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

A version of Littlewood-Paley-Rubio de Francia inequality for bounded multiparameter Vilenkin systems is proved: For any family of disjoint sets I-k =I-k(1) x ... x I-k(D) subset of Z(+)(D) such that I-k(d) are intervals in Z(+) and a family of functions f(k) with Vilenkin-Fourier spectrum inside I(k)t he following holds: parallel to Sigma(k)f(k)parallel to(Lp) <= C parallel to Sigma(k vertical bar)f(k)vertical bar 2(1/2)parallel to(LP), 1 < p <= 2, where C does not depend on the choice of rectangles {I-k} or functions {f(k)}. This result belongs to the 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 <= 1 and an example of a one-parameter inequality for an exotic notion of interval.