Comptes Rendus
Article de recherche - Analyse harmonique
On Sharpness of L log L Criterion for Weak Type (1,1) boundedness of rough operators
[Sur la netteté du critère L log L pour les faibles de type (1,1) continuité des opérateurs rugueux]
Ankit Bhojak1
1 Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal-462066, India.
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1205-1213.

Dans cette note, nous montrons que l’hypothèse Ω∈LlogL est la condition de taille la plus forte sur une fonction Ω sur la sphère unitaire de valeur moyenne zéro, qui assure que l’intégrale singulière correspondante TΩ définie par

TΩf(x)=p.v.∫1|x-y|dΩx-y|x-y|f(y)dy,

est borné de L1(ℝd) dans L1(ℝd) faibles, à condition que TΩ soit bornée dans L2(ℝd).

In this note, we show that the Ω∈LlogL hypothesis is the strongest size condition on a function Ω on the unit sphere with mean value zero, which ensures that the corresponding singular integral TΩ defined by

TΩf(x)=p.v.∫1|x-y|dΩx-y|x-y|f(y)dy,

maps L1(ℝd) to weak L1(ℝd), provided TΩ is bounded in L2(ℝd).

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Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.633
Classification : 42B20
Keywords: Singular Integrals, Orlicz spaces
Mots-clés : Intégrales singulières, espaces d’Orlicz

Ankit Bhojak 1

1 Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal-462066, India.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Ankit Bhojak},
     title = {On {Sharpness} of $L$ log $L$ {Criterion} for {Weak} {Type} $(1,1)$ boundedness of rough operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1205--1213},
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Ankit Bhojak. On Sharpness of $L$ log $L$ Criterion for Weak Type $(1,1)$ boundedness of rough operators. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1205-1213. doi : 10.5802/crmath.633. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.633/

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