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

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

2008-07-18

arxiv.org/abs/0807.2981v1

Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), no. 4, 819-832

Mathematics

Functional Analysis

To appear in The Royal Society of Edinburgh Proc. A (Mathematics)

Scientific paper

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.

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