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

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Let (,d,μ) be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors 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 is bounded on Lᵖ(μ) for all p∈(1,∞), and bounded from L¹(μ) into L¹,∞(μ). These results essentially improve the existing results.