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

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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.