Pointwise ergodic theorems for radial averages on simple Lie groups II
Pointwise ergodic theorems for radial averages on simple Lie groups II
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简单李群 II 上径向平均的点态遍历定理
DOI：
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发表时间：
1994
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通讯作者：
A. Nevo
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作者：
A. Nevo
1.结果陈述、证明方法和一些备注1.1.定义和结果声明。本文是[1]的继续，我们开始简要地回顾了[1]的结构和记法：G Gn SO（n，1）是n维真实的双曲空间H n，n > 2的保向等距群. K是一个固定的极大紧子群，mc-Haar概率测度。A {at]t JR}-双曲平移的单参数群，使得G KA+K是Cartan分解。在G上的双K不变概率测度由at mr *（at* mr，其中表示卷积。注意，or 0先生#t将f表示为ds，即ors的均匀平均值，0 < s < t。我们定义#0 mtc。M（G，K）=由o't，> 0生成的交换卷积代数（G上双K不变复有界Borel测度）。（X，，2）一个标准Borel空间，其Borel可测G-作用保持概率测度2。r（v）f（x）f6 f（o-lx）dr（g）是Lt ′（X）上对应于G上一个概率测度v的Markov算子. Muf（x）=supt> C（t）f（x）l和Mf（x）=supt>olrC（t）f（x）l，L ′（X）中与at和t有关的极大函数，1最后，再看下面的定义。
1. Statement of results, the method of proof, and some remarks 1.1. Definitions and statement of results. The present paper is a continuation of IN1], and we begin by briefly recalling the setup and the notation: G Gn SO(n, 1) is the group of orientation-preserving isometries of ndimensional real hyperbolic space Hn, n > 2. K a fixed maximal compact subgroup, mc Haar probability measure. A {at]t JR}-a one-parameter group of hyperbolic translations such that G KA+K is a Cartan decomposition. at the bi-K-invariant probability measure on G given by at mr * (at* mr, where denotes convolution. Note that or0 mr. #t lit f as ds, the uniform average of ors, 0 < s < t. We define #0 mtc. M(G,K)= the commutative convolution algebra (of bi-K-invariant complex bounded Borel measures on G) generated by o’t, > 0. (X, , 2) a standard Borel space with a Borel measurable G-action which preserves the probability measure 2. r(v)f(x) f6 f(o-lx)dr(g) the Markov operator on Lt’(X) corresponding to a probability measure v on G. Muf(x)=supt>oln(#t)f(x)l and Mf(x)=supt>olrC(trt)f(x)l, maximal functions associated with the action of at and #t in L’(X), 1 < p < oz. Finally, recall also the following definition.