Embeddings between weak Orlicz martingale spaces

Let Φ be an increasing and convex function on [ 0 , ∞ ) with Φ ( 0 ) = 0 satisfying that for any α > 0 , there exists a positive constant C α such that Φ ( α t ) ⩽ C α Φ ( t ) , t > 0 . Let w L Φ denote the corresponding weak Orlicz space. We obtain some embeddings between vector-valued weak Orlicz martingale spaces by establishing the w L Φ -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 = ( f n ) n ⩾ 1 , we claim that ‖ sup n | f n | ‖ w L Φ ≈ ‖ ( ∑ n = 1 ∞ | d f n | 2 ) 1 / 2 ‖ w L Φ , where d f n = f n − f n − 1 and “≈” only depends on Φ.