Rosenthal Type inequalities for free chaos.

By Marius Junge, Javier Parcet, Quanhua Xu. 2005

47 citations recorded by the source corpus.

Let A denote the reduced amalgamated free product of a family A 1 , A 2 An of von Neumann algebras over a von Neumann subalgebra B with respect to normal faithful conditional expectations E k : A k → B. We investigate the norm in Lp (A) of homogeneous polynomials of a given degree d. We first generalize Voiculescu's inequality to arbitrary degree d ≥ 1 and indices 1 ≤ p ≤ ∞. This can be regarded as a free analogue of the classical Rosenthal inequality. Our second result is a length-reduction formula from which we generalize recent results of Pisier, Ricard and the authors. All constants in our estimates are independent of n so that we may consider infinitely many free factors. As applications, we study square functions of free martingales. More precisely, we show that, in contrast with the Khintchine and Rosenthal inequalities, the free analogue of the Burkholder-Gundy inequalities does not hold in L ∞ (A). At the end of the paper we also consider Khintchine type inequalities for Shlyakhtenko's generalized circular systems.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6db90997737c565180c49b739cf90d9387cafbb0

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Noncommutative Burkholder/Rosenthal inequalities II: Applications (2007, background): “e second equivalence remains true for p= 1. Let us point out that a similar statement holds for the generalized circular system in (1.1). In this case, all constants are universal (see [X2]; see also [JPX] for the q-case). We should emphasize that all these Khintchine type inequalities have interest”
Khintchine type inequalities for reduced free products and applications (2005, background): “(K∗(d−k−1) 1 ,L k 1)⊗ℓ∞((A ′′ i)). By virtue of elementary properties of Haagerup tensor product, there is a natural embedding of Xdin Xd. The following is the Khintchine inequality for Σd. Note that [22] contains a variant of this inequality as well as its generalization to noncommutative Lp-spaces”
Pseudo-localization of singular integrals and noncommutative Calderon-Zygmund theory (2007, background): “We find in the recent literature noncommutative analogs of Khintchine and Rosenthal inequalities, a settled noncommutative theory of martingale inequalities, new results on Fourier/Schur multipliers, matrix Ap weights and a sharpened Carleson embedding theorem, see [20, 29, 31, 43, 47, 57] and the r”
Singular integrals in quantum Euclidean spaces (2017, background): “from the operator-valued form of Voiculescu’s inequality [46, 71].”
Pseudo-Localization of Singular Integrals and Noncommutative Littlewood–Paley Inequalities (2008, background): “The iteration of Khintchine inequalities was the basis in [25] for some multilinear generalizations that where further explored in [14, 32].”