Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems

Abstract: We investigate two natural formulations of equal-length Littlewood--Paley--Rubio de Francia (LPR) inequalities on arbitrary Vilenkin systems (\Gm). The canonical formulation uses canonical intervals in (\mathbb N) of equal cardinality, while Vilenkin's formulation uses translates of a common initial interval under the group operation on the dual of (\Gm). In this paper, our main results show that the definitions of these two types of intervals exhibit dramatically different properties. More precisely, for the canonical formulation, we construct a counterexample showing that, for every (2<p<\infty), the Schatten class (S^p) fails the equal-length LPR property, in sharp contrast to the corresponding result on the torus; for Vilenkin's formulation, we prove that, for every (2\le p<\infty), a Banach space (X) has the equal-length LPR property if and only if it is UMD and has type~(2). The sufficiency proof combines a transference argument from the torus to finite cyclic groups with an embedding of Vilenkin intervals into digital rectangular envelopes, yielding bounds independent of the generating sequence.