Doobs inequality for non-commutative martingales

Junge, Marius

doi:10.1515/crll.2002.061

Cited by 182 publications

(349 citation statements)

References 23 publications

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“…These inequalities will be used for the pointwise convergence in the next section. We first recall the definition of the noncommutative maximal norm introduced by Pisier [34] and Junge [13]. Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ.…”

Section: Maximal Inequalitiesmentioning

confidence: 99%

Harmonic Analysis on Quantum Tori

Chen

Yin

2013

Commun. Math. Phys.

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Abstract. This paper is devoted to the study of harmonic analysis on quantum tori. We consider several summation methods on these tori, including the square Fejér means, square and circular Poisson means, and Bochner-Riesz means. We first establish the maximal inequalities for these means, then obtain the corresponding pointwise convergence theorems. In particular, we prove the noncommutative analogue of the classical Stein theorem on Bochner-Riesz means. The second part of the paper deals with Fourier multipliers on quantum tori. We prove that the completely bounded Lp Fourier multipliers on a quantum torus are exactly those on the classical torus of the same dimension. Finally, we present the Littlewood-Paley theory associated with the circular Poisson semigroup on quantum tori. We show that the Hardy spaces in this setting possess the usual properties of Hardy spaces, as one can expect. These include the quantum torus analogue of Fefferman's H 1 -BMO duality theorem and interpolation theorems. Our analysis is based on the recent developments of noncommutative martingale/ergodic inequalities and Littlewood-Paley-Stein theory.

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Section: Maximal Inequalitiesmentioning

confidence: 99%

Harmonic Analysis on Quantum Tori

Chen

Yin

2013

Commun. Math. Phys.

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“…This of course is part of the general development of noncommutative martingale theory for which we refer the reader to [27,15,17,29] for recent history and results. We will work with general hyperfinite finite von Neumann algebra M with increasing filtration of finite dimensional subalgebras (M n ) n≥1 .…”

Section: Introductionmentioning

confidence: 99%

“…Using duality and interpolations, we also obtain boundedness between various noncommutative Lorentz spaces. Moreover, they can be strengthened using the noncommutative maximal functions developed by Junge in [15] (see Theorem 2.2 and Theorem 2.9). These results go beyond Theorem 0.1 in two ways, they provide a unified approach to fractional integrals that are not restricted to dyadic martingales and also the method we use is general enough to include martingales that are not necessarily regular.…”

Section: Introductionmentioning

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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“…Combined with recent noncommutative martingale convergence results due to Junge [17] and to Defant and Junge [11], it gives the following result. …”

Section: Introductionmentioning

confidence: 56%

On ergodic theorems for free group actions on noncommutative spaces

Anantharaman-Delaroche

2006

Probab. Theory Relat. Fields

120

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Abstract. We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres s 2n of even radius. Here we study state preserving actions of free groups on a von Neumann algebra A and the behaviour of (s 2n (x)) for x in noncommutative spaces L p (A). For the Cesàro means 1 n n−1 k=0 s k and p = +∞, this problem was solved by Walker. Our approach is based on ideas of Bufetov. We prove a noncommutative version of Rota "Alternierende Verfahren" theorem. To this end, we introduce specific dilations of the powers of some noncommutative Markov operators.

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Doobs inequality for non-commutative martingales

Abstract: Let 1 ≤ p < ∞ and (x n ) n∈N be a sequence of positive elements in a noncommutative L p space and (E n ) n∈N be an increasing sequence of conditional expectations, then n

Cited by 182 publications

References 23 publications

Harmonic Analysis on Quantum Tori

Harmonic Analysis on Quantum Tori

Noncommutative fractional integrals

On ergodic theorems for free group actions on noncommutative spaces

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