Full torus LPR property for Schatten classes

Determine whether the Schatten class \(S^p(B(\ell^2))\) has the full torus Littlewood–Paley–Rubio de Francia property \(\operatorname{LPR}_{p}^{\mathbb T}\) for every \(2<p<\infty\).

Background

The Schatten class $S^p(B(\ell^2))$ is a central noncommutative example in the paper. The authors distinguish the full torus LPR property, involving arbitrary pairwise disjoint interval lengths, from the equal-length property.

The paper emphasizes that this question is a longstanding special case of the preceding general problem. Its relevance is heightened by the paper’s later counterexample showing that Schatten classes fail the equal-length canonical LPR property on arbitrary Vilenkin systems, despite the corresponding torus equal-length characterization involving UMD and type 2.