On martingale transform inequalities in certain quasi-Banach function spaces

Kikuchi, Masato

doi:10.1007/s40574-018-0184-y

Boll Unione Mat Ital

2018

DOI: 10.1007/s40574-018-0184-y

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On martingale transform inequalities in certain quasi-Banach function spaces

Masato Kikuchi

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Cited by 3 publications

References 24 publications

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Weak martingale Hardy-type spaces associated with quasi-Banach function lattice

2022

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In this paper, the authors introduce weak martingale Hardy-type spaces associated with a quasi-Banach function lattice. The authors then establish the atomic characterizations of these weak martingale Hardy-type spaces. As applications, the authors give the sufficient conditions for the boundedness of σ-sublinear operators from weak martingale Hardy-type spaces to a quasi-Banach function lattice. Furthermore, the authors clarify the relation among different weak martingale Hardy-type spaces in the framework of a rearrangement-invariant quasi-Banach function space. Finally, the authors apply these results to the weighted Lorentz space and the generalized grand Lebesgue space.

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Weak martingale Hardy-type spaces associated with quasi-Banach function lattice

2022

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Martingale transforms in martingale Hardy spaces with variable exponents

Ma,

Lu,

2024

MATH

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On some inequalities for the optional projection and the predictable projection of a discrete parameter process

Kikuchi

2022

Annales Mathématiques Blaise Pascal

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Let (Ω, Σ, P) be a nonatomic probability space. If F = ( F 𝑛 ) 𝑛∈Z+ is a filtration of Ω and if 𝑓 = ( 𝑓 𝑛 ) 𝑛∈Z+ is a stochastic process on Ω such that 𝑓 𝑛 is integrable for all 𝑛 ∈ Z + , the optional projectionOne of the main results gives a necessary and sufficient condition on 𝑋 for the inequality Sur quelques inégalités pour la projection optionnelle et la projection prévisible d'un processus de paramètre discretRésumé Soit (Ω, Σ, P) un espace de probabilité non atomique. Si F = ( F 𝑛 ) est une filtration de Ω et si 𝑓 = ( 𝑓 𝑛 ) 𝑛∈𝑍 est un processus stochastique sur Ω tel que 𝑓 𝑛 est intégrable pour tout 𝑛 ∈ Z + , la projection optionnelle 𝑂 (F) 𝑓 = ( 𝑂 (F) 𝑓 𝑛 ) 𝑛∈Z+ de 𝑓 = ( 𝑓 𝑛 ) 𝑛∈Z+ est définie par 𝑂 (F) 𝑓 𝑛 = E [ 𝑓 𝑛 | F 𝑛 ]. Étant donné un espace de fonction de Banach 𝑋 sur Ω et 𝑟 ∈ [1, ∞), on laisse 𝑋 [ℓ 𝑟 ] désigner l'espace de Banach constitué de tous les processus 𝑓 = ( 𝑓 𝑛 ) 𝑛∈Z+ tels que ( ∞ 𝑛=0 | 𝑓 𝑛 | 𝑟 ) 1/𝑟 ∈ 𝑋 , et on laisse 𝑓 𝑋 [ℓ𝑟 ] = ∞ 𝑛=0 | 𝑓 𝑛 | 𝑟 1/𝑟 𝑋 pour 𝑓 = ( 𝑓 𝑛 ) ∈Z+ ∈ 𝑋 [ℓ 𝑟 ]. L'un des principaux résultats donne une condition nécessaire et suffisante sur 𝑋 pour que l'inégalité 𝑂 (F) 𝑓 𝑋 [ℓ𝑟 ] ≤ 𝐶 𝑓 𝑋 [ℓ𝑟 ] soit valable pour tout 𝑓 = ( 𝑓 𝑛 ) 𝑛∈Z+ ∈ 𝑋 [ℓ 𝑟 ].

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On martingale transform inequalities in certain quasi-Banach function spaces

Cited by 3 publications

References 24 publications

Weak martingale Hardy-type spaces associated with quasi-Banach function lattice

Weak martingale Hardy-type spaces associated with quasi-Banach function lattice

Martingale transforms in martingale Hardy spaces with variable exponents

On some inequalities for the optional projection and the predictable projection of a discrete parameter process

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