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 are intervals in and a family of functions with Vilenkin–Fourier spectrum inside the following holds: where C does not depend on the choice of rectangles or functions . 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 and an example of a one‐parameter inequality for an exotic notion of interval.