The molecular characterization of the Hardy space on non-homogeneous metric measure spaces and its application

Let ( X , d , μ ) be a non-homogeneous metric measure space, which means that ( X , d , μ ) is a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. In this paper, the authors introduce the atomic Hardy space H ˜ atb 1 , p ( μ ) via the discrete coefficient K ˜ B , S ( ρ ) for ρ ∈ ( 1 , ∞ ) and balls B ⊂ S of X . Then, the authors establish the corresponding molecular characterization of H ˜ atb 1 , p ( μ ) via a constructive way. As an application, the authors obtain the boundedness of Calderón–Zygmund operators on H ˜ atb 1 , p ( μ ) . Moreover, the authors give a sufficient condition to guarantee that H ˜ atb 1 , p ( μ ) coincides with the existing atomic Hardy space H atb 1 , p ( μ ) .