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 $$ T:L^{p,1}(μ) \to L^{p,\infty}(ν), \quad \forall p: 1<p_0 < p\leq p_1<\infty, $$ with norm controlled by $Cφ\left(\left[{p_0^{-1}} - p^{-1}\right]^{-1}\right)$ and $φ$ 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 φ\left(1 - \log \frac rt\right)f^*_μ(r)\frac{dr}{r^{1 - \frac 1{p_0}}} + \frac 1{t^\frac 1{p_1}}\int_t^\infty f^*_μ(r)\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.