Revista Matemática Iberoamericana


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Volume 29, Issue 4, 2013, pp. 1127–1157
DOI: 10.4171/RMI/751

Published online: 2013-12-15

Hardy spaces associated with different homogeneities and boundedness of composition operators

Yongsheng Han[1], Chincheng Lin[2], Guozhen Lu[3], Zhuoping Ruan[4] and Eric T. Sawyer[5]

(1) Auburn University, USA
(2) National Central University, Chung-Li, Taiwan
(3) Wayne State University, Detroit, USA
(4) Nanjing University, China
(5) McMaster University, Hamilton, Canada

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 $H^p_{h}(\mathbb{R}^m),$ 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.

Keywords: Hardy spaces, Calderón–Zygmund operators, discrete Calderón’s identity, almost orthogonality estimates, discrete Littlewood–Paley–Stein square functions

Han Yongsheng, Lin Chincheng, Lu Guozhen, Ruan Zhuoping, Sawyer Eric: Hardy spaces associated with different homogeneities and boundedness of composition operators. Rev. Mat. Iberoamericana 29 (2013), 1127-1157. doi: 10.4171/RMI/751