On the sharpness of some quantitative Muckenhoupt-Wheeden inequalities.

Abstract

. In the recent work [Cruz-Uribe et al. (2021)] it was obtained that |{x ∈ R d : w(x)|G(f w−1 )(x)| > α}| ≲ [w] 2 A1 α Z Rd |f |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]A1 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.