A Littlewood-Paley-Rubio de Francia inequality for bounded Vilenkin systems

Abstract

Rubio de Francia proved a one-sided Littlewood-Paley inequality for the square function constructed from an arbitrary system of disjoint intervals. Later, Osipov proved a similar inequality for Walsh systems. We prove a similar inequality for more general Vilenkin systems. Bibliography: 11 titles.

Littlewood–Paley–Rubio de Francia inequality for multi‐parameter Vilenkin systems

“…As for the alternative Fourier transforms, we mention the paper [15] of Osipov, which addressed the version of inequality (3) for the one-parameter Walsh-Fourier series, and the works of Tselishchev [16] and the author [17], which are dedicated to the one-parameter Vilenkin case and the two-parameter Walsh case, respectively, and which we build upon and generalize in this work. 1 It is important to note that we only consider bounded Vilenkin systems in this paper, with proofs heavily relying on the boundedness.…”

“…Questions related to unbounded Vilenkin systems are also studied in the literature (see, e.g., [2,19]) with techniques and results differing considerably. We refer the reader to [20]-where a version of the Rubio de Francia inequality has recently been proven for unbounded one-parameter Vilenkin systems-for a relevant discussion.…”

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

“…Operators E n and ∆ n can be extended to act on martingales by simply putting E n g = g n and treating (19) as the definition of ∆ n g. Obviously, we have E n g = m≤n ∆ m g both for functions and martingales.…”

“…Here we will describe a method of dividing an interval [a, b) ⊆ Z + of integers into subintervals that behave favorably under shifts induced by the operation + defined in Section 2.1. This method, suggested by Tselishchev in [19], is a key combinatorial component of the proof of the main theorem.…”

“…As for the alternative Fourier transforms, we mention the paper [13] of Osipov, which addressed the version of inequality (3) for the one-parameter Walsh-Fourier series, and the works of Tselishchev [19] and the author [1], which are dedicated to the one-parameter Vilenkin case and the two-parameter Walsh case respectively, and which we build upon and generalize in this work.…”