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.

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).

scholarly journals Some spectral formulas for functions generated by differential and integral operators in Orlicz spaces

2021 ◽  
Vol 13 (2) ◽  
pp. 326-339
Author(s):  
H.H. Bang ◽  
V.N. Huy
Keyword(s):  
Fourier Transform ◽  
Orlicz Space ◽  
Differential Operators ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Young Function ◽  
The Fourier Transform

In this paper, we investigate the behavior of the sequence of $L^\Phi$-norm of functions, which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function $f$ is called its spectrum and denoted by sp$(f)$). With $Q$ being a polynomial, we introduce the notion of $Q$-primitives, which will return to the notion of primitives if ${Q}(x)= x$, and study the behavior of the sequence of norm of $Q$-primitives of functions in Orlicz space $L^\Phi(\mathbb R^n)$. We have the following main result: let $\Phi $ be an arbitrary Young function, ${Q}({\bf x} )$ be a polynomial and $(\mathcal{Q}^mf)_{m=0}^\infty \subset L^\Phi(\mathbb R^n)$ satisfies $\mathcal{Q}^0f=f, {Q}(D)\mathcal{Q}^{m+1}f=\mathcal{Q}^mf$ for $m\in\mathbb{Z}_+$. Assume that sp$(f)$ is compact and $sp(\mathcal{Q}^{m}f)= sp(f)$ for all $m\in \mathbb{Z}_+.$ Then $$ \lim\limits_{m\to \infty } \|\mathcal{Q}^m f\|_{\Phi}^{1/m}= \sup\limits_{{\bf x} \in sp(f)} \bigl|1/ {Q}({\bf x}) \bigl|. $$ The corresponding results for functions generated by differential operators and integral operators are also given.

BOUNDEDNESS OF SEVERAL INTEGRAL OPERATORS WITH BOUNDED VARIABLE KERNELS ON HARDY AND WEAK HARDY SPACES

2013 ◽  
Vol 24 (12) ◽  
pp. 1350095 ◽  
Author(s):  
HUA WANG
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Variable Kernel ◽  
Marcinkiewicz Integrals ◽  
Decomposition Theory ◽  
Logarithmic Type ◽  
Lipschitz Conditions

In this paper, by using the atomic decomposition theory of Hardy space H1(ℝn) and weak Hardy space WH1(ℝn), we give the boundedness properties of some operators with variable kernels such as singular integral operators, fractional integrals and parametric Marcinkiewicz integrals on these spaces, under certain logarithmic type Lipschitz conditions assumed on the variable kernel Ω(x, z).

scholarly journals The Boundedness on Mixed Hardy Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Wei Ding ◽  
Meidi Qin ◽  
Yueping Zhu
Keyword(s):  
Singular Integral ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Direct Method ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
A Direct Method

The boundedness of operators on Hardy spaces is usually given by atomic decomposition. In this paper, we obtain the boundedness of singular integral operators in mixed Journé class on mixed Hardy spaces by a direct method.

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 Some weighted estimates for imaginary powers of Laplace operators

2002 ◽  
Vol 65 (1) ◽  
pp. 129-135 ◽  
Author(s):  
Hendra Gunawan
Keyword(s):  
Laplace Operator ◽  
Singular Integral ◽  
Hardy Spaces ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Lebesgue Spaces ◽  
Weighted Hardy Spaces ◽  
Weighted Estimates ◽  
Imaginary Powers ◽  
The Laplace Operator

We study the boundedness of singular integral operators that are imaginary powers of the Laplace operator in Rn, especially from weighted Hardy spaces to weighted Lebesgue spaces where 0 < p ≤ 1. In particular, we prove some estimates for these operators when 0 < p ≤ 1 and w is in the Muckenhoupt's class Aq, for some q > 1.

Sign in / Sign up

Export Citation Format

Share Document