Sufficiency of UMD and type 2 for the full torus LPR property

Establish whether every Banach space that is UMD and has type 2 satisfies the full torus Littlewood–Paley–Rubio de Francia property \(\operatorname{LPR}_{p}^{\mathbb T}\) for every \(2\le p<\infty\), where the property permits pairwise disjoint intervals of arbitrary lengths.

Background

The paper recalls that Hytönen, Torrea, and Yakubovich characterized the equal-length torus LPR property: for every 2≤p<∞2\le p<\infty, a Banach space has LPR⁡p,=T\operatorname{LPR}_{p,=}^{\mathbb T} if and only if it is UMD and has type 2. The full property LPR⁡pT\operatorname{LPR}_{p}^{\mathbb T} is stronger because it allows pairwise disjoint frequency intervals of arbitrary, rather than equal, lengths.

The unresolved issue is whether the geometric conditions that completely characterize the equal-length theory also suffice for the unrestricted-length theory. The paper presents this as an open problem and notes that positive results are known for certain Banach lattices under additional lattice-theoretic hypotheses.

References

It remains open whether the UMD property together with type~2 is sufficient for the full \operatorname{LPR}_{p}{\mathbb T} property, which allows disjoint intervals of arbitrary lengths.

— Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems  (2609.34689 - Chen et al., 28 Sep 2026) in Section 1, subsection “Littlewood–Paley–Rubio de Francia inequalities on the torus,” paragraph following the discussion of \(\operatorname{LPR}_{p,=}^{\mathbb T}\)

Beyond this Banach-lattice result for the Walsh system, the \operatorname{LPR}_p property for general Vilenkin systems and broader classes of Banach spaces remains largely open.

— Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems  (2609.34689 - Chen et al., 28 Sep 2026) in Section 1, subsection “Littlewood–Paley–Rubio de Francia inequalities for Vilenkin systems,” final paragraph before “Further Motivations and Aims”