The John–Nirenberg inequality with sharp constants

Andrei K. Lerner

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

Andrei K. Lerner ¹

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

Comptes Rendus. Mathématique, Volume 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 :

Andrei K. Lerner ¹

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

@article{CRMATH_2013__351_11-12_463_0,
     author = {Andrei K. Lerner},
     title = {The {John{\textendash}Nirenberg} inequality with sharp constants},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {463--466},
     publisher = {Elsevier},
     volume = {351},
     number = {11-12},
     year = {2013},
     doi = {10.1016/j.crma.2013.07.007},
     language = {en},
}

TY  - JOUR
AU  - Andrei K. Lerner
TI  - The John–Nirenberg inequality with sharp constants
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 463
EP  - 466
VL  - 351
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2013.07.007
LA  - en
ID  - CRMATH_2013__351_11-12_463_0
ER  -

%0 Journal Article
%A Andrei K. Lerner
%T The John–Nirenberg inequality with sharp constants
%J Comptes Rendus. Mathématique
%D 2013
%P 463-466
%V 351
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2013.07.007
%G en
%F CRMATH_2013__351_11-12_463_0

Andrei K. Lerner. The John–Nirenberg inequality with sharp constants. Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 463-466. doi : 10.1016/j.crma.2013.07.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.07.007/

Bibliographie
Cité par

[1] F. John; L. Nirenberg On functions of bounded mean oscillation, Commun. Pure Appl. Math., Volume 14 (1961), pp. 415-426

[2] T. Hytönen The $A_{2}$ theorem: Remarks and complements (preprint, available at) | arXiv

[3] I. Klemes A mean oscillation inequality, Proc. Amer. Math. Soc., Volume 93 (1985) no. 3, pp. 497-500

[4] A.A. Korenovskii The connection between mean oscillations and exact exponents of summability of functions, Mat. Sb., Volume 181 (1990) no. 12, pp. 1721-1727 (in Russian); translation in Math. USSR-Sb., 71, 2, 1992, pp. 561-567

[5] A.A. Korenovskii Mean Oscillations and Equimeasurable Rearrangements of Functions, Lect. Notes Unione Mat. Ital., vol. 4, Springer/UMI, Berlin/Bologna, 2007

[6] A.K. Lerner A pointwise estimate for local sharp maximal function with applications to singular integrals, Bull. London Math. Soc., Volume 42 (2010) no. 5, pp. 843-856

[7] L. Slavin; V. Vasyunin Sharp results in the integral-form John–Nirenberg inequality, Trans. Amer. Math. Soc., Volume 363 (2011) no. 8, pp. 4135-4169

[8] V. Vasyunin; A. Volberg Sharp constants in the classical weak form of the John–Nirenberg inequality (preprint, available at) | arXiv

Cité par Sources :

Commentaires - Politique