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

We show that if $v\in A_{\mathrm{\infty }}$ and $u\in A_1$, then there is a constant $c$ depending on the $A_1$ constant of $u$ and the $A_{\mathrm{\infty }}$ constant of $v$ such that $$\parallel \frac{T(fv)}{v}{\parallel }_{L^{1,\mathrm{\infty }}(uv)}\le c\mathrm{\parallel }f{\mathrm{\parallel }}_{L^1(uv)},$$ where $T$ can be the Hardy-Littlewood maximal function or any Calder'on-Zygmund operator. This result was conjectured in [IMRN, (30)2005, 1849–1871] and constitutes the most singular case of some extensions of several problems proposed by E. Sawyer and Muckenhoupt and Wheeden. We also improve and extends several quantitative estimates.