Endpoint mixed weak type extrapolation

Abstract

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 $0<p<\infty$ 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 \[ L^{A,\infty}(uv)=\inf\left\{ λ>0:\sup_{t>0}A(t)w\left(\left\{ x\in\mathbb{R}:|f(x)|>λt\right\} \right)\leq1\right\} \] 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.

14 pages. A reference added. Revised statements