In this talk, I will introduce the Sawyer's conjecture,
which was proposed by Sawyer in 1980s. Then in 2005, Cruz-Uribe,
Martell and Pérez finally solved this conjecture, they also proposed
a new conjecture, which can be viewed as the most singular case of
several extensions of Sawyer's conjecture. I will give a positive
answer to the new conjecture. Several quantitative estimates are also
obtained. In the end, some open questions related to one-sided
maximal functions (proposed by Martín Reyes) will be introduced. This
talk is based on joint work with Sheldy Ombrosi and Carlos Pérez.
