Mathematical Sciences: Oscillation Inequalities, Almost Everywhere Convergence, and Related Questions

这一提议涉及到继续研究遍历理论中与一系列平均算子相关的平方函数和其他振荡测度。在与点遍历定理有关的a.e.收敛问题的理论中，振荡不等式已经发挥了关键作用。特别是，振荡算子给出了关于上交叉和跳跃不等式的新信息。我们希望在单一变换的情况下得到新的结果，并将一些结果推广到多参数的情况。我们还继续搜索子序列上的条件，在单参数和多参数情况下，这将意味着相关平均值的收敛。用于研究提案中问题的技术通常来自谐波分析，包括卡尔德龙-齐格蒙德方法和Littlewood-Paley理论。此外，来自概率论的某些思想，特别是鞅理论，有望发挥关键作用。经典遍历理论是对动力系统长期行为的研究。例如，我们可能对供水系统中污染物的分布，以及污染物的分布如何随时间变化感兴趣。遍历定理可以看作是强大数定律的推广。如果我们考虑抛一枚均匀硬币时观察到的正面和反面的序列，就会出现强大数定律的一个特例。在这种情况下，我们有独立的试验，我们通过观察方差来测量平均值序列的振荡。振荡的测量很重要，因为它提供了一种估计特定平均值与真实平均值（在实验情况下是未知的）接近程度的方法。在民意调查中，我们通常把这种衡量我们之间差距的方法称为误差幅度。在本建议所考虑的更一般的情况下，这些试验不是独立的。例如，污染物在某一时刻的分布肯定会对以后短时间内的分布起决定作用。这与投掷均匀硬币的情况相反，在投掷均匀硬币的情况下，我们在某一次投掷中看到正面的事实并没有告诉我们在下一次投掷中是否会看到正面。这一建议涉及到对遍历平均振荡的研究，以及对诸如平均值上升到某个临界阈值以上的频率等问题的测量。在实验情况下，由于设备故障或人为错误，我们经常会发现一些预期的数据丢失。在早期的工作中已经表明，在某些情况下，这会导致非常具有误导性的结果，而在其他情况下，我们仍然可以恢复正确的结果。这方面的工作还很不完整。我们希望能够进一步分类，在缺乏实验数据的情况下，实验仍然可以抢救的情况。在这些情况下，测量振荡或误差范围尤为关键。

ABSTRACT Jones This proposal involves the continued study of square functions and other measures of the oscillation associated with a sequence of averaging operators in ergodic theory. Oscillation inequalities have already played a key role in the theory of a.e. convergence problems arising in connection with the pointwise ergodic theorem. In particular, oscillation operators give new information about upcrossing and jump inequalities. We hope to obtain new results in the case of a single transformation, and to extend several results to the multi-parameter case. We also continue the search for conditions on subsequences, both in the single and in the multiparameter case, which will imply a.e. convergence of the associated averages. The techniques used to study the questions in the proposal will often be from harmonic analysis, including Calderon-Zygmund methods and Littlewood-Paley theory. In addition, certain ideas from probability, especially martingale theory, can be expected to play a key role. Classical ergodic theory is the study of the long term behavior of dynamical systems. For example, we might be interested in the distribution of a pollutant in a water supply, and how the distribution of that pollutant changes over time. The ergodic theorem can be viewed as a generalization of the strong law of large numbers. A special case of the strong law of large numbers occurs if we consider the sequence of heads and tails observed if we toss a fair coin. In such a case we have independent trials, and we measure the oscillation of the sequence of averages by looking at the variance. A measure of oscillation is important because it provides a way of estimating how close a particular average is to the true (and in experimental situations, unknown) mean. In public opinion polls this measure of how close we are is usually referred to as the margin of error. In the more general setting, considered in this proposal, the trials are not independent. For example, the distribution o f a pollutant at one time will certainly play a role in determining the distribution a short time later. This is in contrast to the case of tossing a fair coin, where the fact that we saw heads on a certain toss gives us no information about whether or not we will see a head on the next toss. This proposal involves the study of the oscillation of ergodic averages, and measures of such things as how often the average values rise above some critical threshold. In experimental situations, because of equipment failure, or human error, we often find that some of the expected data is missing. In earlier work it has been shown that in some cases this leads to very misleading results, and in other cases we can still recover the correct result. The work in this area is very incomplete. We hope to be able to further classify the cases in which an experiment can still be salvaged, despite missing experimental data. In these cases, a measure of the oscillation, or margin of error, is especially critical.