Littlewood–Paley inequality for arbitrary rectangles in $\mathbb R^2$ for $0<p\le2$

The one-sided Littlewood–Paley inequality for pairwise disjoint rectangles in $\mathbb R^2$ is proved for the $L^p$-metric, $0<p\le2$. This result can be treated as an extension of Kislyakov and Parilov's result (they considered the one-dimensional situation) or as an extension of Journé's result (he considered disjoint parallelepipeds in $\mathbb R^n$ but his approach is only suitable for $p\in(1,2]$). We combine Kislyakov and Parilov's methods with methods “dual” to Journé's arguments.