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Journal of Geometric Analysis

, Volume 23, Issue 2, pp 895–932 | Cite as

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

  • The Anh Bui
  • Xuan Thinh Duong
Article

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 

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Copyright information

© Mathematica Josephina, Inc. 2011

Authors and Affiliations

  1. 1.Department of MathematicsMacquarie UniversitySydneyAustralia
  2. 2.Department of MathematicsUniversity of PedagogyHo Chi Minh CityVietnam

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