Comptes Rendus
no. 10
Harmonic analysis/Functional analysis
Hamming cube and martingales

Presented by: Yves Meyer

Paata Ivanisvili1, 2, 3, 4; Fedor Nazarov1; Alexander Volberg5
1 Kent State University, OH 44240, USA
2 MSRI, USA
3 Princeton University, USA
4 University of California, Irvine, USA
5 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Comptes Rendus. Mathématique, Volume 355 (2017) no. 10, pp. 1072-1076.

In the present paper, we show that a correctly chosen Legendre transform of the Bellman functions of martingale problems give us the right tool to prove isoperimetric inequalities on Hamming cube independent of the dimension. We illustrate the power of this “dual function approach” by proving certain Poincaré inequalities on Hamming cube and by improving a particular inequality of Beckner on the Hamming cube.

Dans le présent article, nous montrons qu'une transformée de Legendre adéquate des fonctions de Bellman, issues de problèmes de martingale, fournit le bon outil pour démontrer les inégalités isopérimetriques sur le cube de Hamming, indépendamment de la dimension. Nous illustrons la puissance de cette « approche par fonction duale » en démontrant une inégalité de Poincaré et en améliorant une inégalité de Beckner sur le cube de Hamming.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.09.013

Paata Ivanisvili 1, 2, 3, 4; Fedor Nazarov 1; Alexander Volberg 5

1 Kent State University, OH 44240, USA
2 MSRI, USA
3 Princeton University, USA
4 University of California, Irvine, USA
5 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA 
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Paata Ivanisvili; Fedor Nazarov; Alexander Volberg. Hamming cube and martingales. Comptes Rendus. Mathématique, Volume 355 (2017) no. 10, pp. 1072-1076. doi : 10.1016/j.crma.2017.09.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.09.013/

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Cited by Sources:

☆ We acknowledge the support of the following grants: FN – the grant from DMS of NSF, AV – NSF grant DMS-1600075. The results of this paper were obtained at the MSRI in Berkeley, California, 2017 program Harmonic Analysis under the NSF grant No. DMS-1440140.

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