Rosenthal inequalities in noncommutative symmetric spaces

Dirksen, Sjoerd; Pagter, B. de; Potapov, Denis; Sukochev, Fedor

doi:10.1016/j.jfa.2011.07.015

Cited by 59 publications

(70 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A version of Theorem 5.7 for the case where the Boyd indices satisfy the condition 2 < p E ≤ q E < ∞ was first obtained in [12,Theorem 6.3]. Similar line of result for martingale BMO-norms of sums of noncommuting independent sequences were also considered in [38,Theorem 5.3].…”

Section: If We Setmentioning

confidence: 81%

Noncommutative Burkholder/Rosenthal inequalities associated with convex functions

Randrianantoanina¹,

Wu²

2017

Ann. Inst. H. Poincaré Probab. Statist.

View full text Add to dashboard Cite

Abstract. We prove noncommutative martingale inequalities associated with convex functions. More precisely, we obtain Φ-moment analogues of the noncommutative Burkholder inequalities and the noncommutative Rosenthal inequalities for any convex Orlicz function Φ whose Matuzewska-Orlicz indices pΦ and qΦ are such that 1 < pΦ ≤ qΦ < 2 or 2 < pΦ ≤ qΦ < ∞. These results generalize the noncommutative Burkholder/Rosenthal inequalities due to Junge and Xu.

show abstract

Section: If We Setmentioning

confidence: 81%

Noncommutative Burkholder/Rosenthal inequalities associated with convex functions

Randrianantoanina¹,

Wu²

2017

Ann. Inst. H. Poincaré Probab. Statist.

View full text Add to dashboard Cite

show abstract

“…By the boundedness of martingale transforms on

E (M)

[, Proposition 4.9], there exists a constant

κ_{E}

such that for any given finite martingale

x

E (M)

κ_{E}^{- 1} E {(‖‖, \sum_{n ⩾ 1} ε_{n} d x_{n})}_{E (M)} ⩽ ∥ x {true∥}_{E (M)} ⩽ κ_{E} E {(‖‖, \sum_{n ⩾ 1} ε_{n} d x_{n})}_{E (M)}

where

(ε_{n})

denotes a Rademacher sequence on a given probability space. According to [, Theorem 4.3], we then have,

{(κ_{E}^{'})}^{- 1} ∥ false(d x_{n} false) {true∥}_{E false(scriptM; ℓ_{2}^{c} false) + E false(scriptM; ℓ_{2}^{r} false)} ⩽ ∥ x {true∥}_{E (M)} ⩽ c_{E} ∥ false(d x_{n} false) {true∥}_{E false(scriptM; ℓ_{2}^{c} false) \cap E false(scriptM; ℓ_{2}^{r} false)} .

Applying the noncommutative Stein inequality to the first inequality , we deduce that

C_{E}^{<...>}

…”

Section: Applications To Noncommutative Burkholder/rosenthal Inequalimentioning

confidence: 92%

“…Our strategy for the proof of Theorem is to use the corresponding Burkholder–Gundy alongside our Davis decomposition stated in Theorem . The following version of the Burkholder–Gundy inequalities is implicit in : Proposition Let

1 < p < q < \infty

and

E

be a symmetric Banach function space with the Fatou property such that

E \in Int (L_{p}, L_{q})

. Then there exist positive constants

C_{E}

and

c_{E}

such that: (i)for every

x \in E (M)

, the following inequality holds:

inf \{true∥ x^{c} ∥_{{scriptH}_{E}^{c}} + true∥ x^{r} ∥_{{scriptH}_{E}^{r}}\} ⩽ C_{E} ∥ x {true∥}_{E (M)},

where the infimum is taken over all

x^{c} \in {scriptH}_{E}^{c} false(scriptM false)

, and

x^{r} \in {scriptH}_{E}^{r} false(scriptM false)

such that

x = x^{c} + x^{r}

; (ii)for every

x \in {scriptH}_{E}^{c} false(scriptM false) \cap {scriptH}_{E}^{r} false(scriptM false)

, the following inequality holds:

∥ x {true∥}_{E (M)} ⩽ c_{E} max \{true∥ x ∥_{{scriptH}_{<...>}}\}

…”

Section: Applications To Noncommutative Burkholder/rosenthal Inequalimentioning

confidence: 99%

“…Our strategy for the proof of Theorem 4.2 is to use the corresponding Burkholder-Gundy alongside our Davis decomposition stated in Theorem 3.9. The following version of the Burkholder-Gundy inequalities is implicit in [13]: (i) for every x ∈ E(M), the following inequality holds:…”

Section: The Case Of Noncommutative Symmetric Spacesmentioning

confidence: 99%

“…where (ε n ) denotes a Rademacher sequence on a given probability space. According to [13,Theorem 4.3], we then have,…”

Section: The Case Of Noncommutative Symmetric Spacesmentioning

confidence: 99%

See 2 more Smart Citations