Atomic decompositions for noncommutative martingales

Chen, Zeqian; Randrianantoanina, Narcisse; Xu, Quanhua

doi:10.48550/arxiv.2001.08775

Cited by 3 publications

(20 citation statements)

References 23 publications

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“…Remark 3.6. The order β p ≤ c √ p as p → ∞ improves the previous ones cp in [7] or [4]; and moreover it is sharp, which can be seen from the commutative result (see Theorem 2.50 in [31]).…”

Section: The P-moment Characterizationsupporting

confidence: 63%

“…Note that the characterization (1.3) for 1 ≤ p < ∞ without explicit estimates over α p and β p has been obtained in [4] as a corollary of their atomic decomposition and duality results. While our direct method gives the sharp order β p ≤ c √ p for 2 ≤ p < ∞, and moreover by an unusual real interpolation, the scale of p in (1.3) may extend to 0 < p < 1.…”

Section: Introductionmentioning

confidence: 99%

“…We refer the reader to [11] for noncommutative Doob's maximal inequality, to [13,14] for noncommutative Burkholder/Rosenthal inequalities and ergodic theorems, to [20,22] for the noncommutative Gundy and Davis decompositions, to [24,25] for noncommutative martingale transform and conditioned square functions, to [8,9,10] and the references therein for noncommutative differential subordinates and good-λ inequalities. We should also mention another two works directly related with the objectives of this paper, which are the John-Nirenberg theorems for noncommutative martingale BMO spaces by Mei and the first author [7] (see also [3,15]) and the atomic decomposition for Hardy spaces with small exponents by Chen, Randrianantoanina and Xu [4] (see also [2]).…”

Section: Introductionmentioning

confidence: 99%

“…Recently, for Hardy space-the predual of BMO space, Chen, Randrianantoanina and Xu [4] provide a constructive proof of the atomic characterization; moreover, their constructive proof can be used to establish the duality between Hardy spaces with small exponents and Lipschitz spaces. These inspire us to further understand the John-Nirenberg type theorems and in particular exponential integrability for martingale Lipschitz spaces.…”

Section: Introductionmentioning

confidence: 99%

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Exponential integrability of noncommutative Lipschitz martingales

Hong¹,

Ma²,

Wang³

2022

Preprint

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In this paper, we study the John-Nirenberg theorems for the Lipschitz spaces and the atomic decomposition for Hardy spaces in the noncommutative martingale setting. By adapting Cuculescu's construction for the increasing adapted sequences, we obtain a distribution function inequality form of John-Nirenberg theorem for noncommutative martingale Lipschitz spaces. Using this result, we obtain the exponential integrability and the sharp moment inequalities, which not only extend but also improve the corresponding results for bmo spaces. As an application, we show Lipschitz space is also the dual space of noncommutative Hardy space defined via symmetric atoms.

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Section: The P-moment Characterizationsupporting

confidence: 63%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Exponential integrability of noncommutative Lipschitz martingales

Hong¹,

Ma²,

Wang³

2022

Preprint

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“…14] for a summary of some of the main inequalities from noncommutative martingale theory. Further references relevant to our purpose are [1,7,24,26,32,44]. The main focus of the present article is on Hardy spaces arising from noncommutative martingale theory.…”

Section: Introductionmentioning

confidence: 99%

Interpolation between noncommutative martingale Hardy and BMO spaces: the case $0

Randrianantoanina¹

2021

Preprint

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Let M be a semifinite von Nemann algebra equipped with an increasing filtration (Mn) n≥1 of (semifinite) von Neumann subalgebras of M. For 0 < p < ∞, let h c p (M) denote the noncommutative column conditioned martingale Hardy space and bmo c (M) denote the column "little" martingale BMO space associated with the filtration (Mn) n≥1 .We prove the following real interpolation identity: if 0 < p < ∞ and 0 < θ < 1, then forFor the case of complex interpolation, we obtain that if 0 < p < q < ∞ and 0 < θ < 1, then for 1/r = (1 − θ)/p + θ/q, h c p (M), h c q (M) θ = h c r (M) with equivalent quasi norms.These extend previously known results from p ≥ 1 to the full range 0 < p < ∞. Other related spaces such as spaces of adapted sequences and Junge's noncommutative conditioned Lp-spaces are also shown to form interpolation scale for the full range 0 < p < ∞ 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 Lp-spaces.We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.

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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 for noncommutative martingales

Cited by 3 publications

References 23 publications

Exponential integrability of noncommutative Lipschitz martingales

Exponential integrability of noncommutative Lipschitz martingales

Interpolation between noncommutative martingale Hardy and BMO spaces: the case $0

Interpolation between noncommutative martingale Hardy and BMO spaces: the case

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