We show that any Littlewood--Paley square function $S$ satisfying a minimal local testing condition is dominated by a sparse form, \begin{ equation*} \langle (Sf)^2,g \rangle\le C \sum_{ I \in \mathscr{ S}} \langle \lvert f\rvert\rangle_I^2 \langle \lvert g\rvert\rangle_I \lvert I\rvert . \end{ equation*} This implies strong weighted $L^p$ estimates for all $A_p$ weights with sharp dependence on the $A_p$ characteristic. The proof uses random dyadic grids, decomposition in the Haar basis, and a stopping time argument.
Categories: math.CA