Noncommutative Burkholder/Rosenthal inequalities associated with convex functions

Randrianantoanina, Narcisse; Wu, Lian

doi:10.1214/16-aihp764

Cited by 22 publications

(35 citation statements)

References 45 publications

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“…In this subsection, we focus on noncommutative moment inequalities associated with Orlicz functions, which were considered in . We will assume throughout that

normalΦ

is an Orlicz function satisfying the

Δ_{2}

‐condition, that is, for some constant

C > 0

Φ (2 t) ⩽ C Φ (t), t ⩾ 0 .

We denote by

L_{Φ}

the associated Orlicz function space.…”

Section: Applications To Noncommutative Burkholder/rosenthal Inequalimentioning

confidence: 99%

“…for martingale Hardy spaces

{scriptH}_{L_{normalΦ}}^{c} false(scriptM false)

{sans-serifh}_{L_{normalΦ}}^{c} false(scriptM false)

, etc. We make the observation that if

L_{normalΦ} ⊈ L_{2} + L_{\infty}

, then for an

x \in {sans-serifh}_{normalΦ}^{c} false(scriptM false)

, the

normalΦ

‐moment

τ [Φ false(s_{c} (x) false)]

is understood to be the quantity

τ \otimes tr false[normalΦ false(false| U {scriptD}_{c} (x) false| false) false]

as fully detailed in .…”

Section: Applications To Noncommutative Burkholder/rosenthal Inequalimentioning

confidence: 99%

“…The function

normalΦ

satisfies the

Δ_{2}

‐condition if and only if it is

q

‐concave for some

q < \infty

. Recall the so‐called Matuzewska–Orlicz indices

p_{Φ}

and

q_{Φ}

normalΦ

p_{normalΦ} = \underset{t \to 0^{+}}{trueprefixlim} \frac{log M_{normalΦ} false(t false)}{log t} and q_{normalΦ} = \underset{t \to \infty}{trueprefixlim} \frac{log M_{normalΦ} false(t false)}{log t},

where

M_{normalΦ} false(t false) = \underset{s > 0}{trueprefixsup} \frac{Φ (t s)}{Φ (s)} .

The indices

p_{Φ}

and

q_{Φ}

are used in the previous papers instead of the convexity and concavity indices in the present one. It is easy to see that

p ⩽ p_{normalΦ} ⩽ q_{normalΦ} ⩽ q

normalΦ

p

‐convex and

q

‐concave.…”

Section: Applications To Noncommutative Burkholder/rosenthal Inequalimentioning

confidence: 99%

“…Proof of Theorem The proof is an adaptation of the argument used in so we will only highlight the main points. We begin with the proof of (i).…”

Section: Applications To Noncommutative Burkholder/rosenthal Inequalimentioning

confidence: 99%

“…Moreover, we have:

0true \int_{0}^{\infty} normalΦ ([], t^{- 1 / p}, J, (t^{1 / p - 1 / q}, normalΘ (() u (t)); H_{p}^{c} (M), H_{q}^{c} (M))) d t ⩽ C_{normalΦ, p, q} τ ([], Φ, (S_{c} (x))) .

As in , we need to modify the representation as follows: set

θ = 1 / p - 1 / q

and define:

v false(t false) = \frac{1}{θ} Θ (u false(t^{1 / θ} false)) .

Then

v (\cdot)

is a representation of

x

in the couple

({scriptH}_{p}^{c} false(scriptM false), {scriptH}_{q}^{c} false(scriptM false))

and the preceding inequality becomes:

0true \int_{0}^{\infty} normalΦ ([], t^{- 1 / p}, J, (t^{1 / p - 1 / q}, v (t^{1 / p - 1 / q}); H_{p}^{c} (M), H_{q}^{c} (M))) d t ⩽ C_{...}

…”

Section: Applications To Noncommutative Burkholder/rosenthal Inequalimentioning

confidence: 99%

See 4 more Smart Citations

Noncommutative Davis type decompositions and applications

Randrianantoanina

2018

J. London Math. Soc.

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show abstract

“…In this subsection, we focus on noncommutative moment inequalities associated with Orlicz functions, which were considered in . We will assume throughout that

normalΦ

is an Orlicz function satisfying the

Δ_{2}

‐condition, that is, for some constant

C > 0

Φ (2 t) ⩽ C Φ (t), t ⩾ 0 .

