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arXiv · 2402.05324

Revisiting Yano and Zygmund extrapolation theory

Elona Agora·Jorge Antezana·Sergi Baena-Miret·María J. Carro
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Abstract

We prove a pointwise estimate for the decreasing rearrangement of $Tf$, where $T$ is any sublinear operator satisfying the weak-type boundedness $$ T:L^{p,1}(\mu) \to L^{p,\infty}(\nu), \quad \forall p: 1<p_0 < p\leq p_1<\infty, $$ with norm controlled by $C\varphi\left(\left[{p_0^{-1}} - p^{-1}\right]^{-1}\right)$ and $\varphi$ satisfies some admissibility conditions. The pointwise estimate is: \begin{equation*} \begin{split} (Tf)^*_\nu(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^*_\mu(r)\frac{dr}{r^{1 - \frac 1{p_0}}} + \frac 1{t^\frac 1{p_1}}\int_t^\infty f^*_\mu(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.

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Elona Agora, Jorge Antezana, Sergi Baena-Miret, María J. Carro. 2024-02-07. Revisiting Yano and Zygmund extrapolation theory. https://doi.org/10.1007/s00041-022-09923-9

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