Hardy spaces associated with different homogeneities and boundedness of composition operators

It is well known that standard Calderón-Zygmund singular integral operators with isotropic and nonisotropic homogeneities are bounded on the classical H^p(ℝ^m) and nonisotropic H_h ^p(ℝ^m), respectively. In this paper, we develop a new Hardy space theory and prove that the composition of two Calderón-Zygmund singular integral operators with different homogeneities is bounded on this new Hardy space. Such a Hardy space has a multiparameter structure associated with the underlying mixed homogeneities arising from the two singular integral operators under consideration. The Calderón-Zygmund decomposition and an interpolation theorem hold on these new Hardy spaces.