Mixed Weak type inequalities for pairs of weights related to the Hardy-Littlewood maximal funcion, Calderón-Zygmund operators and their commutators

We study two-weight weak-type estimates for the operator $S_{v}f=\mathcal{T}(fv)\mathrm{/}v$, where $\mathcal{T}$ is the Hardy-Littlewood maximal operator or a Calderón-Zygmund operator (CZO) and $v$ is a weight. Concretely, under certain conditions on the weights involved, we prove that $S_v$ is bounded from $L^1(wv)$ to $L^{1,\mathrm{\infty }}(uv)$. We also consider the corresponding inequalities when $\mathcal{T}$ is a higher-order commutator of a CZO. These types of results are inspired by the article of Sawyer in [21], (see also [17]).