Hardy spaces associated with different homogeneities and boundedness of composition operators

Abstract

It is well known that standard Calderón–Zygmund singular integral operators with isotropic and nonisotropic homogeneities are bounded on the classical $H^p({\mathbb{R}}^m)$ and nonisotropic $H_h^p({\mathbb{R}}^m),$ 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.