Boundedness of Calderón–Zygmund operators on non-homogeneous metric measure spaces: Equivalent characterizations

Let (X,d,μ) be a metric measure space satisfying the upper doubling and the geometrically doubling conditions in the sense of T. Hytönen. In this paper, the authors prove that the boundedness of a Calderón–Zygmund operator T on L2(μ) is equivalent to either of the boundedness of T from the atomic Hardy space H1(μ) to L1,∞(μ) or from H1(μ) to L1(μ). To this end, the authors first establish an interpolation result that a sublinear operator which is bounded from H1(μ) to L1,∞(μ) and from Lp0(μ) to Lp0,∞(μ) for some p0∈(1,∞) is also bounded on Lp(μ) for all p∈(1,p0). A main tool used in this paper is the Calderón–Zygmund decomposition in this setting established by B.T. Anh and X.T. Duong.