John–Nirenberg inequality and atomic decomposition for noncommutative martingales

Hong, Guixiang; Mei, Tao

doi:10.1016/j.jfa.2012.05.013

Cited by 36 publications

(38 citation statements)

References 20 publications

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“…. On the other hand, since a column p-atom in the sense of [1,10] is in particular a column p-atomic block in our sense, we immediately find the following inequality…”

Section: Noncommutative Martingalesmentioning

confidence: 73%

Atomic blocks for noncommutative martingales

Conde-Alonso¹,

Parcet²

2016

Indiana Univ. Math. J.

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Given a probability space (Ω, Σ, µ), the Hardy space H 1 (Ω) which is associated to the martingale square function does not admit a classical atomic decomposition when the underlying filtration is not regular. In this paper we construct a decomposition of H 1 (Ω) into 'atomic blocks' in the spirit of Tolsa, which we will introduce for martingales. We provide three proofs of this result. Only the first one also applies to noncommutative martingales, the main target of this paper. The other proofs emphasize alternative approaches for commutative martingales. One might be well-known to experts, using a weaker notion of atom and approximation by atomic filtrations. The last one adapts Tolsa's argument replacing medians by conditional medians.

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“…. On the other hand, since a column p-atom in the sense of [1,10] is in particular a column p-atomic block in our sense, we immediately find the following inequality…”

Section: Noncommutative Martingalesmentioning

confidence: 73%

Atomic blocks for noncommutative martingales

Conde-Alonso¹,

Parcet²

2016

Indiana Univ. Math. J.

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“…We refer to [1,13] for more details on the concept of atomic decompositions for noncommutative martingales.…”

Section: Now We Apply Lemma 23(i) To Get That Since For Everymentioning

confidence: 99%

Noncommutative fractional integrals

Randrianantoanina

2016

Studia Math.

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Abstract. Let M be a hyperfinite finite von Nemann algebra and (M k ) k≥1 be an increasing filtration of finite dimensional von Neumann subalgebras of M. We investigate abstract fractional integrals associated to the filtration (M k ) k≥1 . For a finite noncommutative martingale x = (x k ) 1≤k≤n ⊆ L1(M) adapted to (M k ) k≥1 and 0 < α < 1, the fractional integral of x of order α is defined by setting:

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“…We have preferred to include this alternative argument using atomic decompositions. Still a third approach is possible using more recent atomic decompositions from [2,6]. This will be needed below for martingale transforms and paraproducts.…”

Section: Proof Of Theorem Aii) It Suffices To Showmentioning

confidence: 99%

“…Row atoms are defined to satisfy a = ea instead and r − atoms are defined similarly. We also refer to [6] for q-analogs of these notions. In the following result, we collect some norm equivalences coming from atomic decompositions and John-Nirenberg type inequalities.…”

Section: Proof Of Theorem Cmentioning

confidence: 99%

Calderón–Zygmund Operators Associated to Matrix-Valued Kernels

Hong¹,

López-Sánchez²,

Martell³

et al. 2012

International Mathematics Research Notices

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Abstract. Calderón-Zygmund operators with noncommuting kernels may fail to be Lp-bounded for p = 2, even for kernels with good size and smoothness properties. Matrix-valued paraproducts, Fourier multipliers on group vNa's or noncommutative martingale transforms are frameworks where we find such difficulties. We obtain weak type estimates for perfect dyadic CZO's and cancellative Haar shifts associated to noncommuting kernels in terms of a row/column decomposition of the function. Arbitrary CZO's satisfy H 1 → L 1 type estimates. In conjunction with L∞ → BMO, we get certain row/column Lp estimates. Our approach also applies to noncommutative paraproducts or martingale transforms with noncommuting symbols/coefficients. Our results complement recent results of Junge, Mei, Parcet and Randrianantoanina.

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John–Nirenberg inequality and atomic decomposition for noncommutative martingales

Cited by 36 publications

References 20 publications

Atomic blocks for noncommutative martingales

Atomic blocks for noncommutative martingales

Noncommutative fractional integrals

Calderón–Zygmund Operators Associated to Matrix-Valued Kernels

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