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

Abstract

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

11 pages, 1 figure