The space 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

DOI:
https://doi.org/10.1090/S0002-9947-02-03131-8

Published electronically:
September 11, 2002

MathSciNet review:
1928090

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Abstract: Let be a Radon measure on , which may be nondoubling. The only condition that must satisfy is the size condition , for some fixed . Recently, some spaces of type and were introduced by the author. These new spaces have properties similar to those of the classical spaces and defined for doubling measures, and they have proved to be useful for studying the boundedness of Calderón-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic Hardy space in terms of a maximal operator is given. It is shown that belongs to if and only if , and , 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

Email:
xtolsa@mat.uab.es

DOI:
https://doi.org/10.1090/S0002-9947-02-03131-8

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