  We study mixed weighted weak-type inequalities for families of functions,
which can be applied to study classical operators in harmonic analysis. Our
main theorem extends the key result from D. Cruz-Uribe, J.M. Martell and C.
Perez, Weighted weak-type inequalities and a conjecture of Sawyer, Int. Math.
Res. Not., V. 30, 2005, 1849-1871.
