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.

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 .

scholarly journals LUSIN AREA FUNCTION AND MOLECULAR CHARACTERIZATIONS OF MUSIELAK–ORLICZ HARDY SPACES AND THEIR APPLICATIONS

2013 ◽  
Vol 15 (06) ◽  
pp. 1350029 ◽  
Author(s):  
SHAOXIONG HOU ◽  
DACHUN YANG ◽  
SIBEI YANG
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Carleson Measure ◽  
Dual Space ◽  
Growth Function ◽  
Area Function ◽  
Type Space ◽  
Lusin Area Function

Let φ : ℝn× [0,∞) → [0,∞) be a growth function such that φ(x, ⋅) is nondecreasing, φ(x, 0) = 0, φ(x, t) > 0 when t > 0, limt→∞φ(x, t) = ∞, and φ(⋅, t) is a Muckenhoupt A∞(ℝn) weight uniformly in t. In this paper, the authors establish the Lusin area function and the molecular characterizations of the Musielak–Orlicz Hardy space Hφ(ℝn) introduced by Luong Dang Ky via the grand maximal function. As an application, the authors obtain the φ-Carleson measure characterization of the Musielak–Orlicz BMO-type space BMOφ(ℝn), which was proved to be the dual space of Hφ(ℝn) by Luong Dang Ky.

A CHARACTERIZATION OF HARDY SPACE ON STRONGLY LIPSCHITZ DOMAINS OF ℝn BY LITTLEWOOD–PALEY–STEIN FUNCTION

2010 ◽  
Vol 12 (01) ◽  
pp. 71-84 ◽  
Author(s):  
JIZHENG HUANG
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Lipschitz Domain ◽  
Lipschitz Domains ◽  
Area Integral ◽  
Divergence Operator ◽  
Lusin Area Integral ◽  
Strongly Lipschitz Domain

Let Ω be a strongly Lipschitz domain of ℝn and define Hardy spaces on Ω by non-tangential maximal function. In this paper, we will give a characterization of the Hardy spaces on Ω by Littlwood–Paley–Stein function associated to L, where L is an elliptic second-order divergence operator. In order to get our result, we also consider the Lusin area integral characterization of the Hardy spaces on Ω.

LITTLEWOOD–PALEY CHARACTERIZATIONS OF ANISOTROPIC WEAK MUSIELAK–ORLICZ HARDY SPACES

2018 ◽  
Vol 237 ◽  
pp. 39-78
Author(s):  
BO LI ◽  
RUIRUI SUN ◽  
MINFENG LIAO ◽  
BAODE LI
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Growth Function ◽  
Anisotropic Growth ◽  
Area Function ◽  
Function Characterization ◽  
Lusin Area Function ◽  
Classical Hardy Space

Let $A$ be an expansive dilation on $\mathbb{R}^{n}$ and $\unicode[STIX]{x1D711}:\mathbb{R}^{n}\times [0,\infty )\rightarrow [0,\infty )$ an anisotropic growth function. In this article, the authors introduce the anisotropic weak Musielak–Orlicz Hardy space $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ via the nontangential grand maximal function and then obtain its Littlewood–Paley characterizations in terms of the anisotropic Lusin-area function, $g$-function or $g_{\unicode[STIX]{x1D706}}^{\ast }$-function, respectively. All these characterizations for anisotropic weak Hardy spaces $\mathit{WH}_{A}^{p}(\mathbb{R}^{n})$ (namely, $\unicode[STIX]{x1D711}(x,t):=t^{p}$ for all $t\in [0,\infty )$ and $x\in \mathbb{R}^{n}$ with $p\in (0,1]$) are new. Moreover, the range of $\unicode[STIX]{x1D706}$ in the anisotropic $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ coincides with the best known range of the $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of classical Hardy space $H^{p}(\mathbb{R}^{n})$ or its weighted variants, where $p\in (0,1]$.

Modulus of continuity and convergence of subsequences of Vilenkin–Fejér means in martingale Hardy spaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Giorgi Tutberidze
Keyword(s):  
Hardy Space ◽  
Modulus Of Continuity ◽  
Hardy Spaces ◽  
Lebesgue Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Fejér Means ◽  
Necessary And Sufficient

Abstract In this paper, we find a necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy space H p {H_{p}} to the Lebesgue space L p {L_{p}} for all 0 < p < 1 2 {0<p<\frac{1}{2}} .

scholarly journals Marcinkiewicz Integrals on Weighted Weak Hardy Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yue Hu ◽  
Yueshan Wang
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Lebesgue Space ◽  
Muckenhoupt Weight ◽  
Marcinkiewicz Integral ◽  
Smoothness Condition ◽  
Weight Class ◽  
Marcinkiewicz Integrals ◽  
Weak Hardy Space ◽  
Weak Lebesgue Space

We prove that, under the conditionΩ∈Lipα, Marcinkiewicz integralμΩis bounded from weighted weak Hardy spaceWHwpRnto weighted weak Lebesgue spaceWLwpRnformaxn/n+1/2,n/n+α<p≤1, wherewbelongs to the Muckenhoupt weight class. We also give weaker smoothness condition assumed on Ω to imply the boundedness ofμΩfromWHw1ℝntoWLw1Rn.

scholarly journals Hardy Spaces on Homogeneous Groups and Littlewood-Paley Functions

2020 ◽  
Vol 71 (1) ◽  
pp. 295-320
Author(s):  
Shuichi Sato
Keyword(s):  
Maximal Function ◽  
Hardy Spaces ◽  
Homogeneous Groups ◽  
Peetre Maximal Function ◽  
Vector Valued

Abstract We establish a characterization of the Hardy spaces on the homogeneous groups in terms of the Littlewood–Paley functions. The proof is based on vector-valued inequalities shown by applying the Peetre maximal function.

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.

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.

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