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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 $\left(X,\mu \right)$ 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 $\left(X,\mu \right)$ 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 $\left({ℝ}^{n},\mu \right)$ to the setting of a general non-homogeneous space $\left(X,\mu \right)$. Our framework of the non-homogeneous space $\left(X,\mu \right)$ 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 $\left(1,1\right)$ 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 $\left(X,\mu \right)$, 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.

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 42B35 Function spaces arising in harmonic analysis
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