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 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).

GENERALIZED MORREY SPACES OVER NONHOMOGENEOUS METRIC MEASURE SPACES

2016 ◽  
Vol 103 (2) ◽  
pp. 268-278 ◽  
Author(s):  
GUANGHUI LU ◽  
SHUANGPING TAO
Keyword(s):  
Integral Operator ◽  
Measure Space ◽  
Fractional Integral ◽  
Morrey Spaces ◽  
Marcinkiewicz Integral ◽  
Metric Measure Spaces ◽  
Generalized Morrey Spaces ◽  
Measure Spaces ◽  
Definition Of ◽  
Marcinkiewicz Integral Operator

Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a nonhomogeneous metric measure space satisfying the so-called upper doubling and the geometric doubling conditions. In this paper, the authors give the natural definition of the generalized Morrey spaces on $({\mathcal{X}},d,\unicode[STIX]{x1D707})$, and then investigate some properties of the maximal operator, the fractional integral operator and its commutator, and the Marcinkiewicz integral operator.

scholarly journals Boundedness of Marcinkiewicz Integrals on RBMO Spaces over Nonhomogeneous Metric Measure Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Ji Cheng ◽  
Guanghui Lu
Keyword(s):  
Marcinkiewicz Integral ◽  
Metric Measure Spaces ◽  
Type Condition ◽  
Doubling Condition ◽  
Marcinkiewicz Integrals ◽  
Measure Spaces ◽  
Dominating Function

LetX,d,μbe a metric measures space satisfying the upper doubling conditions and the geometrically doubling conditions in the sense of Hytönen. Under the assumption that the dominating function satisfies the weak reverse doubling condition, the authors prove that Marcinkiewicz integral with kernel satisfying certain stronger Hörmander-type condition is bounded on RBMOμspace.

scholarly journals Morrey Spaces for Nonhomogeneous Metric Measure Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Cao Yonghui ◽  
Zhou Jiang
Keyword(s):  
Integral Operator ◽  
Maximal Operator ◽  
Fractional Integral ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Fractional Integral Operator ◽  
Marcinkiewicz Integral ◽  
Metric Measure Spaces ◽  
Measure Spaces ◽  
Definition Of

The authors give a definition of Morrey spaces for nonhomogeneous metric measure spaces and investigate the boundedness of some classical operators including maximal operator, fractional integral operator, and Marcinkiewicz integral operators.

scholarly journals Boundedness of maximal Calderón–Zygmund operators on non-homogeneous metric measure spaces

2014 ◽  
Vol 144 (3) ◽  
pp. 567-589 ◽  
Author(s):  
Suile Liu ◽  
Yan Meng ◽  
Dachun Yang
Keyword(s):  
Singular Integral ◽  
Measure Space ◽  
Type Inequality ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Standard Size ◽  
Hörmander Condition ◽  
Zygmund Operator ◽  
Measure Spaces ◽  
Size Condition

Let (X, d, μ) be a metric measure space and let it satisfy the so-called upper doubling condition and the geometrically doubling condition. We show that, for the maximal Calderón–Zygmund operator associated with a singular integral whose kernel satisfies the standard size condition and the Hörmander condition, its Lp(μ)-boundedness with p ∈ (1, ∞) is equivalent to its boundedness from L1(μ) into L1,∞(μ). Moreover, applying this, together with a new Cotlar-type inequality, the authors show that if the Calderón–Zygmund operator T is bounded on L2(μ), then the corresponding maximal Calderón–Zygmund operator is bounded on Lp(μ) for all p ∈ (1, ∞), and bounded from L1(μ) into L1,∞ (μ). These results essentially improve the existing results.

scholarly journals Estimates for Parameter Littlewood-Paleygκ⁎Functions on Nonhomogeneous Metric Measure Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Guanghui Lu ◽  
Shuangping Tao
Keyword(s):  
Hardy Space ◽  
Lebesgue Space ◽  
Measure Space ◽  
Metric Measure Space ◽  
Doubling Measure ◽  
Metric Measure Spaces ◽  
Type Condition ◽  
Lebesgue Spaces ◽  
Measure Spaces ◽  
Atomic Hardy Space

Let(X,d,μ)be a metric measure space which satisfies the geometrically doubling measure and the upper doubling measure conditions. In this paper, the authors prove that, under the assumption that the kernel ofMκ⁎satisfies a certain Hörmander-type condition,Mκ⁎,ρis bounded from Lebesgue spacesLp(μ)to Lebesgue spacesLp(μ)forp≥2and is bounded fromL1(μ)intoL1,∞(μ). As a corollary,Mκ⁎,ρis bounded onLp(μ)for1<p<2. In addition, the authors also obtain thatMκ⁎,ρis bounded from the atomic Hardy spaceH1(μ)into the Lebesgue spaceL1(μ).

The John–Nirenberg Inequality for the Regularized BLO Space on Non-homogeneous Metric Measure Spaces

2019 ◽  
Vol 63 (3) ◽  
pp. 643-654
Author(s):  
Haibo Lin ◽  
Zhen Liu ◽  
Chenyan Wang
Keyword(s):  
Measure Space ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Nirenberg Inequality ◽  
Measure Spaces

AbstractLet $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a metric measure space satisfying the geometrically doubling condition and the upper doubling condition. In this paper, the authors establish the John-Nirenberg inequality for the regularized BLO space $\widetilde{\operatorname{RBLO}}(\unicode[STIX]{x1D707})$.

Generalized Riesz potentials of functions in Morrey spaces L(1,ϕ;κ)(G) over non-doubling measure spaces

2020 ◽  
Vol 32 (2) ◽  
pp. 339-359 ◽  
Author(s):  
Yoshihiro Sawano ◽  
Masaki Shigematsu ◽  
Tetsu Shimomura
Keyword(s):  
Measure Space ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Doubling Measure ◽  
Doubling Condition ◽  
Riesz Potentials ◽  
General Measure ◽  
Open Set ◽  
Measure Spaces

AbstractThis paper proves the boundedness of the generalized Riesz potentials {I_{\rho,\mu,\tau}f} of functions in the Morrey space {L^{(1,\varphi;\kappa)}(G)} over a general measure space X, with G a bounded open set in X (or G is {X)}, as an extension of earlier results. The modification parameter τ is introduced for the purpose of including the case where the underlying measure does not satisfy the doubling condition. What is new in the present paper is that ρ depends on {x\in X}. An example in the end of this article convincingly explains why the modification parameter τ must be introduced.

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