# Hardy Spaces, Regularized BMO Spaces and the Boundedness of Calderón–Zygmund Operators on Non-homogeneous Spaces

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## Abstract

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. 319:89–149, 2011) on the non-homogeneous space (ℝ^{ 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 Hytönen (2011) 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 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.

## Keywords

Non-homogeneous spaces Hardy spaces BMO Calderón–Zygmund operator## Mathematics Subject Classification (2000)2010

42B20 42B35## Preview

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