{"title":"在无穷维流形中取值的绝对连续函数的流形和半李群的正则特性","authors":"Matthieu F. Pinaud","doi":"arxiv-2409.06512","DOIUrl":null,"url":null,"abstract":"For $p\\in [1,\\infty]$, we define a smooth manifold structure on the set of\nabsolutely continuous functions $\\gamma\\colon [a,b]\\to N$ with\n$L^p$-derivatives for each smooth manifold $N$ modeled on a sequentially\ncomplete locally convex topological vector space which admits a local addition.\nSmoothness of natural mappings between spaces of absolutely continuous\nfunctions is discussed. For $1\\leq p <\\infty$ and $r\\in \\mathbb{N}$ we show\nthat the right half-Lie groups $\\text{Diff}_K(\\mathbb{R})$ and $\\text{Diff}(M)$\nare $L^p$-semiregular. Here $K$ is a compact subset of $\\mathbb{R}^n$ and $M$\nis a compact smooth manifold. For the preceding examples, the evolution map\n$\\text{Evol}$ is continuous.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"61 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Manifolds of absolutely continuous functions with values in an infinite-dimensional manifold and regularity properties of half-Lie groups\",\"authors\":\"Matthieu F. Pinaud\",\"doi\":\"arxiv-2409.06512\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For $p\\\\in [1,\\\\infty]$, we define a smooth manifold structure on the set of\\nabsolutely continuous functions $\\\\gamma\\\\colon [a,b]\\\\to N$ with\\n$L^p$-derivatives for each smooth manifold $N$ modeled on a sequentially\\ncomplete locally convex topological vector space which admits a local addition.\\nSmoothness of natural mappings between spaces of absolutely continuous\\nfunctions is discussed. For $1\\\\leq p <\\\\infty$ and $r\\\\in \\\\mathbb{N}$ we show\\nthat the right half-Lie groups $\\\\text{Diff}_K(\\\\mathbb{R})$ and $\\\\text{Diff}(M)$\\nare $L^p$-semiregular. Here $K$ is a compact subset of $\\\\mathbb{R}^n$ and $M$\\nis a compact smooth manifold. For the preceding examples, the evolution map\\n$\\\\text{Evol}$ is continuous.\",\"PeriodicalId\":501036,\"journal\":{\"name\":\"arXiv - MATH - Functional Analysis\",\"volume\":\"61 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Functional Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.06512\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Functional Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06512","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Manifolds of absolutely continuous functions with values in an infinite-dimensional manifold and regularity properties of half-Lie groups
For $p\in [1,\infty]$, we define a smooth manifold structure on the set of
absolutely continuous functions $\gamma\colon [a,b]\to N$ with
$L^p$-derivatives for each smooth manifold $N$ modeled on a sequentially
complete locally convex topological vector space which admits a local addition.
Smoothness of natural mappings between spaces of absolutely continuous
functions is discussed. For $1\leq p <\infty$ and $r\in \mathbb{N}$ we show
that the right half-Lie groups $\text{Diff}_K(\mathbb{R})$ and $\text{Diff}(M)$
are $L^p$-semiregular. Here $K$ is a compact subset of $\mathbb{R}^n$ and $M$
is a compact smooth manifold. For the preceding examples, the evolution map
$\text{Evol}$ is continuous.