Majorization, interpolation and noncommutative Khinchin inequalities

Cadilhac, Léonard

doi:10.4064/sm181217-30-1

Cited by 7 publications

(8 citation statements)

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“…In contrast to that, from Theorem 1.1 in [9] (see also [3,Remark 4.7]) it follows that the couple (L p , L q ) of measurable functions on the semi-axis (0, ∞) with the Lebesgue measure has the Calderón-Mityagin property for all 0 < p < q ≤ ∞. Observe also that some other partial results for the latter couples in the non-Banach case were obtained somewhat earlier in [18,29,2,8].…”

Section: Introductionmentioning

confidence: 83%

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A description of interpolation spaces for quasi-Banach couples by real $K$-method

Асташкин¹,

Nilsson²

2021

Preprint

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The main aim of this paper is to develop a general approach, which allows to extend the basics of Brudnyi-Kruglyak interpolation theory to the realm of quasi-Banach lattices. We prove that all K-monotone quasi-Banach lattices with respect to a L-convex quasi-Banach lattice couple have in fact a stronger property of the so-called K(p, q)-monotonicity for some 0 < q ≤ p ≤ 1, which allows us to get their description by the real K-method. Moreover, we obtain a refined version of the K-divisibility property for Banach lattice couples and then prove an appropriate version of this property for L-convex quasi-Banach lattice couples. The results obtained are applied to refine interpolation properties of couples of sequence l p -and function L p -spaces, considered for the full range 0 < p < ∞.

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Section: Introductionmentioning

confidence: 83%

“…[6]). Recently, a similar investigation has been undertaken in a series of papers (see [3], [9], [8], [2], [18]) in the quasi-Banach setting. Results obtained in the previous sections allow us to push this research somewhat forward.…”

Section: Applicationsmentioning

confidence: 99%

A description of interpolation spaces for quasi-Banach couples by real $K$-method

Асташкин¹,

Nilsson²

2021

Preprint

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“…That is, equipped with the norm . According to [5, Corollary 4.6], . Let be a Banach function space so that .…”

Section: Applications To Martingale Inequalitiesmentioning

confidence: 99%

Interpolation between noncommutative martingale Hardy and BMO spaces: the case

Randrianantoanina¹

2021

Can. J. Math.-J. Can. Math.

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Let $\mathcal {M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal {M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal {M}$ . For $0<p <\infty $ , let $\mathsf {h}_p^c(\mathcal {M})$ denote the noncommutative column conditioned martingale Hardy space and $\mathsf {bmo}^c(\mathcal {M})$ denote the column “little” martingale BMO space associated with the filtration $(\mathcal {M}_n)_{n\geq 1}$ . We prove the following real interpolation identity: if $0<p <\infty $ and $0<\theta <1$ , then for $1/r=(1-\theta )/p$ , $$ \begin{align*} \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}), \end{align*} $$ with equivalent quasi norms. For the case of complex interpolation, we obtain that if $0<p<q<\infty $ and $0<\theta <1$ , then for $1/r =(1-\theta )/p +\theta /q$ , $$ \begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M}) \end{align*} $$ with equivalent quasi norms. These extend previously known results from $p\geq 1$ to the full range $0<p<\infty $ . Other related spaces such as spaces of adapted sequences and Junge’s noncommutative conditioned $L_p$ -spaces are also shown to form interpolation scale for the full range $0<p<\infty $ when either the real method or the complex method is used. Our method of proof is based on a new algebraic atomic decomposition for Orlicz space version of Junge’s noncommutative conditioned $L_p$ -spaces. We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.

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“…In contrast to that, every symmetric quasi-Banach sequence space E is an interpolation space with respect to the couple (ℓ 0 , ℓ ∞ ) (this can be obtained by obvious modifications of reasoning in the papers [26] and [2], where the analogous property is proved for the couple (L 0 , L ∞ ) on (0, ∞) and rearrangement invariant quasi-Banach function spaces). Some other partial results for the couples (ℓ p , ℓ q ), in the non-Banach case, were obtained more recently in [18,35,12,3,11]. Moreover, in [12], the above Arazy-Cwikel property has been proved for the couple (L 0 , L ∞ ) of measurable functions on the semi-axis (0, ∞) with the Lebesgue measure.…”

Section: Introductionmentioning

confidence: 98%

Arazy-Cwikel and Calderón-Mityagin type properties of the couples $(\ell^{p},\ell^{q})$, $0 \le p

Astashkin,

Cwikel,

Nilsson

2021

Preprint

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We establish Arazy-Cwikel type properties for the family of couples (ℓ p , ℓ q ), 0 ≤ p < q ≤ ∞, and show that (ℓ p , ℓ q ) is a Calderón-Mityagin couple if and only if q ≥ 1. Moreover, we identify interpolation orbits of elements with respect to this couple for all p and q such that 0 ≤ p < q ≤ ∞ and obtain a simple positive solution of a Levitina-Sukochev-Zanin problem, clarifying its connections with whether (ℓ p , ℓ q ) has the Calderón-Mityagin property or not.

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Majorization, interpolation and noncommutative Khinchin inequalities

Cited by 7 publications

References 0 publications

A description of interpolation spaces for quasi-Banach couples by real $K$-method

A description of interpolation spaces for quasi-Banach couples by real $K$-method

Interpolation between noncommutative martingale Hardy and BMO spaces: the case

Arazy-Cwikel and Calderón-Mityagin type properties of the couples $(\ell^{p},\ell^{q})$, $0 \le p

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