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.
Categories: math.CA math.FA