Littlewood-Paley-Rubio de Francia inequality for multi-parameter Vilenkin systems. (English) Zbl 07820034
Summary: A version of Littlewood-Paley-Rubio de Francia inequality for bounded multi-parameter Vilenkin systems is proved: For any family of disjoint sets \(I_k = I_k^1 \times \cdots \times I_k^D \subseteq\mathbb{Z}_+^D\) 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:
\[
\left\Vert \sum\nolimits_k f_k \right\Vert_{L^p} \leq C \left\Vert \sum\nolimits_k |f_k|^{2^{1/2}} \right\Vert_{L^p}, \qquad 1 < p \leq 2,
\]
where \(C\) does not depend on the choice of rectangles \(\left\{ I_k \right\}\) or functions \(\left\{ f_k \right\}\). 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 \(0 < p \leq 1\) and an example of a one-parameter inequality for an exotic notion of interval.
© 2023 Wiley-VCH GmbH.
© 2023 Wiley-VCH GmbH.
Keywords:
Gundy’s theorem; Hardy space; Littlewood-Paley inequality; martingale; multi-parameter; Rubio de Francia inequality; singular integral operator; Vilenkin systemSoftware:
MathOverflowReferences:
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