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

Abstract

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 L^p(μ)-boundedness with p ∈ (1, ∞) is equivalent to its boundedness from L¹(μ) into L^1,∞(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 L^p(μ) for all p ∈ (1, ∞), and bounded from L¹(μ) into L^1,∞ (μ). These results essentially improve the existing results.