In this paper, we investigate several two-parameter weak Orlicz–Hardy martingale spaces generated by the $p$-convex and $q$-concave functions, and establish their atomic decomposition theorems. Using the atomic decomposition, we obtain a sufficient condition for the boundedness of a sublinear operator defined on the two-parameter weak Orlicz–Hardy martingale spaces. Furthermore, the dual spaces of the two-parameter weak Orlicz–Hardy martingale spaces are considered.

In this article, the embeddings of Banach space valued (briefly by $ {\bf{B}} $-valued) Lorentz-Karamata martingale spaces are established. Different from the previous approach, we employ the technique of martingale transforms defined by some operator sequences, which can often supply a clear and succinct method in the research of martingale theory and Harmonic analysis. It's worth mentioning that the results obtained here are connected closely with the geometrical properties of the Banach spaces in which the martingales take values, such as $ p $-uniform smoothness and $ q $-uniform convexity.