The Littlewood--Paley--Rubio de Francia property of a Banach space for the case of equal intervals

TL;DR

This paper characterizes UMD Banach spaces of type 2 through a Littlewood--Paley--Rubio de Francia property involving equal intervals for Fourier series and integrals, establishing a key equivalence.

Contribution

It provides a new characterization of UMD spaces of type 2 via a Littlewood--Paley--Rubio de Francia property for equal intervals in Fourier analysis.

Findings

The property holds for all disjoint equal-length intervals and p ≥ 2 if and only if X is UMD of type 2.

The criterion extends from Fourier series to Fourier integrals on the real line.

The paper establishes an equivalence between a square function estimate and the UMD property.

Abstract

Let $X$ be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of $X$-valued functions holds true for all disjoint collections of subintervals of the set of integers of equal length and for all exponents $p$ greater or equal than 2 if and only if the space $X$ is a UMD space of type 2. The same criterion is obtained for the case of subintervals of the real line and Fourier integrals instead of Fourier series.