完备非紧光滑度量测度空间上的权重Sobolev不等式  

The weighted Sobolev type inequalities on complete noncompact smooth metric measure spaces

作　　者：邓严林 侯兰宝[1,2] 杜锋[1] 严政[3] DENG Yanlin;HOU Lanbao;DU Feng;YAN Zheng(School of Mathematics and Physics Science,Jingchu University of Technology,Jingmen 448000,China;Hubei Key Laboratory of Applied Mathematics,Hubei University,Wuhan 430062,China;School of Information and Mathematics,Yangtze University,Jingzhou 434023,China)

机构地区：[1]荆楚理工学院数理学院,湖北荆门448000 [2]湖北大学应用数学湖北省重点实验室,武汉430062 [3]长江大学信息与数学学院,湖北荆州434023

出　　处：《扬州大学学报（自然科学版）》2021年第2期1-6,17,共7页Journal of Yangzhou University：Natural Science Edition

基　　金：国家自然科学基金资助项目(11601041);湖北省教育厅资助项目(D20184301,D20191303,B2016261);应用数学湖北省重点实验室(湖北大学)开放基金资助项目(HBAM202001);荆楚理工学院科研团队资助项目(TD202006);荆楚理工学院资助项目(ZX202002,ZX202006)。

摘　　要：利用完备光滑度量测度空间上的体积比较定理,证明完备非紧光滑度量测度空间上权重Sobolev不等式的常数不小于相同维数欧氏空间上权重Sobolev不等式的最优常数,以及当权重Sobolev不等式在一个Bakry-émery Ricci曲率非负的n维完备非紧光滑度量测度空间上成立时,此完备非紧光滑度量测度空间等距接近于一个n维欧氏空间.Using the volume comparison theorem in the complete smooth metric measure space,it is proved that the constant in the weighted Sobolev inequality cannot be smaller than the optimal one on the Euclidean space of the same dimension.Moreover,when the weighted Sobolev inequality holds on anndimensional complete noncompact smooth metric measure space with non-negative Bakry-émery Ricci curvature,it is proved that the complete noncompact smooth metric measure space is close to an n dimensional Euclidean space of the same dimension.

关 键 词：权重Sobolev不等式 测地球 完备非紧光滑度量测度空间 Bakry-émery Ricci曲率 

分 类 号：O186.12[理学—数学]