Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups
Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups
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半单 Kazhdan 群的谱传递和逐点遍历定理
DOI：
10.4310/mrl.1998.v5.n3.a5
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发表时间：
1998
影响因子：
1
通讯作者：
A. Nevo
中科院分区：
文献类型：
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作者：
A. Nevo
设G是一个中心有限且没有紧化因子的连通半简单李群，且（X，B, m）是一个具有σ-有限G不变测度m的G空间。对于G上的每一个概率测度μ，考虑算子π(μ): L2(X)→L2(X)，由π(μ)f =∫G π(G)fdμ(G)给出。利用M. Cowling [Co2]和R. Howe [H]（参见[H-T][Li][Mo][Oh]）的显式谱估计（“定量性质T”）得到了‖π0（μ）‖的显式估计，其中π0是在与g不变函数空间正交的空间上的表示，假设π0有谱间隙。特别是对于哈萨克团体的行动，规范估计是统一的，不依赖于行动。范数估计可以看作是一种谱传递原理，类似于可服从群的传递原理（参见[W][Ca] [C-W][Hz1][Co3][Co4]）。谱估计用于推导群上自然族平均的指数极大不等式，以及这些平均在Lp中的点遍历定理。均值向遍历均值的逐点收敛速度是指数级的，具有明确的速率。这种现象在双k不变测度的情况下已在[M-N-S]中建立，这里我们讨论非径向平均，它可以是绝对连续的，奇异的或离散的。其他一些讨论的主题是支持格点的平均值，几乎正交性，卷积范数的最佳估计和遍历均值的指数收敛率。1. 半简单哈萨克列群的谱传递原理
Let G be a connected semi-simple Lie group with finite center and no compact factors, and (X,B, m) a G-space with σ-finite G-invariant measure m. For each probability measure μ on G consider the operator π(μ) : L2(X) → L2(X), given by π(μ)f = ∫ G π(g)fdμ(g). The explicit spectral estimates (“quantitative property T”) of M. Cowling [Co2] and R. Howe [H] (see also [H-T][Li][Mo][Oh]) are used to obtain explicit estimates of ‖π0(μ)‖, where π0 is the representation on the space orthogonal to the space of G-invariant functions, provided π0 has a spectral gap. In particular, for actions of Kazhdan groups, the norm estimate is uniform and does not depend on the action. The norm estimates can be viewed as a spectral transfer principle, analogous to the transfer principle for amenable groups (see [W][Ca] [C-W][Hz1][Co3][Co4]). The spectral estimates are used to derive exponential-maximal inequalities for natural families of averages on the group, as well as pointwise ergodic theorems in Lp for these averages. The pointwise convergence of the averages to the ergodic mean is exponentially fast with an explicit rate. This phenomenon in the case of bi-K-invariant measures was established in [M-N-S], and here we discuss non-radial averages, which may be absolutely continuous, singular or discrete. Some other topics discussed are averages supported on lattice points, almost orthogonality, and best possible estimates of convolution norms and exponential rate of convergence to the ergodic mean. 1. The spectral transfer principle for semi-simple Kazhdan Lie groups 1.