Revisiting Yano and Zygmund extrapolation theory

We prove a pointwise estimate for the decreasing rearrangement of $Tf$, where $T$ is any sublinear operator satisfying the weak-type boundedness $$T:L^{p,1}(\mu )\to L^{p,\mathrm{\infty }}(\nu ),\mathrm{\forall }p:1<p_0<p\le p_1<\mathrm{\infty },$$ with norm controlled by $C\phi \left({[p_0^{-1}-p^{-1}]}^{-1}\right)$ and $\phi$ satisfies some admissibility conditions. The pointwise estimate is: \begin{equation*} \begin{split} (Tf)^_ν(t) &\lesssim \frac 1{p_0 - 1}\left(\frac 1{t^\frac 1{p_0}}\int_0^t \varphi\left(1 - \log \frac rt\right)f^_μ®\frac{dr}{r^{1 - \frac 1{p_0}}} + \frac 1{t^\frac 1{p_1}}\int_t^\infty f^_μ®\frac{dr}{r^{1 - \frac 1{p_1}}}\right). \end{split} \end{equation} In particular, this estimate allows to obtain extensions of Yano and Zygmund extrapolation results.