Banach-valued weak martingale Hardy spaces on rearrangement-invariant quasi-Banach function lattice

Abstract

Let $B$ be a Banach space and $X$ a quasi-Banach function lattice. In this paper, we introduce the $B$-valued weak martingale Hardy spaces associated with $X$. We then establish the $\infty$-atomic characterizations of these $B$-valued weak martingale Hardy-type spaces. As an application, we obtain the boundedness of the $\sigma$-sublinear operators from $B$-valued weak martingale Hardy type spaces to weak quasi-Banach function spaces. Using this, we further establish the relationships between different Banach-valued weak martingale Hardy spaces.