Duan Xu Dai, Ji Chao Zhang, Quan Qing Fang, Long Fa Sun, Ben Tuo Zheng
{"title":"粗糙Lipschitz嵌入稳定性的一个普遍不等式","authors":"Duan Xu Dai, Ji Chao Zhang, Quan Qing Fang, Long Fa Sun, Ben Tuo Zheng","doi":"10.1007/s10114-023-2136-4","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>X</i> and <i>Y</i> be two pointed metric spaces. In this article, we give a generalization of the Cheng–Dong–Zhang theorem for coarse Lipschitz embeddings as follows: If <i>f</i>:<i>X</i> → <i>Y</i> is a standard coarse Lipschitz embedding, then for each <i>x</i>* ∈ Lip<sub>0</sub>(<i>X</i>) there exist <i>α, γ</i> > 0 depending only on <i>f</i> and <i>Q</i><sub><i>x</i>*</sub> ∈ Lip<sub>0</sub>(<i>Y</i>) with <span>\\({\\Vert{{Q_{{x^*}}}}\\Vert_{{\\rm{Lip}}}} \\le \\alpha {\\Vert {{x^*}}\\Vert_{{\\rm{Lip}}}}\\)</span> such that </p><div><div><span>$$\\Vert{{Q_{{x^*}}}f(x) - {x^*}(x)}\\Vert\\le \\gamma {\\left\\| {{x^*}} \\right\\|_{{\\rm{Lip}}}},\\;\\;\\;\\;\\;{\\rm{for}}\\;{\\rm{all}}\\;x \\in X.$$</span></div></div><p>Coarse stability for a pair of metric spaces is studied. This can be considered as a coarse version of Qian Problem. As an application, we give candidate negative answers to a 58-year old problem by Lindenstrauss asking whether every Banach space is a Lipschitz retract of its bidual. Indeed, we show that <i>X</i> is not a Lipschitz retract of its bidual if <i>X</i> is a universally left-coarsely stable space but not an absolute cardinality-Lipschitz retract. If there exists a universally right-coarsely stable Banach space with the RNP but not isomorphic to any Hilbert space, then the problem also has a negative answer for a separable space.</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":"0","resultStr":"{\"title\":\"A Universal Inequality for Stability of Coarse Lipschitz Embeddings\",\"authors\":\"Duan Xu Dai, Ji Chao Zhang, Quan Qing Fang, Long Fa Sun, Ben Tuo Zheng\",\"doi\":\"10.1007/s10114-023-2136-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>X</i> and <i>Y</i> be two pointed metric spaces. In this article, we give a generalization of the Cheng–Dong–Zhang theorem for coarse Lipschitz embeddings as follows: If <i>f</i>:<i>X</i> → <i>Y</i> is a standard coarse Lipschitz embedding, then for each <i>x</i>* ∈ Lip<sub>0</sub>(<i>X</i>) there exist <i>α, γ</i> > 0 depending only on <i>f</i> and <i>Q</i><sub><i>x</i>*</sub> ∈ Lip<sub>0</sub>(<i>Y</i>) with <span>\\\\({\\\\Vert{{Q_{{x^*}}}}\\\\Vert_{{\\\\rm{Lip}}}} \\\\le \\\\alpha {\\\\Vert {{x^*}}\\\\Vert_{{\\\\rm{Lip}}}}\\\\)</span> such that </p><div><div><span>$$\\\\Vert{{Q_{{x^*}}}f(x) - {x^*}(x)}\\\\Vert\\\\le \\\\gamma {\\\\left\\\\| {{x^*}} \\\\right\\\\|_{{\\\\rm{Lip}}}},\\\\;\\\\;\\\\;\\\\;\\\\;{\\\\rm{for}}\\\\;{\\\\rm{all}}\\\\;x \\\\in X.$$</span></div></div><p>Coarse stability for a pair of metric spaces is studied. This can be considered as a coarse version of Qian Problem. As an application, we give candidate negative answers to a 58-year old problem by Lindenstrauss asking whether every Banach space is a Lipschitz retract of its bidual. Indeed, we show that <i>X</i> is not a Lipschitz retract of its bidual if <i>X</i> is a universally left-coarsely stable space but not an absolute cardinality-Lipschitz retract. If there exists a universally right-coarsely stable Banach space with the RNP but not isomorphic to any Hilbert space, then the problem also has a negative answer for a separable space.</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\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Sinica-English Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10114-023-2136-4\",\"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":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-023-2136-4","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
A Universal Inequality for Stability of Coarse Lipschitz Embeddings
Let X and Y be two pointed metric spaces. In this article, we give a generalization of the Cheng–Dong–Zhang theorem for coarse Lipschitz embeddings as follows: If f:X → Y is a standard coarse Lipschitz embedding, then for each x* ∈ Lip0(X) there exist α, γ > 0 depending only on f and Qx* ∈ Lip0(Y) with \({\Vert{{Q_{{x^*}}}}\Vert_{{\rm{Lip}}}} \le \alpha {\Vert {{x^*}}\Vert_{{\rm{Lip}}}}\) such that
Coarse stability for a pair of metric spaces is studied. This can be considered as a coarse version of Qian Problem. As an application, we give candidate negative answers to a 58-year old problem by Lindenstrauss asking whether every Banach space is a Lipschitz retract of its bidual. Indeed, we show that X is not a Lipschitz retract of its bidual if X is a universally left-coarsely stable space but not an absolute cardinality-Lipschitz retract. If there exists a universally right-coarsely stable Banach space with the RNP but not isomorphic to any Hilbert space, then the problem also has a negative answer for a separable space.
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.