A noncommutative version of the John-Nirenberg theorem

Junge, Marius; Musat, Magdalena

doi:10.1090/s0002-9947-06-03999-7

Cited by 33 publications

(36 citation statements)

References 42 publications

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“…For more information on noncommutative martingale BM O-spaces, we refer to [27,17,24,16]. It was shown in [27] that the classical Feffermann duality is still valid in the noncommutative settings.…”

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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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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“…The aim of this paper is to prove various John-Nirenberg inequalities on symmetric spaces of noncommutative martingales, extending the results obtained in [JM07,HM12] in the L p -case. This follows the current line of investigations in noncommutative martingale theory.…”

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confidence: 77%

“…If E = L p (0, ∞), these spaces are bmo c p (M), bmo r p (M) and bmo p (M) defined in [HM12] (cf. [JM07]) and in particular, for E = L 2 (0, ∞), they are exactly the spaces bmo c (M), bmo r (M) and bmo(M).…”

Section: This Completes the Proofmentioning

confidence: 98%

“…The John-Nirenberg inequality plays a permanent role in harmonic analysis and martingale theory, which provides a subtle characterization of BMO space. Based on the fundamental work of Pisier and Xu [PX97] on noncommutative martingale Hardy and BMO spaces, a noncommutative version of the John-Nirenberg inequality was established by Junge and Musat [JM07]. Later, Hong and Mei [HM12] proved several versions of the John-Nirenberg inequality for noncommutative martingales and provided an application to the atomic decomposition of noncommutative martingale Hardy space h 1 using q-atoms as building blocks, extending the work of [BCPY10].…”

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confidence: 99%

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Interpolation and the John–Nirenberg inequality on symmetric spaces of noncommutative martingales

et al. 2022

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We prove various John-Nirenberg inequalities on symmetric spaces of noncommutative martingales, including the crude and fine versions, which extend the corresponding results of Junge and Musat ( 2007) and Hong and Mei (2012) in the Lp-case.As an application, we provide the atomic decomposition of a noncommutative martingale Hardy space h1 using symmetric atoms as building blocks, and give the boundedness of paraproducts on symmetric spaces of noncommutative martingales.

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“…For more information on non-commutative martingale BMO-spaces, we refer to the articles [20,10,15,9]. Special attention will be given to the subspace called vanishing mean oscillation, denoted by VMO(M), and defined as the closure (for the BMO-norm) of the linear subspace of those x ∈ BMO(M) for which E n (x) = x for some n ∈ N.…”

Section: Notation and Preliminary Definitionsmentioning

confidence: 99%

Non-commutative martingale VMO-spaces

Randrianantoanina

2009

Studia Math.

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Abstract. We study Banach space properties of non-commutative martingale VMOspaces associated with general von Neumann algebras. More precisely, we obtain a version of the classical Kadets-Pełczyński dichotomy theorem for subspaces of non-commutative martingale VMO-spaces. As application we prove that if M is hyperfinite then the noncommutative martingale VMO-space associated with a filtration of finite-dimensional von Neumannn subalgebras of M has property (u).1. Introduction. The space of functions of bounded mean oscillation generally referred to as BMO-space has been instrumental in several aspects of analysis. Its martingale version plays an equally important role in probability.In this paper, we analyze subspaces of BMO-spaces related to non-commutative martingales. Our main motivation comes primarily from a paper by Müller and Schechtman [14] who studied structural properties of closed subspaces of dyadic martingale VMO (vanishing mean oscillation) as Banach spaces. More precisely, they provided, among other things, a version of the classical Kadets-Pełczyński dichotomy theorem for closed subspaces of dyadic martingale VMO-spaces. In order to explain the details, we first recall the celebrated Kadets-Pełczyński dichotomy theorem, which states that every closed subspace of L p (0, 1), 2 < p < ∞, either is isomorphic to a Hilbert space or contains a subspace which is isomorphic to l p . This dichotomy plays a crucial role in the development of L p -spaces and the theory of function spaces in general. Non-commutative analogues of the Kadets-Pełczyński dichotomy has been considered by several authors with the most general result obtained by Raynaud and Xu (see [21]) in the context of Haagerup L p -spaces when 2 ≤ p < ∞. Clearly, the dichotomy does not extend to closed subspaces of L ∞ (0, 1) or any C(K)-spaces in general. As a substitute, the following result was obtained by Müller and Schechtman:

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A noncommutative version of the John-Nirenberg theorem

Abstract: Abstract. We prove a noncommutative version of the John-Nirenberg theorem for nontracial filtrations of von Neumann algebras. As an application, we obtain an analogue of the classical large deviation inequality for elements of the associated BM O space.

Cited by 33 publications

References 42 publications

Noncommutative fractional integrals

Noncommutative fractional integrals

Interpolation and the John–Nirenberg inequality on symmetric spaces of noncommutative martingales

Non-commutative martingale VMO-spaces

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