scholarly journals The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents

2020 ◽  
Vol 18 (1) ◽  
pp. 434-447
Author(s):  
Qingdong Guo ◽  
Wenhua Wang
Keyword(s):  
Hardy Space ◽  
Molecular Characterization ◽  
Hardy Spaces ◽  
Variable Exponents ◽  
Herz Type Hardy Space

Abstract In this article, the authors establish the characterizations of a class of anisotropic Herz-type Hardy spaces with two variable exponents associated with a non-isotropic dilation on {{\mathbb{R}}}^{n} in terms of molecular decompositions. Using the molecular decompositions, the authors obtain the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy space with two variable exponents.

scholarly journals Molecular Characterization of Hardy Spaces Associated with Twisted Convolution

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Jizheng Huang ◽  
Yu Liu
Keyword(s):  
Hardy Space ◽  
Molecular Characterization ◽  
Hardy Spaces ◽  
Riesz Transform ◽  
Twisted Convolution

We give a molecular characterization of the Hardy space associated with twisted convolution. As an application, we prove the boundedness of the local Riesz transform on the Hardy space.

Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators

2015 ◽  
Vol 67 (5) ◽  
pp. 1161-1200 ◽  
Author(s):  
Junqiang Zhang ◽  
Jun Cao ◽  
Renjin Jiang ◽  
Dachun Yang
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Riesz Transform ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic Operator ◽  
Muckenhoupt Class ◽  
Degenerate Elliptic ◽  
Function Characterization

AbstractLet w be either in the Muckenhoupt class of A2(ℝn) weights or in the class of QC(ℝn) weights, and let be the degenerate elliptic operator on the Euclidean space ℝn, n ≥ 2. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space associated with , and when with , the authors prove that the associated Riesz transform is bounded from to the weighted classical Hardy space .

Molecular characterization of anisotropic Musielak–Orlicz Hardy spaces and their applications

2016 ◽  
Vol 32 (11) ◽  
pp. 1391-1414 ◽  
Author(s):  
Bao De Li ◽  
Xing Ya Fan ◽  
Zun Wei Fu ◽  
Da Chun Yang
Keyword(s):  
Molecular Characterization ◽  
Hardy Spaces

The molecular characterization of weighted Hardy spaces

2001 ◽  
Vol 44 (2) ◽  
pp. 201-211 ◽  
Author(s):  
Xingmin Li ◽  
Lizhong Peng
Keyword(s):  
Molecular Characterization ◽  
Hardy Spaces ◽  
Weighted Hardy Spaces

scholarly journals Fredholm Weighted Composition Operator on Weighted Hardy Space

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Liankuo Zhao
Keyword(s):  
Hardy Space ◽  
Composition Operator ◽  
Hardy Spaces ◽  
Weighted Composition Operator ◽  
Weighted Hardy Space ◽  
Weighted Hardy Spaces ◽  
Weighted Composition

This paper gives a unified characterization of Fredholm weighted composition operator on a class of weighted Hardy spaces.

scholarly journals Anisotropic Hardy Spaces of Musielak-Orlicz Type with Applications to Boundedness of Sublinear Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Baode Li ◽  
Dachun Yang ◽  
Wen Yuan
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Orlicz Function ◽  
Real Variable ◽  
Sublinear Operator ◽  
Atomic Decompositions ◽  
Anisotropic Hardy Space ◽  
Sublinear Operators ◽  
Uniformly Bounded

Letφ:ℝn×[0,∞)→[0,∞)be a Musielak-Orlicz function andAan expansive dilation. In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type,HAφ(ℝn), via the grand maximal function. The authors then obtain some real-variable characterizations ofHAφ(ℝn)in terms of the radial, the nontangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaceHAp(ℝn)withp∈(0,1]and are new even for its weighted variant. Finally, the authors characterize these spaces by anisotropic atomic decompositions. The authors also obtain the finite atomic decomposition characterization ofHAφ(ℝn), and, as an application, the authors prove that, for a given admissible triplet(φ,q,s), ifTis a sublinear operator and maps all(φ,q,s)-atoms withq<∞(or all continuous(φ,q,s)-atoms withq=∞) into uniformly bounded elements of some quasi-Banach spacesℬ, thenTuniquely extends to a bounded sublinear operator fromHAφ(ℝn)toℬ. These results are new even for anisotropic Orlicz-Hardy spaces onℝn.

