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

Abstract

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 L 2 ( μ ) is equivalent to either of the boundedness of T from the atomic Hardy space H 1 ( μ ) to L 1 , ∞ ( μ ) or from H 1 ( μ ) to L 1 ( μ ) . To this end, the authors first establish an interpolation result that a sublinear operator which is bounded from H 1 ( μ ) to L 1 , ∞ ( μ ) and from L p 0 ( μ ) to L p 0 , ∞ ( μ ) for some p 0 ∈ ( 1 , ∞ ) is also bounded on L p ( μ ) for all p ∈ ( 1 , p 0 ) . A main tool used in this paper is the Calderón–Zygmund decomposition in this setting established by B.T. Anh and X.T. Duong.