【作者】 王伟沧；

【导师】 朱赋鎏；

【作者基本信息】 武汉大学 ， 基础数学， 2011， 博士

【摘要】 受到欧氏空间与H-型群上类似问题的启发,首先,我们将关于具有紧支撑的光滑函数的Hardy-Sobolev不等式推广到径向导数的情形,并指出不等式中常数是最佳的.其次,我们将Heisenberg群上的Hardy-Rellich不等式推广到Canrot群上,也得到相应的最佳常数,并指出有关Grushin算子的类似不等式也是成立.NA群包含了一些秩为一的黎曼对称空间,如复双曲空间,四元数双曲空间与Cayley双曲空间.我们仿照秩为一的黎曼对称空间的情形,用球面变换和带状域上解析函数的谱综合的方法,证明了NA群上满足平均值性质可积函数为调和函数的一种充分条件.H-型群是重要的幂零黎曼流形,NA群是典型的可解流形；我们按照黎曼流形和乐群的Berger分类,分别给出了H-型群,NA群的和乐群.运用球面变换,我们证明了NA群上径向群代数的半单性.利用Banach代数的上同调和顺从性等概念,我们得到了H-型群,NA群上群代数,径向群代数的顺从性及上同调群的性质.更多还原

【Abstract】 Having in mind similar problems in the Euclidean and H-type groups setting, first we generalized Hardy-Sobolev inequalities from smooth functions with compact support to radial derivatives in Euclidean spaces, and point out that the constants in such Hardy-Sobolev inequalities are the best. Secondly, we prove Hardy-Rellich inequalities on Canrot groups as the same as on Heisenberg groups, and the constants in these inequalities are sharp. We show the analogue inequalities for Grushin operators also.NA groups include some Riemannian symmetric spaces with rank one as com-plex hyperbolic spaces, quaternionic hyperbolic spaces and Cayley hyperbolic space. We give a sufficient condition for integrable functions satisfy mean value property on NA groups to be harmonic with spherical transformations and spectral synthe-sis on analytic functions in strip.The method comes from the similar result about Riemannian symmetric spaces with rank one.H-type groups as nilpotent manifolds and NA groups as solvable manifolds are important Riemann manifolds. We obtain the holonomy groups of H-type groups and NA groups according to Berger classification of Riemannian holonomy groups. We use the spherical transformation to prove the semisimplity of radial group alge-bras on NA groups, and show these algebras are amenable.更多还原