  In this paper we provide some quantitative mixed-type estimates assuming
conditions that imply that $uv\in A_{\infty}$ for Calder\'on-Zygmund operators,
rough singular integrals and commutators. The main novelty of this paper lies
in the fact that we rely upon sparse domination results, pushing an approach to
endpoint estimates that was introduced by Domingo-Salazar, Lacey and Rey and
extended in works by Lerner, Ombrosi and the second author and Li, Perez, the
second author and Roncal.
