VARIATION IN PROBABILITY, ERGODIC THEORY AND ANALYSIS

摘要

In many areas of analysis, ergodic theory and probability, square functions have proven to be one of the most useful tools to study convergence properties. (See the paper by Stein for a very informative historical discussion of the importance of various square functions in several areas.) For example, the martingale square function was used by Burkholder, Gundy and Silverstein to give the first real variable characterization of H_p. An ergodic square function was used by Bourgain in his proof that the ergodic averages along the sequence of squares converge a.e. In this paper we consider operators that are closely related to the square functions, but have very different properties.