2021/08/31 by Borovitskiy, Viacheslav
#42B20 #42B25 #42B30 #42C10 #60G48 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2108.13891
A version of Littlewood-Paley-Rubio de Francia inequality for bounded multi-parameter Vilenkin systems is proved: for any family of disjoint sets Ik = Ik1 × … × IkD ⊆ ℤ+D such that Ikd are intervals in ℤ+ and a family of functions fk with Vilenkin-Fourier spectrum inside Ik the following holds: ‖∑k fk‖Lp ≤ C ‖(∑k |fk|2)1/2‖Lp , 1 lt; p ≤ 2, where C does not depend on the choice of rectangles \Ik\ or functions \fk\.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 ≤ 1 and an example of a one-parameter inequality for an exotic notion of the interval.