The Littlewood--Paley--Rubio de Francia property of a Banach space for   the case of equal intervals
  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. 
T. P. Hyt\"onen, J. L. Torrea, D. V. Yakubovich