Wavelet characterizations of the atomic Hardy space H1 on spaces of homogeneous type

Let (X,d,μ) be a metric measure space of homogeneous type in the sense of R.R. Coifman and G. Weiss and Hat1(X) be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hytönen, together with obtaining some crucial lower bounds for regular wavelets, the authors give an unconditional basis of Hat1(X) and several equivalent characterizations of Hat1(X) in terms of wavelets, which are proved useful.