Littlewood-Paley inequality for arbitrary rectangles in $\mathbb{R}^2$ for $0 < p \le 2$

The one-sided Littlewood-Paley inequality for pairwise disjoint rectangles in $\mathbb{R}^2$ is proved for the $L^p$-metric, $0 < p \le 2$. 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.