A lower bound for $A_p$ exponents for some weighted weak-type inequalities

We give a weak-type counterpart of the main result in an earlier work of the first author, E. Rela and T. Luque which allows to provide a lower bound for the exponent of the $A_p$ constant in terms of the behaviour of the unweighted inequalities when $p\to \mathrm{\infty }$ and when $p\to 1^{+}$. We also provide some applications to classical operators.