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

By Kangwei Li, S. Ombrosi, C. P'erez. Mathematische Annalen, 2017

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We show that if $$v\in A_\infty $$v∈A∞ and $$u\in A_1$$u∈A1, then there is a constant c depending on the $$A_1$$A1 constant of u and the $$A_{\infty }$$A∞ constant of v such that $$\begin{aligned} \left\| \frac{ T(fv)}{v}\right\| _{L^{1,\infty }(uv)}\le c\, \Vert f\Vert _{L^1(uv)}, \end{aligned}$$T(fv)vL1,∞(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.

Subfields: Applied Mathematics

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In Applied Mathematics

Citations in the corpus

Weighted Norm Inequalities for Rough Singular Integral Operators (2017, background): “12) holds and it is the following (see [40]).”