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 Boundedness of Calderón–Zygmund Operators on Non-homogeneous Metric Measure Spaces

2012 ◽  
Vol 64 (4) ◽  
pp. 892-923 ◽  
Author(s):  
Tuomas Hytönen ◽  
Suile Liu ◽  
Dachun Yang ◽  
Dongyong Yang
Keyword(s):  
Polynomial Growth ◽  
Metric Spaces ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Zygmund Operator ◽  
Borel Measures ◽  
Measure Spaces ◽  
Atomic Condition ◽  
Complex Valued ◽  
Polynomial Growth Condition

Abstract Let (𝒳, d, μ) be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that μ(﹛x﹜) = 0 for all x ∈ 𝒳. In this paper, we show that the boundedness of a Calderón–Zygmund operator T on L2(μ) is equivalent to that of T on Lp(μ) for some p ∈ (1,∞), and that of T from L1(μ) to L1,∞(μ). As an application, we prove that if T is a Calderón–Zygmund operator bounded on L2(μ), then its maximal operator is bounded on Lp(μ) for all p ∈ (1,∞) and from the space of all complex-valued Borel measures on 𝒳 to L1,∞(μ). All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

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.

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 $\widetilde{\operatorname{RBLO}}(\unicode[STIX]{x1D707})$.

Estimates for maximal singular integral operators in non-homogeneous spaces

2006 ◽  
Vol 136 (2) ◽  
pp. 351-364 ◽  
Author(s):  
Guoen Hu ◽  
Yan Meng ◽  
Dachun Yang
Keyword(s):  
Integral Operator ◽  
Singular Integral Operator ◽  
Singular Integral ◽  
Homogeneous Spaces ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Standard Size ◽  
Hörmander Condition ◽  
Size Condition ◽  
Maximal Singular Integral

Under the assumption that the Radon measure μ on Rd satisfies only some growth condition, the authors prove that, for the maximal singular integral operator associated with a singular integral whose kernel only satisfies a standard size condition and the Hörmander condition, its boundedness in Lebesgue spaces Lp(μ) for any p ∈ (1, ∞) is equivalent to its boundedness from L1(μ) into weak L1(μ). As an application, the authors verify that if the truncated singular integral operators are bounded on L2(μ) uniformly, then the associated maximal singular integral operator is also bounded on Lp(μ) for any p ∈ (1, ∞).

scholarly journals Eigenvalues of the drifted Laplacian on complete metric measure spaces

2016 ◽  
Vol 19 (01) ◽  
pp. 1650001 ◽  
Author(s):  
Xu Cheng ◽  
Detang Zhou
Keyword(s):  
Lower Bound ◽  
Measure Space ◽  
Logarithmic Sobolev Inequality ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Dimensional Manifold ◽  
Metric Measure Spaces ◽  
Smooth Metric Measure Space ◽  
Measure Spaces ◽  
Multiplicity Formula

In this paper, first we study a complete smooth metric measure space [Formula: see text] with the ([Formula: see text])-Bakry–Émery Ricci curvature [Formula: see text] for some positive constant [Formula: see text]. It is known that the spectrum of the drifted Laplacian [Formula: see text] for [Formula: see text] is discrete and the first nonzero eigenvalue of [Formula: see text] has lower bound [Formula: see text]. We prove that if the lower bound [Formula: see text] is achieved with multiplicity [Formula: see text], then [Formula: see text], [Formula: see text] is isometric to [Formula: see text] for some complete [Formula: see text]-dimensional manifold [Formula: see text] and by passing an isometry, [Formula: see text] must split off a gradient shrinking Ricci soliton [Formula: see text], [Formula: see text]. This result can be applied to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian [Formula: see text] for properly immersed self-shrinkers in the Euclidean space [Formula: see text], [Formula: see text] and show the discreteness of the spectrum of [Formula: see text] and a logarithmic Sobolev inequality.

scholarly journals The p-Royden and p-Harmonic Boundaries for Metric Measure Spaces

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Marcello Lucia ◽  
Michael J. Puls
Keyword(s):  
Real Number ◽  
Measure Space ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Type Problem ◽  
Locally Compact ◽  
Dirichlet Type ◽  
Algebra Of Functions ◽  
Measure Spaces ◽  
Dirichlet Type Problem

Abstract Let p be a real number greater than one and let X be a locally compact, noncompact metric measure space that satisfies certain conditions. The p-Royden and p-harmonic boundaries of X are constructed by using the p-Royden algebra of functions on X and a Dirichlet type problem is solved for the p-Royden boundary. We also characterize the metric measure spaces whose p-harmonic boundary is empty.

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.

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