  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.

Classical Analysis and ODEs