We show that any Littlewood--Paley square function S satisfying a minimal Carleson condition is dominated by a sparse form. This implies strong weighted L^p estimates for all A_p weights with sharp dependence on the A_p characteristic. In particular, the Carleson condition and the sparse domination are equivalent. The proof uses random dyadic grids, decomposition in the Haar basis, and a stopping time argument.

The author was supported by the UK Engineering and Physical Sciences Research Council (EPSRC).