We consider the Rubio de Francia's Littlewood--Paley square function
associated with an arbitrary family of intervals in R with finite
overlapping. Quantitative weighted estimates are obtained for this operator.
The linear dependence on the characteristic of the weight [w]Ap/2ββ turns
out to be sharp for 3β€p<β, whereas the sharpness in the range 2<p<3
remains as an open question. Weighted weak-type estimates in the endpoint p=2
are also provided. The results arise as a consequence of a sparse domination
shown for these operators, obtained by suitably adapting the ideas coming from
Benea (2015) and Culiuc et al. (2016).Comment: 18 pages. Revised versio