Littlewood-Paley-Rubio de Francia inequality for multi-parameter Vilenkin systems

A version of Littlewood-Paley-Rubio de Francia inequality for bounded multi-parameter Vilenkin systems is proved: for any family of disjoint sets $I_k = I_k^1 \times \ldots \times I_k^D \subseteq {\mathbb{Z}_+^D}$ such that $I_k^d$ are intervals in $\mathbb{Z}_+$ and a family of functions $f_k$ with Vilenkin-Fourier spectrum inside $I_k$ the following holds: $$\Bigl\|\sum_k f_k\Bigr\|_{L^p} \leq C \Bigl\|\bigl(\sum_k |f_k|^2\bigr)^{1/2}\Bigr\|_{L^p} , \qquad 1 < p \leq 2,$$ where $C$ does not depend on the choice of rectangles $\{I_k\}$ or functions $\{f_k\}$.This result belongs to a line of studying of (multi-parameter) generalizations of Rubio de Francia inequality to locally compact abelian groups. The arguments are mainly based on the atomic theory of multi-parameter martingale Hardy spaces and, as a byproduct, yield an easy-to-use multi-parameter version of Gundy's theorem on the boundedness of operators taking martingales to measurable functions. Additionally, some extensions and corollaries of the main result are obtained, including a weaker version of the inequality for exponents $0 < p \leq 1$ and an example of a one-parameter inequality for an exotic notion of the interval.