On Doob’s inequality and Burkholder’s inequality in weak spaces

Kikuchi, Masato

doi:10.1007/s13348-015-0153-z

Cited by 10 publications

(4 citation statements)

References 9 publications

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“…Moreover, more martingale inequalities were recently studied by Ose ¸kowski [61,62], Kikuchi [44,45,46] and Ho [31,30]. Especially, Bañuelos and Ose ¸kowski studied the weighted martingale inequalities in [3,4].…”

Section: Introductionmentioning

confidence: 99%

New martingale inequalities and applications to Fourier analysis

et al. 2019

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Let (Ω, F , P) be a probability space and ϕ : Ω × [0, ∞) → [0, ∞) be a Musielak-Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak-Orlicz space L ϕ (Ω). Using this and extrapolation method, the authors then establish a Fefferman-Stein vector-valued Doob maximal inequality on L ϕ (Ω). As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for L ϕ (Ω), which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak-Orlicz Hardy spaces H s ϕ (Ω), P ϕ (Ω), Q ϕ (Ω), H S ϕ (Ω) and H M ϕ (Ω). From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak-Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on H S ϕ (Ω) and H M ϕ (Ω), the authors obtain the Burkholder-Davis-Gundy inequality associated with Musielak-Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fejér operator is bounded from H ϕ [0, 1) to L ϕ [0, 1), which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.

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Section: Introductionmentioning

confidence: 99%

New martingale inequalities and applications to Fourier analysis

et al. 2019

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“…3). Then by(5.4), E [ 𝑥 | A ] w-𝑋 ≤ 𝐶 𝑋 𝑥 w-𝑋 ≤ 𝐶 𝑋 𝑥 𝑋 .Lemma 3] we see that 𝑘 𝑋 < ∞, and from[20, Theorem 4.1] we see that𝑞 𝑋 = 𝑞 𝜑 𝑋 < 1.It only remains to show that 𝑝𝑋 > 0. As shown above, 𝑞 𝑋 < 1 and 𝑘 𝑋 < ∞.…”

mentioning

confidence: 89%

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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“…The following lemma gives the boundedness of the Doob maximal operator; see, for instance, [26,Corollary 3.6] and [36,Theorem B]. We refer the reader to [16,33,34,35] for martingale inequalities on Banach function space.…”

Section: Weak Martingale Hardy-type Spacesmentioning

confidence: 99%

Dual Spaces for Weak Martingale Hardy Spaces Associated with Rearrangement-Invariant Spaces

Quan,

Silas,

Xie

2023

Potential Anal

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Given a probability space (Ω, F , P) and a rearrangement-invariant quasi-Banach function space X, the authors of this article first prove the α-atomic (α ∈ [1, ∞)) characterization of weak martingale Hardy spaces WH X (Ω) associated with X via simple atoms. The authors then introduce the generalized weak martingale BMO spaces which proves to be the dual spaces of WH X (Ω). Consequently, the authors derive a new John-Nirenberg theorem for these weak martingale BMO spaces. Finally, the authors apply these results to the generalized grand Lebesgue space and the weighted Lorentz space. Even in these special cases, the results obtained in this article are totally new.

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On Doob’s inequality and Burkholder’s inequality in weak spaces

Cited by 10 publications

References 9 publications

New martingale inequalities and applications to Fourier analysis

New martingale inequalities and applications to Fourier analysis

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

Dual Spaces for Weak Martingale Hardy Spaces Associated with Rearrangement-Invariant Spaces

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