On the Nörlund logarithmic means with respect to Vilenkin system in the martingale Hardy space H 1

2017 ◽  
Vol 154 (2) ◽  
pp. 289-301
Author(s):  
L.-E. Persson ◽  
G. Tephnadze ◽  
P. Wall
Keyword(s):  
Hardy Space ◽  
Vilenkin System ◽  
Martingale Hardy Space ◽  
Logarithmic Means

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

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 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.

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 .

The maximal operator of the Marcinkiewicz-Fejér means of the 2-dimensional Vilenkin-Fourier series

2008 ◽  
Vol 45 (3) ◽  
pp. 321-331
Author(s):  
István Blahota ◽  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Hardy Space ◽  
Maximal Operator ◽  
Fejér Means

In this paper we prove that the maximal operator of the Marcinkiewicz-Fejér means of the 2-dimensional Vilenkin-Fourier series is not bounded from the Hardy space H2/3 ( G2 ) to the space L2/3 ( G2 ).

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