A new molecular characterization of diagonal anisotropic Musielak-Orlicz Hardy spaces

2021 ◽  
Vol 166 ◽  
pp. 102939
Author(s):  
Minfeng Liao ◽  
Jinxia Li ◽  
Bo Li ◽  
Baode Li
Keyword(s):  
Molecular Characterization ◽  
Hardy Spaces

scholarly journals Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2216
Author(s):  
Jun Liu ◽  
Long Huang ◽  
Chenlong Yue
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Lebesgue Space ◽  
Linear Operators ◽  
Mixed Norm ◽  
Expansive Matrix

Let p→∈(0,∞)n be an exponent vector and A be a general expansive matrix on Rn. Let HAp→(Rn) be the anisotropic mixed-norm Hardy spaces associated with A defined via the non-tangential grand maximal function. In this article, using the known atomic characterization of HAp→(Rn), the authors characterize this Hardy space via molecules with the best possible known decay. As an application, the authors establish a criterion on the boundedness of linear operators from HAp→(Rn) to itself, which is used to explore the boundedness of anisotropic Calderón–Zygmund operators on HAp→(Rn). In addition, the boundedness of anisotropic Calderón–Zygmund operators from HAp→(Rn) to the mixed-norm Lebesgue space Lp→(Rn) is also presented. The obtained boundedness of these operators positively answers a question mentioned by Cleanthous et al. All of these results are new, even for isotropic mixed-norm Hardy spaces on Rn.

Discrete Littlewood–Paley–Stein characterization of multi-parameter local Hardy spaces

2019 ◽  
Vol 31 (6) ◽  
pp. 1467-1488 ◽  
Author(s):  
Wei Ding ◽  
Guozhen Lu ◽  
Yueping Zhu
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Discrete Version ◽  
Square Functions ◽  
Nonlinear Anal ◽  
Local Hardy Space ◽  
Local Hardy Spaces ◽  
Identity Based ◽  
Calderón’S Identity

AbstractIn our recent work [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380], the multi-parameter local Hardy space {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} has been introduced by using the continuous inhomogeneous Littlewood–Paley–Stein square functions. In this paper, we will first establish the new discrete multi-parameter local Calderón’s identity. Based on this identity, we will define the local multi-parameter Hardy space {h_{\mathrm{dis}}^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} by using the discrete inhomogeneous Littlewood–Paley–Stein square functions. Then we prove that these two multi-parameter local Hardy spaces are actually the same. Moreover, the norms of the multi-parameter local Hardy spaces under the continuous and discrete Littlewood–Paley–Stein square functions are equivalent. This discrete version of the multi-parameter local Hardy space is also critical in establishing the duality theory of the multi-parameter local Hardy spaces.

scholarly journals Discrete Paraproduct Operators on Variable Hardy Spaces

2019 ◽  
Vol 63 (2) ◽  
pp. 304-317 ◽  
Author(s):  
Jian Tan
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Variable Exponent ◽  
Discrete Version ◽  
Variable Exponents ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Hölder Continuous ◽  
Analysis Techniques ◽  
Calderón’S Reproducing Formula

AbstractLet$p(\cdot ):\mathbb{R}^{n}\rightarrow (0,\infty )$be a variable exponent function satisfying the globally log-Hölder continuous condition. In this paper, we obtain the boundedness of paraproduct operators$\unicode[STIX]{x1D70B}_{b}$on variable Hardy spaces$H^{p(\cdot )}(\mathbb{R}^{n})$, where$b\in \text{BMO}(\mathbb{R}^{n})$. As an application, we show that non-convolution type Calderón–Zygmund operators$T$are bounded on$H^{p(\cdot )}(\mathbb{R}^{n})$if and only if$T^{\ast }1=0$, where$\frac{n}{n+\unicode[STIX]{x1D716}}<\text{ess inf}_{x\in \mathbb{R}^{n}}p\leqslant \text{ess sup}_{x\in \mathbb{R}^{n}}p\leqslant 1$and$\unicode[STIX]{x1D716}$is the regular exponent of kernel of$T$. Our approach relies on the discrete version of Calderón’s reproducing formula, discrete Littlewood–Paley–Stein theory, almost orthogonal estimates, and variable exponents analysis techniques. These results still hold for variable Hardy space on spaces of homogeneous type by using our methods.

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