{"title":"卡诺群上Hardy不等式的一个简单证明和一些次椭圆向量场族的证明","authors":"Franccois Vigneron","doi":"10.2140/tunis.2020.2.851","DOIUrl":null,"url":null,"abstract":"We give an elementary proof of the classical Hardy inequality on any Carnot group, using only integration by parts and a fine analysis of the commutator structure, which was not deemed possible until now. We also discuss the conditions under which this technique can be generalized to deal with hypoelliptic families of vector fields, which, in this case, leads to an open problem regarding the symbol properties of the gauge norm.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2019-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.851","citationCount":"1","resultStr":"{\"title\":\"A simple proof of the Hardy inequality on Carnot groups and for some hypoelliptic families of vector fields\",\"authors\":\"Franccois Vigneron\",\"doi\":\"10.2140/tunis.2020.2.851\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We give an elementary proof of the classical Hardy inequality on any Carnot group, using only integration by parts and a fine analysis of the commutator structure, which was not deemed possible until now. We also discuss the conditions under which this technique can be generalized to deal with hypoelliptic families of vector fields, which, in this case, leads to an open problem regarding the symbol properties of the gauge norm.\",\"PeriodicalId\":36030,\"journal\":{\"name\":\"Tunisian Journal of Mathematics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2019-08-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.2140/tunis.2020.2.851\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Tunisian Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2140/tunis.2020.2.851\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tunisian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/tunis.2020.2.851","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
A simple proof of the Hardy inequality on Carnot groups and for some hypoelliptic families of vector fields
We give an elementary proof of the classical Hardy inequality on any Carnot group, using only integration by parts and a fine analysis of the commutator structure, which was not deemed possible until now. We also discuss the conditions under which this technique can be generalized to deal with hypoelliptic families of vector fields, which, in this case, leads to an open problem regarding the symbol properties of the gauge norm.