We denote by

L_{Φ}

the associated Orlicz function space.…”

Section: Applications To Noncommutative Burkholder/rosenthal Inequalimentioning

confidence: 99%

“…for martingale Hardy spaces

{scriptH}_{L_{normalΦ}}^{c} false(scriptM false)

{sans-serifh}_{L_{normalΦ}}^{c} false(scriptM false)

, etc. We make the observation that if

L_{normalΦ} ⊈ L_{2} + L_{\infty}

, then for an

x \in {sans-serifh}_{normalΦ}^{c} false(scriptM false)

, the

normalΦ

‐moment

τ [Φ false(s_{c} (x) false)]

is understood to be the quantity

τ \otimes tr false[normalΦ false(false| U {scriptD}_{c} (x) false| false) false]

as fully detailed in .…”

Section: Applications To Noncommutative Burkholder/rosenthal Inequalimentioning

confidence: 99%

“…The function

normalΦ

satisfies the

Δ_{2}

‐condition if and only if it is

q

‐concave for some

q < \infty

. Recall the so‐called Matuzewska–Orlicz indices

p_{Φ}

and

q_{Φ}

normalΦ

p_{normalΦ} = \underset{t \to 0^{+}}{trueprefixlim} \frac{log M_{normalΦ} false(t false)}{log t} and q_{normalΦ} = \underset{t \to \infty}{trueprefixlim} \frac{log M_{normalΦ} false(t false)}{log t},

where

M_{normalΦ} false(t false) = \underset{s > 0}{trueprefixsup} \frac{Φ (t s)}{Φ (s)} .

The indices

p_{Φ}

and

q_{Φ}

are used in the previous papers instead of the convexity and concavity indices in the present one. It is easy to see that

p ⩽ p_{normalΦ} ⩽ q_{normalΦ} ⩽ q

normalΦ

p

‐convex and

q

‐concave.…”

Section: Applications To Noncommutative Burkholder/rosenthal Inequalimentioning

confidence: 99%

“…Proof of Theorem The proof is an adaptation of the argument used in so we will only highlight the main points. We begin with the proof of (i).…”

Section: Applications To Noncommutative Burkholder/rosenthal Inequalimentioning

confidence: 99%

“…Moreover, we have:

0true \int_{0}^{\infty} normalΦ ([], t^{- 1 / p}, J, (t^{1 / p - 1 / q}, normalΘ (() u (t)); H_{p}^{c} (M), H_{q}^{c} (M))) d t ⩽ C_{normalΦ, p, q} τ ([], Φ, (S_{c} (x))) .

As in , we need to modify the representation as follows: set

θ = 1 / p - 1 / q

and define:

v false(t false) = \frac{1}{θ} Θ (u false(t^{1 / θ} false)) .

Then

v (\cdot)

is a representation of

x

in the couple

({scriptH}_{p}^{c} false(scriptM false), {scriptH}_{q}^{c} false(scriptM false))

and the preceding inequality becomes:

0true \int_{0}^{\infty} normalΦ ([], t^{- 1 / p}, J, (t^{1 / p - 1 / q}, v (t^{1 / p - 1 / q}); H_{p}^{c} (M), H_{q}^{c} (M))) d t ⩽ C_{...}

…”

Section: Applications To Noncommutative Burkholder/rosenthal Inequalimentioning

confidence: 99%

See 3 more Smart Citations

Noncommutative Davis type decompositions and applications

Randrianantoanina

2018

J. London Math. Soc.

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Modular inequalities in martingale Orlicz–Karamata spaces

Zhou

Jiao

2018

Mathematische Nachrichten

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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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Noncommutative Burkholder/Rosenthal inequalities associated with convex functions

Cited by 22 publications

References 45 publications

Noncommutative Davis type decompositions and applications

Noncommutative Davis type decompositions and applications

Modular inequalities in martingale Orlicz–Karamata spaces

Operator-Valued Triebel–Lizorkin Spaces

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