Pointwise ergodic theorems for radial averages on the Heisenberg group
Pointwise ergodic theorems for radial averages on the Heisenberg group
复制标题
海森堡群径向平均的逐点遍历定理
DOI：
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
S. Thangavelu
中科院分区：
文献类型：
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作者：
A. Nevo;S. Thangavelu
LetH=Hn=Cn×R表示Heisenberg群，leσ r0表示球{(z, 0): |z|=r}上的归一化Lebesgue测度。设（X， B, m）是一个标准的Borel概率空间，在该空间上hx通过保测度变换可测遍历地作用，令π(σr)表示与σ ronlp (X)正则关联的算子。在Heisenberg群中，我们证明了径向平均σ的极大和点遍历定理。所得结果最适合于还原海森堡群的作用。证明方法是利用群的径向测度的Banach代数的谱理论及其特征的衰减估计，利用谱方法，特别是littlewood - paly - stein平方函数和解析插值建立极大不等式。
LetH=Hn=Cn×R denote the Heisenberg group, and letσrdenote the normalized Lebesgue measure on the sphere {(z, 0): |z|=r}. Let (X, B, m) be a standard Borel probability space on whichHacts measurably and ergodically by measure preserving transformations, and letπ(σr) denote the operator canonically associated withσronLp(X). We prove maximal and pointwise ergodic theorems inLp, for radial averagesσron the Heisenberg groupHn,n>1. The results are best possible for actions of the reduced Heisenberg group. The method of proof is to use the spectral theory of the Banach algebra of radial measures on the group and decay estimates for its characters to establish maximal inequalities using spectral methods, in particular Littlewood–Paley–Stein square-functions and analytic interpolation.