Φ-moment B-valued martingale inequalities on Lorentz-Karamata spaces

In this article, several Φ-moment Banach spaces valued (briefly by B-valued) martingale in-equalities on Lorentz-Karamata spaces are established by the tool of atomic decompositions, which are new versions of the basic inequalities in B-valued martingale setting associated with concave functions Φ. It is to be mentioned that the results obtained herein are closely connected with the geometric prop-erties of the underlying Banach spaces. In particular, we present several novel characterizations of the geometric properties of Banach spaces by using the Φ-moment B-valued martingale inequalities in the context of Lorentz-Karamata spaces. Our conclusions obtained here generalize the previous conclusions for B-valued martingale inequalities. Moreover, we remove the condition that the slowly varying function b is nondecreasing in [Bull. Malays. Math. Sci. Soc., 2019, 42(5): 2395-2422].