It is well known that standard Calderón–Zygmund singular integral operators with isotropic and nonisotropic homogeneities are bounded on the classical and nonisotropic 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.
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Yongsheng Han, Chincheng Lin, Guozhen Lu, Zhuoping Ruan, Eric T. Sawyer, Hardy spaces associated with different homogeneities and boundedness of composition operators. Rev. Mat. Iberoam. 29 (2013), no. 4, pp. 1127–1157DOI 10.4171/RMI/751