  We show that if $v\in A_\infty$ and $u\in A_1$, then there is a constant $c$
depending on the $A_1$ constant of $u$ and the $A_{\infty}$ constant of $v$
such that $$\Big\|\frac{ T(fv)} {v}\Big\|_{L^{1,\infty}(uv)}\le c\,
\|f\|_{L^1(uv)},$$ 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.
