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

Bhojak, Ankit

Comptes Rendus. Mathématique, Tome 362 (2024) no. G10, pp. 1205-1213

In this note, we show that the $\Omega \in LlogL$ 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

$$T_{\Omega }f\left(x\right)=p.v.\int \frac{1}{{|x-y|}^d}\Omega \left(\frac{x-y}{|x-y|}\right)f\left(y\right)\mathrm{d}y,$$

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

Dans cette note, nous montrons que l’hypothèse $\Omega \in LlogL$ est la condition de taille la plus forte sur une fonction $\Omega$ sur la sphère unitaire de valeur moyenne zéro, qui assure que l’intégrale singulière correspondante $T_{\Omega }$ définie par

$$T_{\Omega }f\left(x\right)=p.v.\int \frac{1}{{|x-y|}^d}\Omega \left(\frac{x-y}{|x-y|}\right)f\left(y\right)\mathrm{d}y,$$

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

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

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