Analogs of Wiener's ergodic theorems for semisimple Lie groups. II
Analogs of Wiener's ergodic theorems for semisimple Lie groups. II
复制标题
半单李群的维纳遍历定理的类似物。
DOI：
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发表时间：
2000
期刊：
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通讯作者：
E. Stein
中科院分区：
文献类型：
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作者：
G. Margulis;A. Nevo;E. Stein
设G是真实的秩为1的连通有限中心单李群，K是G的极大紧子群.设A = {atItCR}是双曲平移的单参数子群，使得G = KA+K是Cartan分解.用ot表示G上的概率测度，由ot = mK * 6at * MK给出，其中MK是K上的归一化Haar测度。设It表示密度vol（Bt）>1XB，（g）的概率测度，其中Bt = {g：d（gK，K）< t}，d表示G/K上的标准G不变黎曼度量.设（X，B，A）是标准Borel概率空间，G通过保测度变换作用于其上，且G是可测遍历的.令7 r：G -* Iso（LP（X））是相关联的等距表示，并且令7 r（ut）和7 r（！3 t）表示与Ct和It正则相关的算子，设n（G）= dimRG/K.本文证明了类于N的单实秩一李群上的球平均It的极大遍历定理和点态遍历定理。中球平均的维纳极大遍历定理和点态遍历定理。(The半单群上球平均的一般情况在[M-N-S]中完成。我们的主要结果如下：
Let G be a connected finite-center simple Lie group of real rank one, and let K be a maximal compact subgroup of G. Let A = {at I t C R} be a 1-parameter subgroup of hyperbolic translations such that G = KA+K is a Cartan decomposition. Denote by ot the probability measure on G given by ot = mK * 6at * MK, where MK is the normalized Haar measure on K. Let It denote the probablity measure with density vol(Bt)>1XB,(g), where Bt = {g: d(gK, K) < t} and d denotes the canonical G-invariant Riemannian metric on G/K. Let (X, B, A) be a standard Borel probability space on which G acts measurably and ergodically by measure preserving transformations. Let 7r: G -* Iso(LP(X)) be the associated isometric representation, and let 7r(ut) and 7r(!3t) denote the operators canonically associated with Ct and It. Let n(G) = dimR G/K. We prove maximal and pointwise ergodic theorems for the ball averages It on simple real-rank one Lie groups which are analogous to N. Wiener's maximal and pointwise ergodic theorems for ball averages on in. (The general case of ball averages on semisimple groups is completed in [M-N-S].) Our main result is the following: