Proof of an extension of E. Sawyer’s conjecture about weighted mixed weak-type estimates

Abstract

We show that if v∈ A∞ and u∈ A1, then there is a constant c depending on the A1 constant of u and the A∞ constant of v such that ∥T(fv)v∥L1,∞(uv)≤c‖f‖L1(uv),where T can be the Hardy–Littlewood maximal function or any Calderón–Zygmund operator. This result was conjectured in Cruz-Uribe et al. (Int Math Res Not 30:1849–1871, 2005) and constitutes the most singular case of some extensions of several problems proposed by Sawyer and Muckenhoupt and Wheeden. We also improve and extends several quantitative estimates.

We show that if v∈A∞ and u∈A1 , then there is a constant c depending on the A1 constant of u and the A∞ constant of v such that T ( f v) v L1,∞(uv) ≤ c f L1(uv), where T can be the Hardy–Littlewood maximal function or any Calderón–Zygmund operator. This result was conjectured in Cruz-Uribe et al. (Int Math Res Not 30:1849–1871, 2005) and constitutes the most singular case of some extensions of several problems proposed by Sawyer and Muckenhoupt and Wheeden. We also improve and extends several quantitative estimates.