On the sharpness of some quantitative Muckenhoupt-Wheeden inequalities

In a recent work by Cruz-Uribe et al. was obtained that $$\mathrm{\mid }\{x\in {\mathbb{R}}^d:w(x)\mathrm{\mid }G(f{w}^{-1})(x)\mathrm{\mid }>\alpha \}\mathrm{\mid }\lesssim \frac{[w{]}_{A_1}^2}{\alpha }{\int }_{{\mathbb{R}}^d}\mathrm{\mid }f\mathrm{\mid }dx$$ both in the matrix and scalar settings, where $G$ is either the Hardy-Littlewood maximal function or any Calderón-Zygmund operator. In this note we show that the quadratic dependence on $[w{]}_{A_1}$ is sharp. This is done by constructing a sequence of scalar-valued weights with blowing up characteristics so that the corresponding bounds for the Hilbert transform and maximal function are exactly quadratic.