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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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.633/ |
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| author | Bhojak, Ankit |
| author_facet | Bhojak, Ankit |
| author_sort | Bhojak, Ankit |
| collection | DOAJ |
| description | 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 $L^2(\mathbb{R}^d)$. |
| format | Article |
| id | doaj-art-21f1d72b979e4036ae369ccfd64e3dd3 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-21f1d72b979e4036ae369ccfd64e3dd32025-02-07T11:23:31ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G101205121310.5802/crmath.63310.5802/crmath.633On Sharpness of $L$ log $L$ Criterion for Weak Type $(1,1)$ boundedness of rough operatorsBhojak, Ankit0Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal-462066, India.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 $L^2(\mathbb{R}^d)$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.633/Singular IntegralsOrlicz spaces |
| spellingShingle | Bhojak, Ankit On Sharpness of $L$ log $L$ Criterion for Weak Type $(1,1)$ boundedness of rough operators Comptes Rendus. Mathématique Singular Integrals Orlicz spaces |
| title | On Sharpness of $L$ log $L$ Criterion for Weak Type $(1,1)$ boundedness of rough operators |
| title_full | On Sharpness of $L$ log $L$ Criterion for Weak Type $(1,1)$ boundedness of rough operators |
| title_fullStr | On Sharpness of $L$ log $L$ Criterion for Weak Type $(1,1)$ boundedness of rough operators |
| title_full_unstemmed | On Sharpness of $L$ log $L$ Criterion for Weak Type $(1,1)$ boundedness of rough operators |
| title_short | On Sharpness of $L$ log $L$ Criterion for Weak Type $(1,1)$ boundedness of rough operators |
| title_sort | on sharpness of l log l criterion for weak type 1 1 boundedness of rough operators |
| topic | Singular Integrals Orlicz spaces |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.633/ |
| work_keys_str_mv | AT bhojakankit onsharpnessoflloglcriterionforweaktype11boundednessofroughoperators |