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The space $H^1$ for nondoubling measures in terms of a grand maximal operator

Author: Xavier Tolsa
Journal: Trans. Amer. Math. Soc. 355 (2003), 315-348
MSC (2000): Primary 42B20; Secondary 42B30
Published electronically: September 11, 2002
MathSciNet review: 1928090
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Abstract: Let $\mu$ be a Radon measure on $\mathbb{R} ^d$, which may be nondoubling. The only condition that $\mu$ must satisfy is the size condition $\mu(B(x,r))\leq C\,r^n$, for some fixed $0<n\leq d$. Recently, some spaces of type $BMO(\mu)$ and $H^1(\mu)$ were introduced by the author. These new spaces have properties similar to those of the classical spaces ${B\!M\!O}$ and $H^1$defined for doubling measures, and they have proved to be useful for studying the $L^p(\mu)$ boundedness of Calderón-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic Hardy space $H^1(\mu)$ in terms of a maximal operator $M_\Phi$ is given. It is shown that $f$ belongs to $H^1(\mu)$ if and only if $f\in L^1(\mu)$, $\int f\, d\mu=0$ and $M_\Phi f \in L^1(\mu)$, as in the usual doubling situation.

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Additional Information

Xavier Tolsa
Affiliation: Département de Mathématique, Bâtiment 425, Université de Paris-Sud, 91405 Orsay-Cedex, France
Address at time of publication: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

Keywords: BMO, atomic spaces, Hardy spaces, Calder\'on-Zygmund operators, nondoubling measures, maximal functions, grand maximal operator
Received by editor(s): October 31, 2000
Published electronically: September 11, 2002
Additional Notes: Supported by a postdoctoral grant from the European Commission for the TMR Network “Harmonic Analysis”. Also partially supported by grants DGICYT PB96-1183 and CIRIT 1998-SGR00052 (Spain)
Article copyright: © Copyright 2002 American Mathematical Society

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