BMO spaces associated with semigroups of operators

Junge, Marius; Mei, Tao

doi:10.1007/s00208-011-0657-0

Cited by 57 publications

(118 citation statements)

References 33 publications

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“…The above definitions are motivated by Hardy spaces of noncommutative martingales ( [22,37]) and of quantum Markov semigroups ( [15,17,27]). The main results of this section are summarized in the following statement which shows that the Hardy spaces on T d θ possess the properties of the usual Hardy spaces, as expected.…”

Section: Hardy Spacesmentioning

confidence: 99%

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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: Hardy Spacesmentioning

confidence: 99%

“…[15,27,28,16,17]). One can also include in this topic the very fresh promising direction of research on the Calderón-Zygmund singular integral operators in the noncommutative setting (cf.…”

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confidence: 99%

Harmonic Analysis on Quantum Tori

Chen

Yin

2013

Commun. Math. Phys.

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“…We take the natural definition using a faithful normal state which is not necessarily tracial anymore as a starting point. We extend interpolation results from [JuMe12,Theorem 5.2] to the arbitrary setting under a modularity assumption on the Markov semi-group. The modularity assumption is necessary to carry out our proof through Haagerup's reduction method and due to the fact that the probabilistic martingale BMO-spaces in [JuPe14] are studied (in principle only) in the tracial setting.…”

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confidence: 88%

“…In [JuMe12] Junge and Mei pursued the theory of non-commutative semi-group BMO-spaces associated with non-commutative measure spaces. They introduce several notions of BMO starting from a Markov semi-group on a tracial von Neumann algebra.…”

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confidence: 99%

Harmonic analysis and BMO-spaces of free Araki--Woods factors

Caspers

2019

Studia Math.

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“…Non-commutative finite BMO-spaces. We recall the following from [JuMe12]. Fix a finite von Neumann algebra (M, τ ).…”

Section: Preliminariesmentioning

confidence: 99%

BMO-estimates for non-commutative vector valued Lipschitz functions

Caspers

Junge

Sukochev

et al. 2020

Journal of Functional Analysis

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We construct Markov semi-groups T and associated BMO-spaces on a finite von Neumann algebra (M, τ ) and obtain results for perturbations of commutators and non-commutative Lipschitz estimates. In particular, we prove that for any A ∈ M self-adjoint and f :We obtain an analogue of this result for more general von Neumann valuedfunctions f : R n → N by imposing Hörmander-Mikhlin type assumptions on f .In establishing these result we show that Markov dilations of Markov semigroups have certain automatic continuity properties. We also show that Markov semi-groups of double operator integrals admit (standard and reversed) Markov dilations.

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BMO spaces associated with semigroups of operators

Cited by 57 publications

References 33 publications

Harmonic Analysis on Quantum Tori

Harmonic Analysis on Quantum Tori

Harmonic analysis and BMO-spaces of free Araki--Woods factors

BMO-estimates for non-commutative vector valued Lipschitz functions

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