度量测度空间上的模与容量等式  

Modulus and Capacity Equalities in Metric Measure Spaces

作　　者：吴炯圻[1] 

机构地区：[1]漳州师范学院数学系,福建漳州363000

出　　处：《厦门大学学报（自然科学版）》2004年第6期757-761,共5页Journal of Xiamen University：Natural Science

基　　金：国家自然科学基金(19871068);福建省自然科学基金(K00018)资助

摘　　要：假定X是Heinonen和Koskela意义下的非紧致度量测度空间,X的闭子集F和紧致子集E不相交.本文在X的紧致化X 上建立了p 模与p 容量的等式,并得到了使得X上的等式cappC(E,F,X)=modp(E,F,X)成立的新的充分条件,其中右边的(E,F,X)是X上连接E和F的曲线.这给出了Heinonen和Koskela提出的有关等式成立条件的公开问题的部分回答,加深了此前的相关研究.Suppose that X is a non-compact metric measure space,E and F are two disjoint closed subsets of X and E is also compact and that X is proper and φ-convex. At first,by means of the Alexandroff compactification X~* of X we obtain a sufficient condition P for the equalities of p-modulus and p-capacity with continuous test-functions on X~*. Then we consider the p-capacity with continuous test-functions on X and prove that if condition P is satisfied and mod_p(Γ_∞) = 0,then cap_(pC)(E,F,X)=mod_p(E,F,X)(1) where Γ_∞ is the family of all locally rectifiable curves in X\F each of which starts form any point of E and tends to ∞.If,moreover,X is locally quasiconvex,(1) holds also for p-capacity with locally Lipschitz test-functions on the left-side. Our conclusion (1) answers partly to the open problem on p-modulus and p-capacity posed by Heinonen and Koskela and improves some results obtained by the other authors.

关 键 词：测度空间 等式 闭子集 非紧致度 公开问题 紧致子集 充分条件 容量 量测 度量 

分 类 号：N945.12[自然科学总论—系统科学] O178[理学—数学]