In this paper we provide some quantitative mixed-type estimates assuming conditions that imply that uv ∈ A∞ for Calderón-Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in [8] and extended in [23] and [25]Sawyer also conjectured that (1.1) should hold as well for the Hilbert transform. Cruz-Uribe, Martell and Pérez [7] generalized (1.1) to higher dimensions and actually proved that Sawyer's conjecture holds for Calderón-Zygmund operators via the following extrapolation argument.Theorem C. Assume that for every w ∈ A ∞ and some 0 < p < ∞,Then for every u ∈ A 1 and everyThe conditions on the weights in that extrapolation result lead them to conjecture that (1.1), and consequently the corresponding estimate for Calderón-Zygmund operators should hold as well with u ∈ A 1 and v ∈ A ∞ . That conjecture was positively answered recently in [24] where several quantitative estimates were provided as well. At this point we would like to mention, as well, a recent generalization provided for Orlicz maximal operators in [2].In [7], besides the aforementioned results, it was shown that (1.1) holds if u ∈ A 1 and v ∈ A ∞ (u) (see Section 2.2 for the precise definition of A p (u)). The advantage of that condition is that the product uv is an A ∞ weight. Over the past few years, there have been new contributions under those assumptions such as [3] for the case of fractional integrals and related operators, [28,29] for related quantitative estimates and [26] for multilinear extensions.

Abstract. We show that the Hilbert transform does not mapfor every Young function Φ growing more slowly than t log log(e e + t). Our proof is based on a construction of M.C. Reguera and C. Thiele.

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