Revisiting Yano and Zygmund extrapolation theory

Abstract

We prove a pointwise estimate for the decreasing rearrangement of $Tf$, where $T$ is any sublinear operator satisfying the weak-type boundedness $$T:L^{p,1}(\mu )\to L^{p,\infty }(\nu ),\forall p:1<p_0<p\le p_1<\infty ,$$ with norm controlled by $C\phi \left({\left[p_0^{-1}-p^{-1}\right]}^{-1}\right)$ 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 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.

12 pages

References in corpus (1)

• From weak type weighted inequality to pointwise estimate for the decreasing rearrangement