Atomic Decomposition and Boundedness of Operators on Weighted Hardy Spaces

2012 ◽  
Vol 55 (2) ◽  
pp. 303-314 ◽  
Author(s):  
Yongsheng Han ◽  
Ming-Yi Lee ◽  
Chin-Cheng Lin
Keyword(s):  
Linear Operator ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Weighted Hardy Spaces ◽  
Distribution Sense ◽  
Uniformly Bounded

AbstractIn this article, we establish a new atomic decomposition for , where the decomposition converges in -norm rather than in the distribution sense. As applications of this decomposition, assuming that T is a linear operator bounded on and 0 < p ≤ 1, we obtain (i) if T is uniformly bounded in -norm for all w-p-atoms, then T can be extended to be bounded from to ; (ii) if T is uniformly bounded in -norm for all w-p-atoms, then T can be extended to be bounded on ; (iii) if T is bounded on , then T can be extended to be bounded from to .

scholarly journals On Convexity of Composition and Multiplication Operators on Weighted Hardy Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
Karim Hedayatian ◽  
Lotfollah Karimi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Composition Operator ◽  
Hardy Spaces ◽  
Bounded Linear Operator ◽  
Sufficient Conditions ◽  
Multiplication Operators ◽  
Weighted Hardy Spaces ◽  
Bounded Linear ◽  
Necessary And Sufficient

A bounded linear operatorTon a Hilbert spaceℋ, satisfying‖T2h‖2+‖h‖2≥2‖Th‖2for everyh∈ℋ, is called a convex operator. In this paper, we give necessary and sufficient conditions under which a convex composition operator on a large class of weighted Hardy spaces is an isometry. Also, we discuss convexity of multiplication operators.

Fourier operators on weighted Hardy spaces

1987 ◽  
Vol 101 (1) ◽  
pp. 113-121
Author(s):  
Hans P. Heinig
Keyword(s):  
Fourier Transform ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Integral Operators ◽  
Muckenhoupt Weight ◽  
Hardy’S Inequality ◽  
Weighted Hardy Spaces ◽  
Hardy's Inequality ◽  
The Fourier Transform

AbstractIn this note we utilize the atomic decomposition of weighted Hardy spaces to prove weighted versions of Hardy's inequality for the Fourier transform with Muckenhoupt weight. The result extends to certain integral operators with homogeneous kernels of degree −1.

scholarly journals The Boundedness of Some Integral Operators on Weighted Hardy Spaces Associated with Schrödinger Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Hua Wang
Keyword(s):  
Integral Operator ◽  
Hardy Spaces ◽  
Fractional Integral ◽  
Atomic Decomposition ◽  
Integrable Function ◽  
Integral Operators ◽  
Weighted Hardy Spaces ◽  
Imaginary Power ◽  
Locally Integrable Function

LetL=-Δ+Vbe a Schrödinger operator acting onL2(Rn),n≥1, whereV≢0is a nonnegative locally integrable function onRn. In this paper, we will first define molecules for weighted Hardy spacesHLp(w)  (0<p≤1)associated withLand establish their molecular characterizations. Then, by using the atomic decomposition and molecular characterization ofHLp(w), we will show that the imaginary powerLiγis bounded onHLp(w)forn/(n+1)<p≤1, and the fractional integral operatorL-α/2is bounded fromHLp(w)toHLq(wq/p), where0<α<min{n/2,1},n/(n+1)<p≤n/(n+α), and1/q=1/p-α/n.

A note on Hardy spaces and bounded linear operators

2018 ◽  
Vol 25 (1) ◽  
pp. 73-76
Author(s):  
Pablo Rocha
Keyword(s):  
Linear Operator ◽  
Hardy Spaces ◽  
Bounded Linear Operator ◽  
Atomic Decomposition ◽  
Bounded Operator ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear

AbstractIn this note we show that if{f\in H^{p}(\mathbb{R}^{n})\cap L^{s}(\mathbb{R}^{n})}, where{0<p\leq 1<s<\infty}, then there exists a{(p,\infty)}-atomic decomposition which converges tofin{L^{s}(\mathbb{R}^{n})}. From this result, we obtain that a bounded linear operatorTon{L^{s}(\mathbb{R}^{n})}can be extended to a bounded operator from{H^{p}(\mathbb{R}^{n})}into{L^{p}(\mathbb{R}^{n})}if and only ifTis bounded uniformly in{L^{p}}norm on all{(p,\infty)}-atoms. A similar result is also obtained from{H^{p}(\mathbb{R}^{n})}into{H^{p}(\mathbb{R}^{n})}.

scholarly journals Variable Anisotropic Hardy Spaces with Variable Exponents

2021 ◽  
Vol 9 (1) ◽  
pp. 65-89
Author(s):  
Zhenzhen Yang ◽  
Yajuan Yang ◽  
Jiawei Sun ◽  
Baode Li
Keyword(s):  
Hardy Spaces ◽  
Variable Exponent ◽  
Atomic Decomposition ◽  
Integral Operators ◽  
Moment Condition ◽  
Variable Exponents ◽  
Hölder Continuous ◽  
Multi Level ◽  
The Moment ◽  
Variable Anisotropy

Abstract Let p(·) : ℝ n → (0, ∞] be a variable exponent function satisfying the globally log-Hölder continuous and let Θ be a continuous multi-level ellipsoid cover of ℝ n introduced by Dekel et al. [12]. In this article, we introduce highly geometric Hardy spaces Hp (·)(Θ) via the radial grand maximal function and then obtain its atomic decomposition, which generalizes that of Hardy spaces Hp (Θ) on ℝ n with pointwise variable anisotropy of Dekel et al. [16] and variable anisotropic Hardy spaces of Liu et al. [24]. As an application, we establish the boundedness of variable anisotropic singular integral operators from Hp (·)(Θ) to Lp (·)(ℝ n ) in general and from Hp (·)(Θ) to itself under the moment condition, which generalizes the previous work of Bownik et al. [6] on Hp (Θ).

Hardy spaces on ZN

2002 ◽  
Vol 132 (1) ◽  
pp. 25-43 ◽  
Author(s):  
Santiago Boza ◽  
María J. Carro
Keyword(s):  
Maximal Function ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Classical Case ◽  
Positive Answer ◽  
Riesz Transforms ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Definition Of ◽  
Open Question

The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of Hp(ZN) in terms of an atomic decomposition.Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms.In this paper, we give a positive answer to this question.

scholarly journals Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales

2010 ◽  
Vol 258 (7) ◽  
pp. 2483-2505 ◽  
Author(s):  
Turdebek N. Bekjan ◽  
Zeqian Chen ◽  
Mathilde Perrin ◽  
Zhi Yin
Keyword(s):  
Hardy Spaces ◽  
Atomic Decomposition
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