scholarly journals An equivalent quasinorm for the Lipschitz space of noncommutative martingales

2020 ◽  
Vol 18 (1) ◽  
pp. 1281-1291
Author(s):  
Congbian Ma ◽  
Yanbo Ren
Keyword(s):  
Hardy Space ◽  
Duality Theorem ◽  
Lipschitz Space ◽  
Martingale Hardy Space ◽  
Noncommutative Martingale

Abstract In this paper, an equivalent quasinorm for the Lipschitz space of noncommutative martingales is presented. As an application, we obtain the duality theorem between the noncommutative martingale Hardy space {h}_{p}^{c}( {\mathcal M} ) (resp. {h}_{p}^{r}( {\mathcal M} ) ) and the Lipschitz space {\lambda }_{\beta }^{c}( {\mathcal M} ) (resp. {\lambda }_{\beta }^{r}( {\mathcal M} ) ) for 0\lt p\lt 1 , \beta =\tfrac{1}{p}-1 . We also prove some equivalent quasinorms for {h}_{p}^{c}( {\mathcal M} ) and {h}_{p}^{r}( {\mathcal M} ) for p=1 or 2\lt p\lt \infty .

Interpolation on weak martingale Hardy space

2009 ◽  
Vol 25 (8) ◽  
pp. 1297-1304 ◽  
Author(s):  
Yong Jiao ◽  
Wei Chen ◽  
Pei De Liu
Keyword(s):  
Hardy Space ◽  
Martingale Hardy Space

UNCONDITIONAL CONVERGENCE AND UNCONDITIONAL BASES IN HARDY SPACES

2013 ◽  
Vol 11 (04) ◽  
pp. 1350016
Author(s):  
RONG-QING JIA
Keyword(s):  
Hardy Space ◽  
Unconditional Basis ◽  
Lipschitz Space ◽  
Unconditional Convergence ◽  
Projection Operators ◽  
General Criterion ◽  
Positive Order ◽  
Piecewise Constant ◽  
Unconditional Bases ◽  
High Smoothness

This paper is devoted to a study of unconditional convergence of series in the Hardy space H1(ℝs) and unconditional bases for H1(ℝs). First, we use quasi-projection operators from approximation theory to give a very general criterion for unconditional convergence in H1. Second, we prove that a system of wavelets forms an unconditional basis of the Hardy space H1, provided the dual wavelet lies in a Lipschitz space of positive order. In particular, for H1(ℝ) we construct an unconditional basis consisting of piecewise constant functions. Third, we demonstrate that our conditions for unconditional bases are sharp by showing that, if the dual refinable function is the characteristic function of the interval [0, 1), then the corresponding system of wavelets does not form an unconditional basis for H1(ℝ), even though the wavelet itself could have arbitrarily high smoothness.

scholarly journals Hoeffding decomposition in $$H^1$$ spaces

2020 ◽  
Author(s):  
Maciej Rzeszut ◽  
Michał Wojciechowski
Keyword(s):  
Hilbert Space ◽  
Hardy Space ◽  
Martingale Hardy Space ◽  
Hoeffding Decomposition

Abstract The well known result of Bourgain and Kwapień states that the projection $$P_{\le m}$$ P ≤ m onto the subspace of the Hilbert space $$L^2\left( \Omega ^\infty \right) $$ L 2 Ω ∞ spanned by functions dependent on at most m variables is bounded in $$L^p$$ L p with norm $$\le c_p^m$$ ≤ c p m for $$1<p<\infty $$ 1 < p < ∞ . We will be concerned with two kinds of endpoint estimates. We prove that $$P_{\le m}$$ P ≤ m is bounded on the space $$H^1\left( {\mathbb {D}}^\infty \right) $$ H 1 D ∞ of functions in $$L^1\left( {\mathbb {T}}^\infty \right) $$ L 1 T ∞ analytic in each variable. We also prove that $$P_{\le 2}$$ P ≤ 2 is bounded on the martingale Hardy space associated with a natural double-indexed filtration and, more generally, we exhibit a multiple indexed martingale Hardy space which contains $$H^1\left( {\mathbb {D}}^\infty \right) $$ H 1 D ∞ as a subspace and $$P_{\le m}$$ P ≤ m is bounded on it.

scholarly journals Interpolation between noncommutative martingale Hardy and BMO spaces: the case

2021 ◽  
pp. 1-45
Author(s):  
Narcisse Randrianantoanina
Keyword(s):  
Atomic Decomposition ◽  
Full Range ◽  
Bmo Space ◽  
Complex Interpolation ◽  
Complex Method ◽  
Von Neumann ◽  
Martingale Hardy Space ◽  
Noncommutative Martingale ◽  
Real Method ◽  
Bmo Spaces

Abstract 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.

Bilinear decompositions of products of local Hardy and Lipschitz or BMO spaces through wavelets

2018 ◽  
Vol 20 (03) ◽  
pp. 1750025 ◽  
Author(s):  
Jun Cao ◽  
Luong Dang Ky ◽  
Dachun Yang
Keyword(s):  
Hardy Space ◽  
Orlicz Function ◽  
Lipschitz Space ◽  
Bmo Space ◽  
Product Spaces ◽  
Dual Spaces ◽  
Local Hardy Space ◽  
Bmo Spaces

Let [Formula: see text] and [Formula: see text] be the local Hardy space in the sense of D. Goldberg. In this paper, the authors establish two bilinear decompositions of the product spaces of [Formula: see text] and their dual spaces. More precisely, the authors prove that [Formula: see text] and, for any [Formula: see text], [Formula: see text], where [Formula: see text] denotes the local BMO space, [Formula: see text], for any [Formula: see text] and [Formula: see text], the inhomogeneous Lipschitz space and [Formula: see text] a variant of the local Orlicz–Hardy space related to the Orlicz function [Formula: see text] for any [Formula: see text] which was introduced by Bonami and Feuto. As an application, the authors establish a div-curl lemma at the endpoint case.

scholarly journals On the duality of some martingale spaces

1992 ◽  
Vol 45 (1) ◽  
pp. 43-52 ◽  
Author(s):  
N.L. Bassily ◽  
A.M. Abdel-Fattah
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Dual Space ◽  
Young Function ◽  
Martingale Hardy Space ◽  
Finite Power ◽  
Young Functions

Fefferman has proved that the dual space of the martingale Hardy space H1 is the BMO1-space. Garsia went further and proved that the dual of Hp is the so-called martingale Kp-space, where p and q are two conjugate numbers and 1 ≤ p < 2.The martingale Hardy spaces HΦ with general Young function Φ, were investigated by Bassily and Mogyoródi. In this paper we show that the dual of the martingale Hardy space HΦ is the martingale Hardy space HΦ where (Φ, Ψ) is a pair of conjugate Young functions such that both Φ and Ψ have finite power. Moreover, two other remarkable dualities are presented.

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