Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

2018 ◽  
Vol 61 (9) ◽  
pp. 1647-1664 ◽  
Author(s):  
Jiecheng Chen ◽  
Jiawei Dai ◽  
Dashan Fan ◽  
Xiangrong Zhu
Keyword(s):  
Hardy Spaces ◽  
Lebesgue Spaces ◽  
Hausdorff Operators

New variable martingale Hardy spaces

2021 ◽  
pp. 1-29
Author(s):  
Yong Jiao ◽  
Dan Zeng ◽  
Dejian Zhou
Keyword(s):  
Brownian Motion ◽  
Hardy Space ◽  
Hardy Spaces ◽  
Stochastic Integral ◽  
Lebesgue Spaces ◽  
Type Space ◽  
Variable Lebesgue Spaces ◽  
Martingale Inequalities

We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces $\mathcal {L}_{p(\cdot )}$ defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space $\mathcal {H}_{p(\cdot )}^{s}$ with $0<p_{-}\leq p_{+}\leq 1$ can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion.

scholarly journals Variable Exponent Spaces of Analytic Functions

2020 ◽  
Author(s):  
Gerardo A. Chacón ◽  
Gerardo R. Chacón
Keyword(s):  
Hardy Spaces ◽  
Measurable Function ◽  
Analytic Functions ◽  
Variable Exponent ◽  
Bergman Spaces ◽  
Lebesgue Spaces ◽  
Basic Properties ◽  
Variable Exponent Spaces ◽  
Real Functions ◽  
Spaces Of Analytic Functions

Variable exponent spaces are a generalization of Lebesgue spaces in which the exponent is a measurable function. Most of the research done in this topic has been situated under the context of real functions. In this work, we present two examples of variable exponent spaces of analytic functions: variable exponent Hardy spaces and variable exponent Bergman spaces. We will introduce the spaces together with some basic properties and the main techniques used in the context. We will show that in both cases, the boundedness of the evaluation functionals plays a key role in the theory. We also present a section of possible directions of research in this topic.

scholarly journals Hardy spaces in Reinhardt domains, and Hausdorff operators

2009 ◽  
Vol 53 (4) ◽  
pp. 1033-1049 ◽  
Author(s):  
L. Aizenberg ◽  
E. Liflyand
Keyword(s):  
Hardy Spaces ◽  
Reinhardt Domains ◽  
Hausdorff Operators

scholarly journals Hausdorff operators on holomorphic Hardy spaces and applications

2019 ◽  
Vol 150 (3) ◽  
pp. 1095-1112 ◽  
Author(s):  
Ha Duy Hung ◽  
Luong Dang Ky ◽  
Thai Thuan Quang
Keyword(s):  
Half Plane ◽  
Hardy Spaces ◽  
Hausdorff Operator ◽  
Upper Half Plane ◽  
Hausdorff Operators ◽  
Operator Norms ◽  
Image Position

AbstractThe aim of this paper is to characterize the non-negative functions φ defined on (0,∞) for which the Hausdorff operator $${\rm {\cal H}}_\varphi f(z) = \int_0^\infty f \left( {\displaystyle{z \over t}} \right)\displaystyle{{\varphi (t)} \over t}{\rm d}t$$is bounded on the Hardy spaces of the upper half-plane ${\rm {\cal H}}_a^p ({\open C}_ + )$, $p\in [1,\infty ]$. The corresponding operator norms and their applications are also given.

scholarly journals On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

2020 ◽  
Vol 32 (1) ◽  
pp. 111-119 ◽  
Author(s):  
Adolf R. Mirotin
Keyword(s):  
Present Article ◽  
Spectral Representation ◽  
Research Field ◽  
Hausdorff Operator ◽  
Lebesgue Spaces ◽  
Summation Methods ◽  
Hausdorff Operators ◽  
Active Research ◽  
Matrix Symbol

AbstractHausdorff operators originated from some classical summation methods. Now this is an active research field. In the present article, a spectral representation for multidimensional normal Hausdorff operator is given. We show that normal Hausdorff operator in {L^{2}(\mathbb{R}^{n})} is unitary equivalent to the operator of multiplication by some matrix-valued function (its matrix symbol) in the space {L^{2}(\mathbb{R}^{n};\mathbb{C}^{2^{n}})}. Several corollaries that show that properties of a Hausdorff operator are closely related to the properties of its symbol are considered. In particular, the norm and the spectrum of such operators are described in terms of the symbol.

The fractional Hausdorff operators on the Hardy spaces H p (ℝ n )

2016 ◽  
Vol 42 (1) ◽  
pp. 1-17 ◽  
Author(s):  
J. Chen ◽  
D. Fan ◽  
X. Lin ◽  
J. Ruan
Keyword(s):  
Hardy Spaces ◽  
Hausdorff Operators

scholarly journals Commutators of Littlewood-Paley gκ∗ $g_{\kappa}^{*} $-functions on non-homogeneous metric measure spaces

2017 ◽  
Vol 15 (1) ◽  
pp. 1283-1299 ◽  
Author(s):  
Guanghui Lu ◽  
Shuangping Tao
Keyword(s):  
Hardy Spaces ◽  
Lebesgue Space ◽  
Metric Measure Spaces ◽  
Type Condition ◽  
Lebesgue Spaces ◽  
Measure Spaces ◽  
Weak Lebesgue Space ◽  
Weak Lebesgue Spaces

Abstract The main purpose of this paper is to prove that the boundedness of the commutator $\mathcal{M}_{\kappa,b}^{*} $ generated by the Littlewood-Paley operator $\mathcal{M}_{\kappa}^{*} $ and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of $\mathcal{M}_{\kappa}^{*} $ satisfies a certain Hörmander-type condition, the authors prove that $\mathcal{M}_{\kappa,b}^{*} $ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from the space L log L(μ) to the weak Lebesgue space L1,∞(μ), and is bounded from the atomic Hardy spaces H1(μ) to the weak Lebesgue spaces L1,∞(μ).

scholarly journals Littlewood-Paley spaces

2011 ◽  
Vol 108 (1) ◽  
pp. 77 ◽  
Author(s):  
Kwok-Pun Ho
Keyword(s):  
Hardy Spaces ◽  
Orlicz Spaces ◽  
Morrey Spaces ◽  
Function Spaces ◽  
Unified Framework ◽  
Lebesgue Spaces ◽  
Banach Function Spaces

We introduce the Littlewood-Paley spaces in which the Lebesgue spaces, the Hardy spaces, the Orlicz spaces, the Lorentz-Karamata spaces, the r.-i. quasi-Banach function spaces and the Morrey spaces reside. The Littlewood-Paley spaces provide a unified framework for the study of some important function spaces arising in analysis.

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