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

Let (X, d, mu) be a metric measure space satisfying the upper doubling and the geometrically doubling conditions in the sense of T. Hytonen. In this paper, the authors prove that the boundedness of a Calderon-Zygmund operator T on L-2(mu) is equivalent to either of the boundedness of T from the atomic Hardy space H-1(mu) to L-1,L-infinity(mu) or from H-1(mu) to L-1(mu). To this end, the authors first establish an interpolation result that a sublinear operator which is bounded from H-1(mu) to L-1,L-infinity(mu) and from L-p0(mu) to L-p0,L-infinity(mu) for some p(0) is an element of (1, infinity) is also bounded on L-p(mu) for all p is an element of (1, p(0)). A main tool used in this paper is the Calderon-Zygmund decomposition in this setting established by B.T. Anh and X.T. Duong. (C) 2011 Elsevier Inc. All rights reserved.