Endpoint mixed weak type extrapolation

The purpose of this note is to extend the extrapolation result by by Cruz-Uribe Martell and P\'erez as follows. Given a family $\mathcal{F}$ of pairs of functions suppose that for some $0<p<\infty$ and for every $w\in A_{\infty}$ \begin{equation} \int f^{p}w\leq c_{w}\int g^{p}w\qquad(f,g)\in\mathcal{F}\label{eq:Hip-1} \end{equation} provided the left-hand side of the estimate is finite. If we have that $A(t)=t\log(e+t)^{\rho}$ for some $\rho>0$, then, for every $u\in A_{1}$ and every $v\in A_{\infty}$ we have that $$\left\Vert \frac{f}{v}\right\Vert_{L^{A,\infty}(uv)}\lesssim\left\Vert \frac{g}{v}\right\Vert_{L^{A,\infty}(uv)},$$ where $$L^{A,\infty}(uv)=\inf\left\{ \lambda>0:\sup_{t>0}A(t)w\left(\left\{ x\in\mathbb{R}:|f(x)|>\lambda t\right\} \right)\leq1\right\}$$ is the weak Orlicz type introduced by Iaffei. As a corollary of this extrapolation result we derive a mixed weak type inequality for Coifman-Rochberg-Weiss commutators.