Relative weak compactness in infinite-dimensional Fefferman-Meyer duality

TL;DR

This paper explores the relationship between relative weak compactness in vector-valued martingale Hardy spaces and convex compactness criteria, extending classical results and providing new decompositions and convergence theorems.

Contribution

It establishes a dynamic convex compactness criterion in martingale Hardy spaces and proves a Kadec-Pe{ }lczy{\u}nski dichotomy for bounded sequences, extending known results to a broader setting.

Findings

Connected weak compactness in $H^{1}(1),E$ to convex compactness in weaker topologies.

Proved a Kadec-Pe{ }lczy{\u}nski dichotomy for $H^{1}(1),E$-bounded sequences.

Investigated a parameterized vector-valued KomlF3s theorem without boundedness assumption.

Abstract

Let $E$ be a Banach space such that $E'$ has the Radon-Nikod\'ym 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-Pe{\l}czy\'nski 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 Koml\'os theorem without the assumption of $H^{1}(\mu,E)$-boundedness.