estimates of k–functional for integrable functions with respect to a vector measure

IIn this paper, we study the Banach lattices Lp (m) and Lp(∥m∥) with respect to a vector measure We show that the most natural attempts to introduce new properties by considering some inequalities for k–functionals. On the other hand there are equation that lead to the generalization of Steffen’s inequality for Lp(∥m∥) spaces.