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Hardy spaces, regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces. (English) Zbl 1267.42013
Summary: One defines a non-homogeneous space (X,μ) as a metric space equipped with a non-doubling measure μ 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,μ) 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 ( n ,μ) to the setting of a general non-homogeneous space (X,μ). Our framework of the non-homogeneous space (X,μ) 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 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,μ), 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.

42B20Singular and oscillatory integrals, several variables
42B35Function spaces arising in harmonic analysis
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