Two boundedness criteria for a class of operators on Musielak–Orlicz Hardy spaces and applications

2019 ◽  
Vol 63 (1) ◽  
pp. 13-35
Author(s):  
Xiaoli Qiu ◽  
Baode Li ◽  
Xiong Liu ◽  
Bo Li
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Critical Case ◽  
Orlicz Function ◽  
Linear Operators ◽  
Riesz Means ◽  
Weak Hardy Space ◽  
Sublinear Operators

AbstractLet φ : ℝn × [0, ∞) → [0, ∞) satisfy that φ(x, · ), for any given x ∈ ℝn, is an Orlicz function and φ( · , t) is a Muckenhoupt A∞ weight uniformly in t ∈ (0, ∞). The (weak) Musielak–Orlicz Hardy space Hφ(ℝn) (WHφ(ℝn)) generalizes both the weighted (weak) Hardy space and the (weak) Orlicz Hardy space and hence has wide generality. In this paper, two boundedness criteria for both linear operators and positive sublinear operators from Hφ(ℝn) to Hφ(ℝn) or from Hφ(ℝn) to WHφ(ℝn) are obtained. As applications, we establish the boundedness of Bochner–Riesz means from Hφ(ℝn) to Hφ(ℝn), or from Hφ(ℝn) to WHφ(ℝn) in the critical case. These results are new even when φ(x, t): = Φ(t) for all (x, t) ∈ ℝn × [0, ∞), where Φ is an Orlicz function.

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 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.

Sublinear operators on weighted Hardy spaces with variable exponents

2019 ◽  
Vol 31 (3) ◽  
pp. 607-617 ◽  
Author(s):  
Kwok-Pun Ho
Keyword(s):  
Hardy Spaces ◽  
Variable Exponents ◽  
Square Functions ◽  
Weighted Hardy Spaces ◽  
Riesz Means ◽  
Marcinkiewicz Integrals ◽  
Sublinear Operators ◽  
Intrinsic Square Functions

Abstract We establish the mapping properties for some sublinear operators on weighted Hardy spaces with variable exponents by using extrapolation. In particular, we study the Calderón–Zygmund operators, the maximal Bochner–Riesz means, the intrinsic square functions and the Marcinkiewicz integrals on weighted Hardy spaces with variable exponents.

scholarly journals Musielak–Orlicz Hardy Spaces Associated with Divergence Form Elliptic Operators Without Weight Assumptions

2014 ◽  
Vol 216 ◽  
pp. 71-110 ◽  
Author(s):  
Tri Dung Tran
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Carleson Measure ◽  
Dual Space ◽  
Orlicz Function ◽  
Adjoint Operator ◽  
Riesz Transform ◽  
Divergence Form ◽  
Measurable Coefficients ◽  
Nirenberg Inequality

AbstractLet L be a divergence form elliptic operator with complex bounded measurable coefficients, let ω be a positive Musielak-Orlicz function on (0, ∞) of uniformly strictly critical lower-type pω ∈ (0, 1], and let ρ(x,t) = t−1/ω−1 (x,t−1) for x ∈ ℝn, t ∊ (0, ∞). In this paper, we study the Musielak-Orlicz Hardy space Hω,L(ℝn) and its dual space BMOρ,L* (ℝ n), where L* denotes the adjoint operator of L in L2 (ℝ n). The ρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L (ℝn) are also established. Finally, as applications, we show that the Riesz transform ∇L−1/2 and the Littlewood–Paley g-function gL map Hω,L(ℝn) continuously into L(ω).

scholarly journals Musielak–Orlicz Hardy Spaces Associated with Divergence Form Elliptic Operators Without Weight Assumptions

2014 ◽  
Vol 216 ◽  
pp. 71-110
Author(s):  
Tri Dung Tran
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Carleson Measure ◽  
Dual Space ◽  
Orlicz Function ◽  
Adjoint Operator ◽  
Riesz Transform ◽  
Divergence Form ◽  
Measurable Coefficients ◽  
Nirenberg Inequality

AbstractLetLbe a divergence form elliptic operator with complex bounded measurable coefficients, letωbe a positive Musielak-Orlicz function on (0, ∞) of uniformly strictly critical lower-typepω∈ (0, 1], and letρ(x,t) = t−1/ω−1(x,t−1) forx∈ ℝn, t∊ (0, ∞). In this paper, we study the Musielak-Orlicz Hardy spaceHω,L(ℝn) and its dual space BMOρ,L* (ℝn), whereL*denotes the adjoint operator ofLinL2(ℝn). Theρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L(ℝn) are also established. Finally, as applications, we show that the Riesz transform ∇L−1/2and the Littlewood–Paleyg-functiongLmapHω,L(ℝn) continuously intoL(ω).

scholarly journals Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces

2010 ◽  
Vol 8 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Xiangxing Tao ◽  
Xiao Yu ◽  
Songyan Zhang
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Marcinkiewicz Integrals ◽  
Dini Condition ◽  
Weak Hardy Space

In this article, we consider the Marcinkiewicz integrals with variable kernels defined byμΩ(f)(x)=(∫0∞|∫|x−y|≤tΩ(x,x−y)|x−y|n−1f(y)dy|2dtt3)1/2, whereΩ(x,z)∈L∞(ℝn)×Lq(Sn−1)forq> 1. We prove that the operatorμΩis bounded from Hardy space,Hp(ℝn), toLp(ℝn)space; and is bounded from weak Hardy space,Hp,∞(ℝn), to weakLp(ℝn)space formax{2n2n+1,nn+α}<p<1, ifΩsatisfies theL1,α-Dini condition with any0<α≤1.

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.

scholarly journals Numerical Range on Weighted Hardy Spaces as Semi Inner Product Spaces

2017 ◽  
Vol 25 (1) ◽  
pp. 87-98
Author(s):  
Mohammad Taghi Heydari
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Numerical Range ◽  
Linear Operators ◽  
Inner Product ◽  
Product Spaces ◽  
Weighted Hardy Space ◽  
Weighted Hardy Spaces ◽  
Bounded Linear ◽  
Inner Product Spaces

AbstractThe semi-inner product, in the sense of Lumer, on weighted Hardy space which generate the norm is unique. Also we will discuss some properties of the numerical range of bounded linear operators on weighted Hardy spaces.

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.

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 .

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