Some notes on the vector-valued extension of Littlewood--Paley--Rubio de Francia inequality for Walsh functions

J. L. Rubio de Francia proved the one-sided Littlewood–Paley inequality for arbitrary intervals in $L^p$, $2\le p<\mathrm{\infty }$ and later N. N. Osipov proved the similar inequality for Walsh functions. In this paper we investigate some properties of Banach spaces $X$ such that the latter inequality holds for $X$-valued functions.