Two-parameter B-valued martingale Hardy–Lorentz–Karamata spaces: Inequalities and dualities

In this paper, we introduce five two-parameter $B$-valued martingale Hardy–Lorentz–Karamata spaces and establish some atomic decomposition theorems via atomic martingales as well as atomic functions. With the aid of atomic decompositions, we obtain some martingale inequalities and characterize the duals of these spaces. Our conclusions strongly depend on the geometric properties of the underlying Banach spaces and remove the restrictive condition that the slowly varying function $b$ is nondecreasing as in the previous literature.