The John–Nirenberg inequality with sharp constants

Lerner, Andrei K.

doi:10.1016/j.crma.2013.07.007

Harmonic Analysis

The John–Nirenberg inequality with sharp constants
[Meilleures constantes dans lʼinégalité de John–Nirenberg]

Présenté par : Yves Meyer

Lerner, Andrei K.¹

¹ Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israel

Comptes Rendus. Mathématique, Tome 351 (2013) no. 11-12, pp. 463-466.

Résumé
Abstract

On considère lʼinégalité de John–Nirenberg unidimensionnelle :

| {x \in I_{0} : | f (x) - f_{I_{0}} | > α} | ⩽ C_{1} | I_{0} | \exp (- \frac{C_{2}}{{‖ f ‖}_{⁎}} α) .

A. Korenovskii a montré que la meilleure constante

C_{2}

était égale à

2 / e

. Dans cette Note, on montre que si

C_{2} = 2 / e

, alors la meilleure constante possible pour

C_{1}

est

C_{1} = \frac{1}{2} e^{4 / e}

We consider the one-dimensional John–Nirenberg inequality:

| {x \in I_{0} : | f (x) - f_{I_{0}} | > α} | ⩽ C_{1} | I_{0} | \exp (- \frac{C_{2}}{{‖ f ‖}_{⁎}} α) .

A. Korenovskii found that the sharp

C_{2}

here is

C_{2} = 2 / e

. It is shown in this paper that if

C_{2} = 2 / e

, then the best possible

C_{1}

C_{1} = \frac{1}{2} e^{4 / e}

Reçu le : 2013-03-14
Accepté le : 2013-07-03
Publié le : 2013-06-01

DOI : 10.1016/j.crma.2013.07.007

Affiliations des auteurs :

Lerner, Andrei K. ¹

¹ Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israel

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Lerner, Andrei K. The John–Nirenberg inequality with sharp constants. Comptes Rendus. Mathématique, Tome 351 (2013) no. 11-12, pp. 463-466. doi : 10.1016/j.crma.2013.07.007. http://www.numdam.org/articles/10.1016/j.crma.2013.07.007/

Bibliographie
Cité par

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