Embeddings between weak Orlicz martingale spaces

Let Φ   be an increasing and convex function on [0,∞)[0,∞) with Φ(0)=0Φ(0)=0 satisfying that for any α>0α>0, there exists a positive constant CαCα such that Φ(αt)⩽CαΦ(t)Φ(αt)⩽CαΦ(t), t>0t>0. Let wLΦwLΦ denote the corresponding weak Orlicz space. We obtain some embeddings between vector-valued weak Orlicz martingale spaces by establishing the wLΦwLΦ-inequalities for martingale transform operators with operator-valued multiplying sequences. These embeddings are closely related to the geometric properties of the underlying Banach space. In particular, for any scalar valued martingale f=(fn)n⩾1f=(fn)n⩾1, we claim that‖supn|fn|‖wLΦ≈‖(∑n=1∞|dfn|2)1/2‖wLΦ, where dfn=fn−fn−1dfn=fn−fn−1 and “≈” only depends on Φ.