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

Abstract

A version of Littlewood-Paley-Rubio de Francia inequality for bounded multi-parameter Vilenkin systems is proved: for any family of disjoint sets+ such that I d k are intervals in Z+ and a family of functions f k with Vilenkin-Fourier spectrum inside I k the following holds:where C does not depend on the choice of rectangles {I k } or functions {f k }.The arguments are based on the atomic theory of multi-parameter martingale Hardy spaces. In the course of the proof, a multi-parameter version of Gundy's theorem on the boundedness of operators taking martingales to measurable functions is formulated. This theorem might be of independent interest outside the scope of the paper. Some extensions are obtained, including a weaker version of the inequality for exponents 0 < p ≤ 1 and a one-parameter inequality for an exotic notion of the interval.