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.

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 Conjugations in $$L^2(\mathcal {H})$$

2020 ◽  
Vol 92 (6) ◽  
Author(s):  
M. Cristina Câmara ◽  
Kamila Kliś-Garlicka ◽  
Bartosz Łanucha ◽  
Marek Ptak
Keyword(s):  
Hilbert Space ◽  
Hardy Space ◽  
Separable Hilbert Space ◽  
Model Space ◽  
Inner Function

AbstractConjugations commuting with $$\mathbf {M}_z$$ M z and intertwining $$\mathbf {M}_z$$ M z and $$\mathbf {M}_{{\bar{z}}}$$ M z ¯ in $$L^2(\mathcal {H})$$ L 2 ( H ) , where $$\mathcal {H}$$ H is a separable Hilbert space, are characterized. We also investigate which of them leave invariant the whole Hardy space $$H^2(\mathcal {H})$$ H 2 ( H ) or a model space $$K_\Theta =H^2(\mathcal {H})\ominus \Theta H^2(\mathcal {H})$$ K Θ = H 2 ( H ) ⊖ Θ H 2 ( H ) , where $$\Theta $$ Θ is a pure operator valued inner function.

scholarly journals Power series of several variables with condition of logarithmical convexity

2021 ◽  
Vol 8 (1) ◽  
pp. 49-62
Author(s):  
Alexandr V. Zheleznyak ◽  
Keyword(s):  
Hilbert Space ◽  
Hardy Space ◽  
Power Series ◽  
Unit Disc ◽  
Reproducing Kernel ◽  
Space Of Analytic Functions ◽  
Several Variables ◽  
Positive Coefficients ◽  
Classical Hardy Space ◽  
Reproducing Kernel Function

We obtain a new version of Hardy theorem about power series of several variables reciprocal to the power series with positive coefficients. We prove that if the sequence {as} = as1,s2,...,sn, ||s|| ≥ K satisfies condition of logarithmically convexity and the first coefficient a0 is sufficiently large then reciprocal power series has only negative coefficients {bs} = bs1,s2,...,sn, except b0,0,...,0 for any K. The classical Hardy theorem corresponds to the case K = 0, n = 1. Such results are useful in Nevanlinna - Pick theory. For example, if function k(x, y) can be represented as power series Σn≥0 an(x-y)n, an > 0, and reciprocal function 1/k(x,y) can be represented as power series Σn≥0 bn(x-y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna-Pick property. The reproducing kernel 1/1-x-y of the classical Hardy space H2(D) is a prime example for our theorems.

scholarly journals Fixed point theorems for generalized Lipschitzian semigroups in Banach spaces

1999 ◽  
Vol 22 (1) ◽  
pp. 119-129
Author(s):  
Balwant Singh Thakur ◽  
Jong Soo Jung
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Hardy Space ◽  
Banach Spaces ◽  
Sobolev Spaces ◽  
Fixed Point Theorems ◽  
Uniformly Convex ◽  
Uniformly Convex Banach Spaces

Fixed point theorems for generalized Lipschitzian semigroups are proved inp-uniformly convex Banach spaces and in uniformly convex Banach spaces. As applications, its corollaries are given in a Hilbert space, inLpspaces, in Hardy spaceHp, and in Sobolev spacesHk,p, for1<p<∞andk≥0.

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 Toeplitz Operators on the Symmetrized Bidisc

2020 ◽  
Author(s):  
Tirthankar Bhattacharyya ◽  
B Krishna Das ◽  
Haripada Sau
Keyword(s):  
Hilbert Space ◽  
Hardy Space ◽  
Unit Disc ◽  
Toeplitz Operators ◽  
Reproducing Kernel ◽  
Multiplication Operators ◽  
Algebraic Characterization ◽  
Hilbert Modules ◽  
Symmetrized Bidisc

Abstract The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results. (1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules. (2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators. (3) Theorem III gives several characterizations of an analytic Toeplitz operator. (4) Theorem IV characterizes asymptotic Toeplitz operators. (5) Theorem V is a commutant lifting theorem. (6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

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