On Sharpness of $L$ log $L$ Criterion for Weak Type $(1,1)$ boundedness of rough operators

Ankit Bhojak

doi:10.5802/crmath.633

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 Bhojak ¹

¹ Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal-462066, India.

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1205-1213.

Résumé
Abstract

Dans cette note, nous montrons que l’hypothèse $Ω \in L log L$ 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 $T_{Ω}$ définie par

T_{Ω} f (x) = p . v . \int \frac{1}{{| x - y |}^{d}} Ω (\frac{x - y}{| x - y |}) f (y) d y,

est borné de $L^{1} (ℝ^{d})$ dans $L^{1} (ℝ^{d})$ faibles, à condition que $T_{Ω}$ soit bornée dans $L^{2} (ℝ^{d})$ .

In this note, we show that the $Ω \in L log L$ 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 $T_{Ω}$ defined by

T_{Ω} f (x) = p . v . \int \frac{1}{{| x - y |}^{d}} Ω (\frac{x - y}{| x - y |}) f (y) d y,

maps $L^{1} (ℝ^{d})$ to weak $L^{1} (ℝ^{d})$ , provided $T_{Ω}$ is bounded in $L^{2} (ℝ^{d})$ .

Reçu le : 2023-07-24
Révisé le : 2024-01-22
Accepté le : 2024-03-23
Publié le : 2024-11-05

DOI : 10.5802/crmath.633

Classification : 42B20
Keywords: Singular Integrals, Orlicz spaces
Mot clés : Intégrales singulières, espaces d’Orlicz

Affiliations des auteurs :

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},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.633},
     language = {en},
}

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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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