On Sharpness of $L$ log $L$ Criterion for Weak Type $(1,1)$ boundedness of rough operators
In this note, we show that the $\Omega \in L\log L$ hypothesis is the strongest size condition on a function $\Omega $ on the unit sphere with mean value zero, which ensures that the corresponding singular integral $T_\Omega $ defined by \[ T_{\Omega }f(x)=p.v.\int \frac{1}{|x-y|^d}\Omega \Bigl (\fr...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.633/ |
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| Summary: | In this note, we show that the $\Omega \in L\log L$ hypothesis is the strongest size condition on a function $\Omega $ on the unit sphere with mean value zero, which ensures that the corresponding singular integral $T_\Omega $ defined by
\[ T_{\Omega }f(x)=p.v.\int \frac{1}{|x-y|^d}\Omega \Bigl (\frac{x-y}{|x-y|}\Big )f(y)\, \mathrm{d} y,\]
maps $L^1(\mathbb{R}^d)$ to weak $L^1(\mathbb{R}^d)$, provided $T_\Omega $ is bounded in $L^2(\mathbb{R}^d)$. |
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| ISSN: | 1778-3569 |