On the weighted weak-type constant for the dyadic square function

We present an explicit construction of examples showing that the estimate $\mathrm{\parallel }S(f){\mathrm{\parallel }}_{L^2(w)\to L^{2,\mathrm{\infty }}(w)}\lesssim ([w{]}_{A_2}\log (1+[w{]}_{A_2}){)}^{1\mathrm{/}2}$ for the dyadic square function is sharp in terms of the characteristic $[w{]}_{A_2}$.