{"title":"具有 $${{v\\arepsilon }$$ 范围的 Bakry-Émery-Ricci 曲率下限下的一些函数不等式","authors":"Yasuaki Fujitani","doi":"10.1007/s00229-024-01537-3","DOIUrl":null,"url":null,"abstract":"<p>For <i>n</i>-dimensional weighted Riemannian manifolds, lower <i>m</i>-Bakry–Émery–Ricci curvature bounds with <span>\\({\\varepsilon }\\)</span>-range, introduced by Lu-Minguzzi-Ohta (Anal Geom Metr Spaces 10(1):1–30, 2022), integrate constant lower bounds and certain variable lower bounds in terms of weight functions. In this paper, we prove a Cheng type inequality and a local Sobolev inequality under lower <i>m</i>-Bakry–Émery–Ricci curvature bounds with <span>\\({\\varepsilon }\\)</span>-range. These generalize those inequalities under constant curvature bounds for <span>\\(m \\in (n,\\infty )\\)</span> to <span>\\(m\\in (-\\infty ,1]\\cup \\{\\infty \\}\\)</span>.\n</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Some functional inequalities under lower Bakry–Émery–Ricci curvature bounds with $${\\\\varepsilon }$$ -range\",\"authors\":\"Yasuaki Fujitani\",\"doi\":\"10.1007/s00229-024-01537-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For <i>n</i>-dimensional weighted Riemannian manifolds, lower <i>m</i>-Bakry–Émery–Ricci curvature bounds with <span>\\\\({\\\\varepsilon }\\\\)</span>-range, introduced by Lu-Minguzzi-Ohta (Anal Geom Metr Spaces 10(1):1–30, 2022), integrate constant lower bounds and certain variable lower bounds in terms of weight functions. In this paper, we prove a Cheng type inequality and a local Sobolev inequality under lower <i>m</i>-Bakry–Émery–Ricci curvature bounds with <span>\\\\({\\\\varepsilon }\\\\)</span>-range. These generalize those inequalities under constant curvature bounds for <span>\\\\(m \\\\in (n,\\\\infty )\\\\)</span> to <span>\\\\(m\\\\in (-\\\\infty ,1]\\\\cup \\\\{\\\\infty \\\\}\\\\)</span>.\\n</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-04-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00229-024-01537-3\",\"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://doi.org/10.1007/s00229-024-01537-3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Some functional inequalities under lower Bakry–Émery–Ricci curvature bounds with $${\varepsilon }$$ -range
For n-dimensional weighted Riemannian manifolds, lower m-Bakry–Émery–Ricci curvature bounds with \({\varepsilon }\)-range, introduced by Lu-Minguzzi-Ohta (Anal Geom Metr Spaces 10(1):1–30, 2022), integrate constant lower bounds and certain variable lower bounds in terms of weight functions. In this paper, we prove a Cheng type inequality and a local Sobolev inequality under lower m-Bakry–Émery–Ricci curvature bounds with \({\varepsilon }\)-range. These generalize those inequalities under constant curvature bounds for \(m \in (n,\infty )\) to \(m\in (-\infty ,1]\cup \{\infty \}\).