论文标题
关于多种千古均值的收敛性
On the convergence of multiple ergodic means
论文作者
论文摘要
给定度量序列保存转换$ \ {u_k：\，k = 1,2，\ ldots，n \} $在可测量的空间$（x，μ）$上。我们证明了A.E.沿贡献的融合 \ begin {equation} \ frac {1} {s_1 \ cdots s_ {n}} \ sum_ {j_1 = 0}^{s_1-1} \ cdots \ cdots \ sum_ {j_n = 0}^0}^{s_n-n-1} f \ lesg \ end {equation}作为$ \ min_j s_j \ to \ infty $，对于任何函数$ f \ in l \ log^{d-1}（x）$，其中$ d \ le n $是转换的等级。结果给出了N. Dunford和A. Zygmund对定理的概括，声称在较窄的函数类别$ l \ log^{n-1}（x）$中的均值收敛。
Given sequence of measure preserving transformations $\{U_k:\,k=1,2,\ldots, n\}$ on a measurable space $(X,μ)$. We prove a.e. convergence of the ergodic means \begin{equation} \frac{1}{s_1\cdots s_{n}}\sum_{j_1=0}^{s_1-1}\cdots\sum_{j_n=0}^{s_n-1}f\left(U_1^{j_1}\cdots U_n^{j_n} x \right) \end{equation} as $\min_j s_j\to\infty $, for any function $f\in L\log^{d-1}(X)$, where $d\le n$ is the rank of the transformations. The result gives a generalization of a theorem by N. Dunford and A. Zygmund, claiming the convergence of the means in a narrower class of functions $L\log^{n-1}(X)$.
