On the vector-valued Littlewood-Paley-Rubio de Francia inequality

TL;DR

This paper investigates Banach spaces with the Littlewood-Paley-Rubio de Francia property, establishing conditions under which Banach lattices possess this property and how it extends across different p-values.

Contribution

It characterizes Banach lattices with the LPR_p property via 2-concavification and demonstrates the extension of LPR_q to LPR_p for q  p.

Findings

Banach lattices with 2-concavification as UMD have LPR_p.

LPR_q implies LPR_p for q  p.

The property extends to larger p-values within the specified class.

Abstract

The paper studies Banach spaces satisfying the Littlewood-Paley-Rubio de Francia property LPR_p, 2 \leq p < \infty. The paper shows that every Banach lattice whose 2-concavification is a UMD Banach lattice has this property. The paper also shows that every space having LPR_q also has LPR_p with q \leq p < \infty.