Atomic decompositions and asymmetric Doob inequalities in noncommutative symmetric spaces

Randrianantoanina, Narcisse; Wu, Lian; Zhou, Dejian

doi:10.1016/j.jfa.2020.108794

Cited by 11 publications

(27 citation statements)

References 24 publications

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“…It strengthens the results from [8,22,42,44] and completes our understanding of noncommutative Burkholder inequalities. It also improves [45,Corollary 4.11] greatly. In fact, the asymmetric form of Burkholder inequalities involving column and row maximal diagonals provided in [45,Corollary 4.11] can be easily inferred from our result.…”

Section: Introductionmentioning

confidence: 86%

“…It also improves [45,Corollary 4.11] greatly. In fact, the asymmetric form of Burkholder inequalities involving column and row maximal diagonals provided in [45,Corollary 4.11] can be easily inferred from our result.…”

Section: Introductionmentioning

confidence: 86%

“…x ∈ E(M), then (7) x E ≃ E max s c (x) E , s r (x) E , (dx n ) n≥1 E(M;ℓ∞) , Following the above line of research, the main purpose of this paper is to study noncommutative Burkholder inequalities with asymmetric maximal diagonal term in symmetric operator spaces. This is strongly motivated by two recent papers [13,45], where asymmetric maximal inequalities in noncommutative L p spaces and in symmetric operator spaces were established, respectively. We should also mention that the first asymmetric forms of Doob inequalities was found by Junge in [24].…”

Section: Introductionmentioning

confidence: 99%

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Asymmetric Burkholder inequalities in noncommutative symmetric spaces

Lian¹,

Xia²

2022

Preprint

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Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

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Asymmetric Burkholder inequalities in noncommutative symmetric spaces

Lian¹,

Xia²

2022

Preprint

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“…In this section, we will give applications of the results obtained in the previous sections to atomic decomposition and paraproducts on noncommutative martingales. We refer to [CRX20,RWZ21] for some recent results on the atomic decomposition of noncommutative martingales.…”

Section: Conversely Asmentioning

confidence: 99%

Interpolation and the John–Nirenberg inequality on symmetric spaces of noncommutative martingales

et al. 2022

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We prove various John-Nirenberg inequalities on symmetric spaces of noncommutative martingales, including the crude and fine versions, which extend the corresponding results of Junge and Musat ( 2007) and Hong and Mei (2012) in the Lp-case.As an application, we provide the atomic decomposition of a noncommutative martingale Hardy space h1 using symmetric atoms as building blocks, and give the boundedness of paraproducts on symmetric spaces of noncommutative martingales.

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“…We refer to [25], [34], and [43] for various forms of noncommutative Davis decompositions. We refer to [44] for formal definition of the noncommutative vector-valued space Φ (M; ℓ 1 ) used below. Theorem 4.8 Assume that 0 < < < 2 and Φ is an Orlicz function that is -convex and -concave.…”

Section: Andmentioning

confidence: 99%

Interpolation between noncommutative martingale Hardy and BMO spaces: the case

Randrianantoanina¹

2021

Can. J. Math.-J. Can. Math.

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Let $\mathcal {M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal {M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal {M}$ . For $0<p <\infty $ , let $\mathsf {h}_p^c(\mathcal {M})$ denote the noncommutative column conditioned martingale Hardy space and $\mathsf {bmo}^c(\mathcal {M})$ denote the column “little” martingale BMO space associated with the filtration $(\mathcal {M}_n)_{n\geq 1}$ . We prove the following real interpolation identity: if $0<p <\infty $ and $0<\theta <1$ , then for $1/r=(1-\theta )/p$ , $$ \begin{align*} \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}), \end{align*} $$ with equivalent quasi norms. For the case of complex interpolation, we obtain that if $0<p<q<\infty $ and $0<\theta <1$ , then for $1/r =(1-\theta )/p +\theta /q$ , $$ \begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M}) \end{align*} $$ with equivalent quasi norms. These extend previously known results from $p\geq 1$ to the full range $0<p<\infty $ . Other related spaces such as spaces of adapted sequences and Junge’s noncommutative conditioned $L_p$ -spaces are also shown to form interpolation scale for the full range $0<p<\infty $ when either the real method or the complex method is used. Our method of proof is based on a new algebraic atomic decomposition for Orlicz space version of Junge’s noncommutative conditioned $L_p$ -spaces. We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.

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Atomic decompositions and asymmetric Doob inequalities in noncommutative symmetric spaces

Cited by 11 publications

References 24 publications

Asymmetric Burkholder inequalities in noncommutative symmetric spaces

Asymmetric Burkholder inequalities in noncommutative symmetric spaces

Interpolation and the John–Nirenberg inequality on symmetric spaces of noncommutative martingales

Interpolation between noncommutative martingale Hardy and BMO spaces: the case

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