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

Abstract

In this article, several $\Phi$-moment Banach spaces valued (briefly by \textbf{B}-valued) martingale inequalities on Lorentz-Karamata spaces are established by the tool of atomic decompositions, which are new versions of the basic inequalities in \textbf{B}-valued martingale setting associated with concave functions $\Phi$.  It is to be mentioned that the results  obtained herein are closely connected with the geometric properties of the underlying Banach spaces. In particular, we present several novel characterizations of the geometric properties of Banach spaces by using the $\Phi$-moment $\mathbf{B}$-valued martingale inequalities in the context of Lorentz-Karamata spaces.   Our conclusions obtained here generalize  the previous conclusions for $\mathbf{B}$-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].