Abstract 9705228 Rosenblatt This proposal is for research in a number of new directions in harmonic analysis and ergodic theory. The main principle at work here is that there are fundamental connections between classes of operators arising in harmonic analysis, ergodic theory, and probability theory. For example, there is a very strong connection between the behavior of martingales and the averages in ergodic theory. This is clear where maximal functions are concerned, but it also is an important principle in regards to upcrossings of both processes. Using estimates on the degree to which ergodic averages are a perturbation of suitable martingales, precise estimates on oscillation of ergodic sums can be analyzed. There has been a good deal of success in this direction in the work of Jones, Kaufman, Rosenblatt, and Wierdl, but additional studies of these properties are much needed . In particular, it is proposed here to show that the best possible jump and upcrossing inequalities in ergodic theory are the ones that can be obtained by relating the ergodic averages to martingales. In this same spirit, the variational behavior of ergodic averages is linked closely to the behavior of martingales. In recent work in ergodic theory, the importance of square functions for measuring the oscillation of general averages of ergodic transformations has become very clear. This work has also led to obtaining estimates on the square function of the difference between ergodic sums and martingales. Such analysis is yet in progress.However, the analysis of square functions of moving averages and of related operator sequences is not yet at all complete. Also, other aspects of differences in ergodic theory, like unconditionality, and how they mimic the behavior of martingale difference sequences, is well worth more investigation. This grant is to fund the study of the long term behavior of dynamical systems and stochastic processes. Many important physical phenomena, as varied as the beha vior of the stock market, lines in the supermarket, and data being received from a distant transmission source, exhibit locally erratic behavior which can be understood better by looking at the long term average behavior instead. The study of such stochastic processes and their connections with one another, as well as their connections in terms of their fine structure with other less random processes, is a central one in modern mathematics. It is only through such studies that we will be able to understand better and thus be able to control or predict the evolution of the physical phenomena which this mathematics can model.
