{"title":"序列变量勒贝格空间之间的嵌入，严格和有限严格奇点","authors":"Jan Lang, Aleš Nekvinda","doi":"10.1002/mana.12031","DOIUrl":null,"url":null,"abstract":"<p>For two variable Lebesgue spaces <span></span><math>\n <semantics>\n <msub>\n <mi>ℓ</mi>\n <msub>\n <mi>p</mi>\n <mi>n</mi>\n </msub>\n </msub>\n <annotation>$\\ell _{p_n}$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <msub>\n <mi>ℓ</mi>\n <msub>\n <mi>q</mi>\n <mi>n</mi>\n </msub>\n </msub>\n <annotation>$\\ell _{q_n}$</annotation>\n </semantics></math>, with <span></span><math>\n <semantics>\n <mrow>\n <mn>0</mn>\n <mo><</mo>\n <msub>\n <mi>p</mi>\n <mi>n</mi>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>q</mi>\n <mi>n</mi>\n </msub>\n <mo><</mo>\n <mi>∞</mi>\n </mrow>\n <annotation>$0<p_n, q_n <\\infty$</annotation>\n </semantics></math>, we provide necessary and sufficient conditions under which the natural embeddings <span></span><math>\n <semantics>\n <mrow>\n <mi>i</mi>\n <mi>d</mi>\n <mo>:</mo>\n <msub>\n <mi>ℓ</mi>\n <msub>\n <mi>p</mi>\n <mi>n</mi>\n </msub>\n </msub>\n <mo>→</mo>\n <msub>\n <mi>ℓ</mi>\n <msub>\n <mi>q</mi>\n <mi>n</mi>\n </msub>\n </msub>\n </mrow>\n <annotation>$id:\\ell _{p_n} \\rightarrow \\ell _{q_n}$</annotation>\n </semantics></math> are strictly or finitely strictly singular. We also provide estimates for the Bernstein numbers of the natural embedding <span></span><math>\n <semantics>\n <mrow>\n <mi>i</mi>\n <mi>d</mi>\n </mrow>\n <annotation>$id$</annotation>\n </semantics></math> and show how they depend on the exponents <span></span><math>\n <semantics>\n <msub>\n <mi>p</mi>\n <mi>n</mi>\n </msub>\n <annotation>$p_n$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <msub>\n <mi>q</mi>\n <mi>n</mi>\n </msub>\n <annotation>$q_n$</annotation>\n </semantics></math>.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 9","pages":"2926-2941"},"PeriodicalIF":0.8000,"publicationDate":"2025-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.12031","citationCount":"0","resultStr":"{\"title\":\"Embeddings between sequence variable Lebesgue spaces, strict and finitely strict singularity\",\"authors\":\"Jan Lang, Aleš Nekvinda\",\"doi\":\"10.1002/mana.12031\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For two variable Lebesgue spaces <span></span><math>\\n <semantics>\\n <msub>\\n <mi>ℓ</mi>\\n <msub>\\n <mi>p</mi>\\n <mi>n</mi>\\n </msub>\\n </msub>\\n <annotation>$\\\\ell _{p_n}$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <msub>\\n <mi>ℓ</mi>\\n <msub>\\n <mi>q</mi>\\n <mi>n</mi>\\n </msub>\\n </msub>\\n <annotation>$\\\\ell _{q_n}$</annotation>\\n </semantics></math>, with <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>0</mn>\\n <mo><</mo>\\n <msub>\\n <mi>p</mi>\\n <mi>n</mi>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mi>q</mi>\\n <mi>n</mi>\\n </msub>\\n <mo><</mo>\\n <mi>∞</mi>\\n </mrow>\\n <annotation>$0<p_n, q_n <\\\\infty$</annotation>\\n </semantics></math>, we provide necessary and sufficient conditions under which the natural embeddings <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>i</mi>\\n <mi>d</mi>\\n <mo>:</mo>\\n <msub>\\n <mi>ℓ</mi>\\n <msub>\\n <mi>p</mi>\\n <mi>n</mi>\\n </msub>\\n </msub>\\n <mo>→</mo>\\n <msub>\\n <mi>ℓ</mi>\\n <msub>\\n <mi>q</mi>\\n <mi>n</mi>\\n </msub>\\n </msub>\\n </mrow>\\n <annotation>$id:\\\\ell _{p_n} \\\\rightarrow \\\\ell _{q_n}$</annotation>\\n </semantics></math> are strictly or finitely strictly singular. We also provide estimates for the Bernstein numbers of the natural embedding <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>i</mi>\\n <mi>d</mi>\\n </mrow>\\n <annotation>$id$</annotation>\\n </semantics></math> and show how they depend on the exponents <span></span><math>\\n <semantics>\\n <msub>\\n <mi>p</mi>\\n <mi>n</mi>\\n </msub>\\n <annotation>$p_n$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <msub>\\n <mi>q</mi>\\n <mi>n</mi>\\n </msub>\\n <annotation>$q_n$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":49853,\"journal\":{\"name\":\"Mathematische Nachrichten\",\"volume\":\"298 9\",\"pages\":\"2926-2941\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2025-08-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.12031\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematische Nachrichten\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mana.12031\",\"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":"Mathematische Nachrichten","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.12031","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
对于两个变量勒贝格空间，p n $\ell _{p_n}$和q n $\ell _{q_n}$，0 ＆lt; p n, q n ＆lt;∞$0<p_n, q_n <\infty$，我们提供了自然嵌入的充分必要条件：p n→q n $id:\ell _{p_n} \rightarrow \ell _{q_n}$是严格或有限严格奇异。我们还提供了自然嵌入id $id$的Bernstein数的估计，并展示了它们如何依赖于指数p n $p_n$和q n$q_n$。
Embeddings between sequence variable Lebesgue spaces, strict and finitely strict singularity
For two variable Lebesgue spaces and , with , we provide necessary and sufficient conditions under which the natural embeddings are strictly or finitely strictly singular. We also provide estimates for the Bernstein numbers of the natural embedding and show how they depend on the exponents and .
期刊介绍：
Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index
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