January 17, 2004

Guy Cohen

Title: Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory

Abstract:

We prove extensions of Menchoff's inequality and the Menchoff-Rademacher theorem for sequences $ \{f_n\} \subset L_p$, based on the size of the norms of sums of sub-blocks of the first $ n$ functions.

The results are applied to the study of a.e. convergence of series $ \displaystyle{ \sum_n \frac{a_n T^ng}{n^\alpha} }$ when $ T$ is an $ L_2$-contraction, $ g \in L_2$, and $ \{a_n\}$ is an appropriate sequence.

Given a sequence $ \{f_n\} \subset L_p(\Omega,\mu)$ of independent symmetric random variables, we study conditions for the existence of a set of $ x$ of $ \mu$-probability 1, such that for every contraction $ T$ on $ L_2({\mathcal Y},\pi)$ and $ g \in L_2(\pi)$, the random power series $ \sum_n f_n(x)T^n g$ converges $ \pi$-a.e., and for every $ T$ Dunford-Schwartz on $ L_1({\mathcal Y},\pi)$ of a probability space, the series $ \displaystyle{ \sum_n \frac{f_n(x) T^ng}{n^{\alpha_s}} }$ converges $ \pi$-a.e. for $ g \in L_s(\pi)$, $ 1<s < 2$ (where $ \alpha_s <1$ is a constant which does not depend on $ \{f_n\}$).

This is a joint work with M. Lin.