Strong weighted and restricted weak weighted estimates of the square function

Abstract: 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}}$.