Revisiting Yano and Zygmund extrapolation theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911773489102848 |
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| author | Agora, Elona Antezana, Jorge Baena-Miret, Sergi Carro, María J. |
| author_facet | Agora, Elona Antezana, Jorge Baena-Miret, Sergi Carro, María J. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_05324 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Revisiting Yano and Zygmund extrapolation theory Agora, Elona Antezana, Jorge Baena-Miret, Sergi Carro, María J. Classical Analysis and ODEs 46E30, 28A10, 47A30 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. |
| title | Revisiting Yano and Zygmund extrapolation theory |
| topic | Classical Analysis and ODEs 46E30, 28A10, 47A30 |
| url | https://arxiv.org/abs/2402.05324 |