Bui, The Anh; Duong, Xuan Thinh Hardy spaces, regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces. (English) Zbl 1267.42013 J. Geom. Anal. 23, No. 2, 895-932 (2013). Summary: One defines a non-homogeneous space \((X,\mu)\) as a metric space equipped with a non-doubling measure \(\mu\) so that the volume of the ball with center \(x\), radius \(r\) has an upper bound of the form \(r^n\) for some \(n>0\). The aim of this paper is to study the boundedness of Calderón-Zygmund singular integral operators \(T\) on various function spaces on \((X,\mu)\) such as the Hardy spaces, the \(L^p\) spaces, and the regularized BMO spaces. This article thus extends the work of X. Tolsa [Math. Ann. 319, No. 1, 89–149 (2001; Zbl 0974.42014)] on the non-homogeneous space \((\mathbb R^n,\mu)\) to the setting of a general non-homogeneous space \((X,\mu)\). Our framework of the non-homogeneous space \((X,\mu)\) is similar to that of T. Hytönen [Publ. Mat., Barc. 54, No. 2, 485–504 (2010; Zbl 1246.30087)] and we are able to obtain quite a few properties similar to those of Calderón-Zygmund operators on doubling spaces such as the weak type \((1,1)\) estimate, boundedness from the Hardy space into \(L^1\), boundedness from \(L^\infty\) into the regularized BMO, and an interpolation theorem. Furthermore, we prove that the dual space of the Hardy space is the regularized BMO space, obtain a Calderón-Zygmund decomposition on the non-homogeneous space \((X,\mu)\), and use this decomposition to show the boundedness of the maximal operators in the form of a Cotlar inequality as well as the boundedness of commutators of Calderón-Zygmund operators and BMO functions. Cited in 4 ReviewsCited in 54 Documents MSC: 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B35 Function spaces arising in harmonic analysis Keywords:non-homogeneous spaces; Hardy spaces; BMO; Calderón-Zygmund operator Citations:Zbl 0974.42014; Zbl 1246.30087 PDFBibTeX XMLCite \textit{T. A. Bui} and \textit{X. T. Duong}, J. Geom. Anal. 23, No. 2, 895--932 (2013; Zbl 1267.42013) Full Text: DOI arXiv References: [1] Bui, T.A., Duong, X.T.: Endpoint estimates for maximal operator and boundedness of maximal commutators. Preprint [2] Coifman, R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977) · Zbl 0358.30023 [3] Di Fazio, G., Gutiérrez, C.E., Lanconelli, E.: Covering theorems, inequalities on metric spaces and applications to PDE’s. Math. Ann. 341(2), 255–291 (2008) · Zbl 1149.46029 [4] Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. 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