Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces

2012 ◽  
Vol 55 (3) ◽  
pp. 646-662 ◽  
Author(s):  
Jiang Zhou ◽  
Bolin Ma
Keyword(s):  
Hardy Spaces ◽  
Lipschitz Function ◽  
Homogeneous Spaces ◽  
Lipschitz Functions ◽  
Marcinkiewicz Integral ◽  
Type Condition ◽  
Lebesgue Spaces ◽  
Lipschitz Spaces ◽  
Nondoubling Measure

AbstractUnder the assumption that μ is a nondoubling measure, we study certain commutators generated by the Lipschitz function and the Marcinkiewicz integral whose kernel satisfies a Hörmander type condition. We establish the boundedness of these commutators on the Lebesgue spaces, Lipschitz spaces, and Hardy spaces. Our results are extensions of known theorems in the doubling case.

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 Boundedness of Commutators of Marcinkiewicz Integrals on Nonhomogeneous Metric Measure Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Guanghui Lu ◽  
Shuangping Tao
Keyword(s):  
Hardy Spaces ◽  
Measure Space ◽  
Morrey Spaces ◽  
Marcinkiewicz Integral ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Lebesgue Spaces ◽  
Marcinkiewicz Integrals ◽  
Measure Spaces ◽  
Weak Lebesgue Spaces

Let(X,d,μ)be a metric measure space satisfying the upper doubling condition and geometrically doubling condition in the sense of Hytönen. The aim of this paper is to establish the boundedness of commutatorMbgenerated by the Marcinkiewicz integralMand Lipschitz functionb. The authors prove thatMbis bounded from the Lebesgue spacesLp(μ)to weak Lebesgue spacesLq(μ)for1≤p<n/β, from the Lebesgue spacesLp(μ)to the spacesRBMO(μ)forp=n/β, and from the Lebesgue spacesLp(μ)to the Lipschitz spacesLip(β-n/p)(μ)forn/β<p≤∞. Moreover, some results in Morrey spaces and Hardy spaces are also discussed.

scholarly journals Existence and Boundedness of Parametrized Marcinkiewicz Integral with Rough Kernel on Campanato Spaces

2006 ◽  
Vol 181 ◽  
pp. 103-148 ◽  
Author(s):  
Yong Ding ◽  
Qingying Xue ◽  
Kôzô Yabuta
Keyword(s):  
Single Point ◽  
Morrey Spaces ◽  
Rough Kernel ◽  
Marcinkiewicz Integral ◽  
Type Condition ◽  
Lipschitz Spaces ◽  
Area Function ◽  
Bmo Function ◽  
Lusin Area Function ◽  
Lipschitz Kernel

AbstractLet g(f), S(f), g*λ(f) be the Littlewood-Paley g function, Lusin area function, and Littlewood-Paley g*λ(f) function of f, respectively. In 1990 Chen Jiecheng and Wang Silei showed that if, for a BMO function f, one of the above functions is finite for a single point in ℝn, then it is finite a.e. on ℝn, and BMO boundedness holds. Recently, Sun Yongzhong extended this result to the case of Campanato spaces (i.e. Morrey spaces, BMO, and Lipschitz spaces). One of us improved his g*λ(f) result further, and treated parametrized Marcinkiewicz functions with Lipschitz kernel μρ(f), μρs(f) and μλ*,ρ(f). In this paper, we show that the same results hold also in the case of rough kernel satisfying Lp-Dini type condition.

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.

Marcinkiewicz integral on hardy spaces

2002 ◽  
Vol 42 (2) ◽  
pp. 174-182 ◽  
Author(s):  
Yong Ding ◽  
Shanzhen Lu ◽  
Quigying Xue
Keyword(s):  
Hardy Spaces ◽  
Marcinkiewicz Integral

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.

A note on fractional integral operators defined by weights and non-doubling measures

2010 ◽  
Vol 106 (2) ◽  
pp. 283 ◽  
Author(s):  
Oscar Blasco ◽  
Vicente Casanova ◽  
Joaquín Motos
Keyword(s):  
Measure Space ◽  
Fractional Integral ◽  
Integral Operators ◽  
Metric Measure Space ◽  
Integral Type ◽  
Lebesgue Spaces ◽  
Lipschitz Spaces ◽  
Fractional Integral Operators ◽  
Doubling Measures

Given a metric measure space $(X,d,\mu)$, a weight $w$ defined on $(0,\infty)$ and a kernel $k_w(x,y)$ satisfying the standard fractional integral type estimates, we study the boundedness of the operators $K_w f(x)=\int_X k_w(x,y)f(y)\,d\mu(y)$ and $\tilde K_w f(x)=\int_X (k_w(x,y)-k_w(x_0,y))f(y)\,d\mu(y)$ on Lebesgue spaces $L^p(\mu)$ and generalized Lipschitz spaces $\mathrm{Lip}_\phi$, respectively, for certain range of the parameters depending on the $n$-dimension of $\mu$ and some indices associated to the weight $w$.

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