Relative weak compactness in infinite-dimensional Fefferman-Meyer duality

Abstract

Let E be a Banach space such that $E^{\prime }$ has the Radon-Nikodým property. The aim of this work is to connect relative weak compactness in the E-valued martingale Hardy space $H^1(\mu ,E)$ to a convex compactness criterion in a weaker topology, such as the topology of uniform convergence on compacts in measure. These results represent a dynamic version of the deep result of Diestel, Ruess, and Schachermayer on relative weak compactness in $L^1(\mu ,E)$. In the reflexive case, we obtain a Kadec-Pełczyński dichotomy for $H^1(\mu ,E)$-bounded sequences, which decomposes a subsequence into a relatively weakly compact part, a pointwise weakly convexly convergent part, and a null part converging to zero uniformly on compacts in measure. As a corollary, we investigate a parameterized version of the vector-valued Komlós theorem without the assumption of $H^1(\mu ,E)$-boundedness.