Weighted estimates for θ-type Calderón-Zygmund operator and its commutator on metric measure spaces

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.

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Weighted estimates for iterated commutators of multilinear Calderón-Zygmund operators on non-homogeneous metric measure spaces

Science China Mathematics ◽

10.1007/s11425-018-9471-7 ◽

2020 ◽

Author(s):

Yuan Zhao ◽

Haibo Lin ◽

Yan Meng

Keyword(s):

Metric Measure Spaces ◽

Weighted Estimates ◽

Measure Spaces ◽

Iterated Commutators

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210512000054 ◽

2014 ◽

Vol 144 (3) ◽

pp. 567-589 ◽

Cited By ~ 3

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.

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