  In this note we give a sharp weighted estimate for square function from
$L^2(w)$ to $L^2(w)$, $w\in A_2$. This has been known. But we also give a
sharpening of this weighted estimate in the spirit of $T1$-type testing
conditions. Finally we show that for any weight $w\in A^d_2$ and any
characteristic function of a measurable set $\|S_w\chi_E\|_{L^{2,
\infty}(w^{-1})} \le C \sqrt{[w]_{A^d_2}}\, \|\chi_E\|_w$, and this estimate is
sharp. So on characteristic functions of measurable sets at least, no
logarithmic correction is needed for the weak type of the dyadic square
function.The sharp estimate for the restricted weak type is at most $
\sqrt{[w]_{A^d_2}}$.
