Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales

Bekjan, Turdebek N.; Chen, Zeqian; Perrin, Mathilde; Yin, Zhi

doi:10.1016/j.jfa.2009.12.006

Cited by 39 publications

(101 citation statements)

References 24 publications

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“…By approximation, we assume that the operators

w_{n}

are invertible. As in the case of martingales, the following inequality holds:

0true \sum_{n ⩾ 1} true∥ ξ_{n} w_{n}^{- 1} ∥_{2}^{2} ⩽ \frac{2}{p} true∥ ξ ∥_{L_{p} (M; ℓ_{2}^{c})}^{p} .

This is implicit in . Indeed, for every

n ⩾ 1

∥ ξ_{n} w_{n}^{- 1} {true∥}_{2}^{2} = τ (| ξ_{n} {false|}^{2} w_{n}^{- 2}) = τ (ς_{n}^{p - 2} (ς_{n}^{2} - ς_{n - 1}^{2})) ⩽ \frac{2}{p} τ (ς_{n}^{p} - ς_{n - 1}^{p}),

where the last inequality comes from .…”

Section: Davis‐type Decompositionsmentioning

confidence: 94%

“…As in the case of martingales, the following inequality holds:

0true \sum_{n ⩾ 1} true∥ ξ_{n} w_{n}^{- 1} ∥_{2}^{2} ⩽ \frac{2}{p} true∥ ξ ∥_{L_{p} (M; ℓ_{2}^{c})}^{p} .

This is implicit in . Indeed, for every

n ⩾ 1

∥ ξ_{n} w_{n}^{- 1} {true∥}_{2}^{2} = τ (| ξ_{n} {false|}^{2} w_{n}^{- 2}) = τ (ς_{n}^{p - 2} (ς_{n}^{2} - ς_{n - 1}^{2})) ⩽ \frac{2}{p} τ (ς_{n}^{p} - ς_{n - 1}^{p}),

where the last inequality comes from . Consider now the following decomposition of

ξ

: for

n ⩾ 1

, we set

({, \begin{matrix} y_{n} & = ξ_{n} w_{n}^{- 1} false(w_{n} - w_{n - 1} false), \\ z_{n} & = ξ_{n} w_{n}^{- 1} w_{n - 1}, \end{matrix})

where we have taken

w_{0} = 0

.…”

Section: Davis‐type Decompositionsmentioning

confidence: 99%

See 1 more Smart Citation

Noncommutative Davis type decompositions and applications

Randrianantoanina

2018

J. London Math. Soc.

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show abstract

“…By approximation, we assume that the operators

w_{n}

are invertible. As in the case of martingales, the following inequality holds:

0true \sum_{n ⩾ 1} true∥ ξ_{n} w_{n}^{- 1} ∥_{2}^{2} ⩽ \frac{2}{p} true∥ ξ ∥_{L_{p} (M; ℓ_{2}^{c})}^{p} .

This is implicit in . Indeed, for every

n ⩾ 1

∥ ξ_{n} w_{n}^{- 1} {true∥}_{2}^{2} = τ (| ξ_{n} {false|}^{2} w_{n}^{- 2}) = τ (ς_{n}^{p - 2} (ς_{n}^{2} - ς_{n - 1}^{2})) ⩽ \frac{2}{p} τ (ς_{n}^{p} - ς_{n - 1}^{p}),

where the last inequality comes from .…”

Section: Davis‐type Decompositionsmentioning

confidence: 94%

“…As in the case of martingales, the following inequality holds:

0true \sum_{n ⩾ 1} true∥ ξ_{n} w_{n}^{- 1} ∥_{2}^{2} ⩽ \frac{2}{p} true∥ ξ ∥_{L_{p} (M; ℓ_{2}^{c})}^{p} .

This is implicit in . Indeed, for every

n ⩾ 1

∥ ξ_{n} w_{n}^{- 1} {true∥}_{2}^{2} = τ (| ξ_{n} {false|}^{2} w_{n}^{- 2}) = τ (ς_{n}^{p - 2} (ς_{n}^{2} - ς_{n - 1}^{2})) ⩽ \frac{2}{p} τ (ς_{n}^{p} - ς_{n - 1}^{p}),

where the last inequality comes from . Consider now the following decomposition of

ξ

: for

n ⩾ 1

, we set

({, \begin{matrix} y_{n} & = ξ_{n} w_{n}^{- 1} false(w_{n} - w_{n - 1} false), \\ z_{n} & = ξ_{n} w_{n}^{- 1} w_{n - 1}, \end{matrix})

where we have taken

w_{0} = 0

.…”

Section: Davis‐type Decompositionsmentioning

confidence: 99%

Noncommutative Davis type decompositions and applications

Randrianantoanina

2018

J. London Math. Soc.

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“…Our approach below is very different from Wang's proof. It was inspired by an argument used in [4] to describe an equivalent quasi-norm on h c p (M) when 0 < p < 2 which in turn was adapted from an argument due to Herz [12] for the classical case. We now state the main result of this subsection: where c ′ p = 2…”

Section: Comparisons Of Normsmentioning

confidence: 99%

“…Since [34], the theory of noncommutative martingale has been steadily progressing to a point where many classical inequalities now have noncommutative analogues. The articles [4,13,14,15,19,20,23,21,28,31,32,36] contain samples of various noncommutative analogues of some of the most well known classical inequalities and techniques in the literature. We also refer to the book [33,Chap.…”

Section: Introductionmentioning

confidence: 99%

Square Functions for Noncommutative Differentially Subordinate Martingales

Jiao

Randrianantoanina

et al. 2019

Commun. Math. Phys.

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We prove inequalities involving noncommutative differentially subordinate martingales. More precisely, we prove that if x is a self-adjoint noncommutative martingale and y is weakly differentially subordinate to x then y admits a decomposition dy = a + b + c (resp. dy = z + w) where a, b, and c are adapted sequences (resp. z and w are martingale difference sequences) such that:(an) n≥1 L 1,∞ (M⊗ℓ∞)

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“…(0.1)      supp ϕ ⊂ {ξ : 1 2 ≤ |ξ| ≤ 2}. ϕ > 0 on {ξ : 1 2 < |ξ| < 2}, k∈Z ϕ(2 −k ξ) = 1, ∀ ξ = 0.…”

mentioning

confidence: 99%

Operator-Valued Triebel–Lizorkin Spaces

Xia

Xiong

2018

Integr. Equ. Oper. Theory

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This paper is devoted to the study of operator-valued Triebel-Lizorkin spaces. We develop some Fourier multiplier theorems for square functions as our main tool, and then study the operator-valued Triebel-Lizorkin spaces on R d . As in the classical case, we connect these spaces with operator-valued local Hardy spaces via Bessel potentials. We show the lifting theorem, and get interpolation results for these spaces. We obtain Littlewood-Paley type, as well as the Lusin type square function characterizations in the general way. Finally, we establish smooth atomic decompositions for the operator-valued Triebel-Lizorkin spaces. These atomic decompositions play a key role in our recent study of mapping properties of pseudo-differential operators with operator-valued symbols.

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Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales

Abstract: We prove that atomic decomposition for the Hardy spaces h 1 and H 1 is valid for noncommutative martingales. We also establish that the conditioned Hardy spaces of noncommutative martingales h p and bmo form interpolation scales with respect to both complex and real interpolations.

Cited by 39 publications

References 24 publications

Noncommutative Davis type decompositions and applications

Noncommutative Davis type decompositions and applications

Square Functions for Noncommutative Differentially Subordinate Martingales

Operator-Valued Triebel–Lizorkin Spaces

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