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 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.

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).

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.

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 .

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 Factorizations of outer functions and extremal problems

1996 ◽  
Vol 39 (3) ◽  
pp. 535-546 ◽  
Author(s):  
Takahiko Nakazi
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Extremal Problems ◽  
The Other ◽  
Outer Function ◽  
Inner Functions ◽  
Special Case ◽  
Class Of Functions

The author has proved that an outer function in the Hardy space H1 can be factored into a product in which one factor is strongly outer and the other is the sum of two inner functions. In an endeavor to understand better the latter factor, we introduce a class of functions containing sums of inner functions as a special case. Using it, we describe the solutions of extremal problems in the Hardy spaces Hp for 1≦p<∞.

scholarly journals THE RATE OF INCREASE OF MEAN VALUES OF FUNCTIONS IN HARDY SPACES

2009 ◽  
Vol 86 (2) ◽  
pp. 199-204
Author(s):  
JAVAD MASHREGHI
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Mean Values ◽  
Rate Of Increase

AbstractThe norm of a function f in the Hardy space Hp(𝔻) is by definition the limit of $\|f_r\|_p$ as r→1. We show that $d \|f_r\|_p/dr$ grows at most like o(1/log r) as r→1.

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