space Lp),θ(Ω) whenever g belongs to Lp),θ(Ω).
We also study continuity, uniform continuity, Hölder and Lipschitz continuity of the composition operator Tf[·] in Lp),θ(Ω).
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