{"title":"Lipschitz 地图和单调地图扩展的连续性","authors":"Krzysztof J. Ciosmak","doi":"10.1112/jlms.70014","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> be a subset of a Hilbert space. We prove that if <span></span><math>\n <semantics>\n <mrow>\n <mi>v</mi>\n <mo>:</mo>\n <mi>X</mi>\n <mo>→</mo>\n <msup>\n <mi>R</mi>\n <mi>m</mi>\n </msup>\n </mrow>\n <annotation>$v\\colon X\\rightarrow \\mathbb {R}^m$</annotation>\n </semantics></math> is such that\n\n </p><p>Moreover, if either <span></span><math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>∈</mo>\n <mo>{</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mn>2</mn>\n <mo>,</mo>\n <mn>3</mn>\n <mo>}</mo>\n </mrow>\n <annotation>$m\\in \\lbrace 1,2,3\\rbrace$</annotation>\n </semantics></math> or <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> is convex, we prove the converse: We show that a map <span></span><math>\n <semantics>\n <mrow>\n <mi>v</mi>\n <mo>:</mo>\n <mi>X</mi>\n <mo>→</mo>\n <msup>\n <mi>R</mi>\n <mi>m</mi>\n </msup>\n </mrow>\n <annotation>$v\\colon X\\rightarrow \\mathbb {R}^m$</annotation>\n </semantics></math> that allows for a 1-Lipschitz, uniform distance preserving extension of any 1-Lipschitz map on a subset of <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> also satisfies the above set of inequalities. We also prove a similar continuity result concerning extensions of monotone maps. Our results hold true also for maps taking values in infinite-dimensional spaces.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70014","citationCount":"0","resultStr":"{\"title\":\"Continuity of extensions of Lipschitz maps and of monotone maps\",\"authors\":\"Krzysztof J. Ciosmak\",\"doi\":\"10.1112/jlms.70014\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> be a subset of a Hilbert space. We prove that if <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>v</mi>\\n <mo>:</mo>\\n <mi>X</mi>\\n <mo>→</mo>\\n <msup>\\n <mi>R</mi>\\n <mi>m</mi>\\n </msup>\\n </mrow>\\n <annotation>$v\\\\colon X\\\\rightarrow \\\\mathbb {R}^m$</annotation>\\n </semantics></math> is such that\\n\\n </p><p>Moreover, if either <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>m</mi>\\n <mo>∈</mo>\\n <mo>{</mo>\\n <mn>1</mn>\\n <mo>,</mo>\\n <mn>2</mn>\\n <mo>,</mo>\\n <mn>3</mn>\\n <mo>}</mo>\\n </mrow>\\n <annotation>$m\\\\in \\\\lbrace 1,2,3\\\\rbrace$</annotation>\\n </semantics></math> or <span></span><math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> is convex, we prove the converse: We show that a map <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>v</mi>\\n <mo>:</mo>\\n <mi>X</mi>\\n <mo>→</mo>\\n <msup>\\n <mi>R</mi>\\n <mi>m</mi>\\n </msup>\\n </mrow>\\n <annotation>$v\\\\colon X\\\\rightarrow \\\\mathbb {R}^m$</annotation>\\n </semantics></math> that allows for a 1-Lipschitz, uniform distance preserving extension of any 1-Lipschitz map on a subset of <span></span><math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> also satisfies the above set of inequalities. We also prove a similar continuity result concerning extensions of monotone maps. Our results hold true also for maps taking values in infinite-dimensional spaces.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-11-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70014\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70014\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70014","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
摘要
让 X $X$ 是一个希尔伯特空间的子集。我们证明，如果 v : X → R m $v\colon X\rightarrow \mathbb {R}^m$ 是这样的，而且，如果 m ∈ { 1 , 2 , 3 } 或者 X $X$ 是凸的，我们证明反过来：如果 v : X → R m $v\colon X\rightarrow \mathbb {R}^m$ 是这样的。 或 X $X$ 是凸的，我们证明反过来：我们证明了一个映射 v : X → R m $v\colon X\rightarrow \mathbb {R}^m$ 可以在 X $X$ 的子集上对任何 1-Lipschitz 映射进行 1-Lipschitz、均匀距离保持的扩展，这个映射也满足上述不等式集。我们还证明了关于单调映射扩展的类似连续性结果。我们的结果也适用于在无限维空间取值的映射。
Continuity of extensions of Lipschitz maps and of monotone maps
Let be a subset of a Hilbert space. We prove that if is such that
Moreover, if either or is convex, we prove the converse: We show that a map that allows for a 1-Lipschitz, uniform distance preserving extension of any 1-Lipschitz map on a subset of also satisfies the above set of inequalities. We also prove a similar continuity result concerning extensions of monotone maps. Our results hold true also for maps taking values in infinite-dimensional spaces.
期刊介绍：
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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