Endpoint mixed weak type extrapolation

The purpose of this note is to extend the extrapolation result by by Cruz-Uribe Martell and Pérez as follows. Given a family $\mathcal{F}$ of pairs of functions suppose that for some $0<p<\mathrm{\infty }$ and for every $w\in A_{\mathrm{\infty }}$ \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 $v\in A_{\mathrm{\infty }}$ we have that $${\parallel \frac{f}{v}\parallel }_{L^{A,\mathrm{\infty }}(uv)}\lesssim {\parallel \frac{g}{v}\parallel }_{L^{A,\mathrm{\infty }}(uv)},$$ where $$L^{A,\mathrm{\infty }}(uv)=\inf \{\lambda >0:\underset{t>0}{\sup }A(t)w\left(\{x\in \mathbb{R}:\mathrm{\mid }f(x)\mathrm{\mid }>\lambda t\}\right)\le 1\}$$ 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.