Equivalence of norms on finite linear combinations of atoms

Let (M, ρ, μ) be a space of homogeneous type and denote by FcC(M) the space of finite linear combinations of continuous (1, ∞)-atoms with compact support. In this note we give a simple function theoretic proof of the equivalence on FcC(M) of the H1-norm and the norm defined in terms of finite linear combinations of atoms. The result holds also for the class of nondoubling metric measure spaces considered in previous works of the authors and Carbonaro. © 2010 Springer-Verlag.