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 $I_k=I_k^1\times \dots \times I_k^D\subseteq {\mathbb{Z}}_{+}^D$ 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: $$\Vert \sum _k{f}_k{\Vert }_{L^p}\le C\Vert (\sum _k|f_k{|}^2{)}^{1/2}{\Vert }_{L^p},1<p\le 2,$$ 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 $0<p\le 1$ and an example of a one-parameter inequality for an exotic notion of the interval.

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