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Mathematische Annalen

, Volume 319, Issue 1, pp 89–149 | Cite as

$BMO, H^1$, and Calderón-Zygmund operators for non doubling measures

  • Xavier Tolsa
Original article

Abstract.

Given a Radon measure \(\mu\) on \({\mathbb R}^d\), which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties which hold for the classical space \(BMO(\mu)\) when \(\mu\) is a doubling measure remain valid for the space of type BMO introduced in this paper, without assuming \(\mu\) doubling. For instance, Calderón-Zygmund operators which are bounded on \(L^2(\mu)\) are also bounded from \(L^\infty(\mu)\) into the new BMO space. Moreover, this space also satisfies a John-Nirenberg inequality, and its predual is an atomic space \(H^1\). Using a sharp maximal operator it is shown that operators which are bounded from \(L^\infty(\mu)\) into the new BMO space and from its predual \(H^1\) into \(L^1(\mu)\) must be bounded on \(L^p(\mu)\), \(1< p< infty\). From this result one can obtain a new proof of the T(1) theorem for the Cauchy transform for non doubling measures. Finally, it is proved that commutators of Calderón-Zygmund operators bounded on \(L^2(\mu)\) with functions of the new BMO are bounded on \(L^p(\mu), 1< p < \infty\).

Mathematics Subject Classification (1991): 42B20, 42B30 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Xavier Tolsa
    • 1
  1. 1.Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain ES

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