时空遍历平均的精确Orlicz端点
Sharp Orlicz Endpoints for Spatial-Temporal Ergodic Averaging
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中文总结 AI 辅助
本文研究时空遍历平均复合的精确Orlicz端点，确定了不同时间序列下的端点结果，基于局部稳定性原理等方法证明上下界，回答了Young的两个相关问题。
中文摘要 AI 辅助
我们研究了时间遍历平均与收缩度量球上空间平均的复合，并确定了相应无限制联合极限的精确Orlicz端点。对于以$N\Lambda_q(N)$（$q\ge 0$）归一化的连续Birkhoff平均，其精确Orlicz端点为$L\log_{q+1}L$。特别地，$q=0$的普通情形仅在勒贝格微分性质下就得到了$L\log L$局部联合收敛定理，而$L^1$即使在欧氏区间上也不成立，这回答了Young的两个问题。对于所有$q\ge 1$，素数平均也满足相同的$L\log_{q+1}L$端点。对于任意时间序列，在相同的归一化$N\Lambda_q(N)$（$q\ge 1$）下，$L\log_qL$始终是充分的，且该端点在Orlicz意义下对于固定底指数序列$k^n$（$k\ge 2$）以及包括$n!$在内的具有多项式比例间隔的序列是精确的。\n正面结果基于一个局部稳定性原理：在球的勒贝格微分性质下，只要相关的时间极大函数存在$L^1$控制函数，逐点时间收敛就可以提升到局部联合极限。正则时间情形中额外的对数项来自于将受限对数极大估计从集合提升到一般函数。任意序列定理则采用了二进分解方法。下界通过局部$\infty$-扫除构造得到，其中多项式增长正则时间的下界通过加权局部扫除构造，多项式比例间隔序列的下界通过剩余类构造，固定底指数序列的下界通过数字构造。
英文摘要
We study the composition of temporal ergodic averaging with spatial averaging over shrinking metric balls, and determine sharp Orlicz endpoints for the corresponding unrestricted joint limit. For consecutive Birkhoff averages normalized by $NΛ_q(N)$ ($q\ge 0$), the sharp Orlicz endpoint is $L\log_{q+1}L$. In particular, the ordinary case $q=0$ yields an $L\log L$ local joint convergence theorem under the Lebesgue differentiation property alone, while $L^1$ fails even on the Euclidean interval, answering two questions of Young. The same $L\log_{q+1}L$ endpoint holds for prime averages for every $q\ge 1$. For arbitrary time sequences, $L\log_qL$ always suffices at the same normalization $NΛ_q(N)$ ($q\ge 1$), and this endpoint is sharp in the Orlicz sense for fixed-base exponential sequences $k^n$ ($k\ge 2$) and for sequences with polynomial ratio separation, including $n!$. The positive results rest on a local stability principle: under the ball Lebesgue differentiation property, pointwise temporal convergence lifts to the local joint limit whenever the associated temporal maximal function admits an $L^1$ majorant. The additional logarithm in the regular-time case comes from lifting restricted logarithmic maximal estimates from sets to general functions. The arbitrary-sequence theorem uses a dyadic decomposition instead. The lower bounds are local $\infty$-sweeping out constructions, obtained by weighted local sweeping out for polynomial-growth regular times, by residue constructions for polynomially ratio-separated sequences, and by digit constructions for fixed-base exponentials.
