Comptes Rendus
no. 1-2
Probability Theory
Hardy–Littlewoodʼs inequalities in the case of a capacity
[Inégalités de Hardy–Littlewood dans le cas dʼune capacité]

Présenté par : the Editorial Board

Miryana Grigorova1
1 LPMA, CNRS–UMR 7599, université Denis-Diderot – Paris-7, 175, rue du Chevaleret, 75013 Paris, France
Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 73-76.

Sous des hypothèses appropriées, nous généralisons les inégalités de Hardy–Littlewood, bien connues dans le cas où lʼespace mesurable sous-jacent est muni dʼune probabilité, au cas dʼune fonction dʼensembles monotone, appelée capacité. Le résultat fait usage de la théorie de lʼintégration au sens de Choquet.

Hardy–Littlewoodʼs inequalities, well known in the case of a probability measure, are extended to the case of a monotone (but not necessarily additive) set function, called a capacity. The upper inequality is established in the case of a capacity assumed to be continuous and submodular, the lower — under assumptions of continuity and supermodularity.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.01.008
Miryana Grigorova 1

1 LPMA, CNRS–UMR 7599, université Denis-Diderot – Paris-7, 175, rue du Chevaleret, 75013 Paris, France 
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Miryana Grigorova. Hardy–Littlewoodʼs inequalities in the case of a capacity. Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 73-76. doi : 10.1016/j.crma.2013.01.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.01.008/

[1] D. Denneberg Non-Additive Measure and Integral, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994

[2] H. Föllmer; A. Schied Stochastic Finance. An Introduction in Discrete Time, De Gruyter Studies in Mathematics, 2004

[3] M. Grigorova, Stochastic orderings with respect to a capacity and an application to a financial optimization problem, Working paper, hal-00614716, 2011.

[4] M. Grigorova, Stochastic dominance with respect to a capacity and risk measures, Working paper, hal-00639667, 2011.

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