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The Hardy space H1 on non-homogeneous metric spaces

Published online by Cambridge University Press:  08 December 2011

TUOMAS HYTÖNEN ,
DACHUN YANG  and
DONGYONG YANG
TUOMAS HYTÖNEN
Affiliation:
Department of Mathematics and Statistics, P.O.B. 68, (Gustaf Hällströmin katu 2b), FI-00014 University of Helsinki, Finland. e-mail: [email protected]
DACHUN YANG*
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex systems, Ministry of Education, Beijing 100875, People's Republic of China. e-mail: [email protected]
DONGYONG YANG
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China. e-mail: [email protected]
*
Corresponding author
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Abstract

Let (, d, μ) be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. We introduce the atomic Hardy space H1(μ) and prove that its dual space is the known space RBMO(μ) in this context. Using this duality, we establish a criterion for the boundedness of linear operators from H1(μ) to any Banach space. As an application of this criterion, we obtain the boundedness of Calderón–Zygmund operators from H1(μ) to L1(μ).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 1 , July 2012 , pp. 9 - 31
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Chen, W., Meng, Y. and Yang, D.Calderón-Zygmund operators on Hardy spaces without the doubling condition. Proc. Amer. Math. Soc. 133 (2005), 2671–2680.CrossRefGoogle Scholar
[2]Coifman, R. R. and Weiss, G.Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes in Math. 242 (Springer-Verlag, 1971).CrossRefGoogle Scholar
[3]Coifman, R. R. and Weiss, G.Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc. 83 (1977), 569–645.CrossRefGoogle Scholar
[4]Heinonen, J.Lectures on Analysis on Metric Spaces (Springer-Verlag, 2001).CrossRefGoogle Scholar
[5]Hu, G., Meng, Y. and Yang, D.New atomic characterization of H 1 space with non-doubling measures and its applications. Math. Proc. Camb. Phil. Soc. 138 (2005), 151–171.CrossRefGoogle Scholar
[6]Hytönen, T.A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa. Publ. Mat. 54 (2010), 485–504.CrossRefGoogle Scholar
[7]Hytönen, T. and Martikainen, H. Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces. J. Geom. Anal., doi:10.1007/s12220-011-9230-z.CrossRefGoogle Scholar
[8]Journé, J.-L.Calderón–Zygmund operators, pseudodifferential operators and the Cauchy integral of Calderón. Lecture Notes in Math. 994 (Springer-Verlag, 1983).CrossRefGoogle Scholar
[9]Liu, L., Yang, Da. and Yang, Do.Atomic Hardy-type spaces between H 1 and L 1 on metric spaces with non-doubling measures. Acta Math. Sin. (Engl. Ser.) 27 (2011), 2445–2468.CrossRefGoogle Scholar
[10]Luukkainen, J. and Saksman, E.Every complete doubling metric space carries a doubling measure. Proc. Amer. Math. Soc. 126 (1998), 531–534.CrossRefGoogle Scholar
[11]Meda, S., Sjögren, P. and Vallarino, M.On the H 1-L 1 boundedness of operators. Proc. Amer. Math. Soc. 136 (2008), 2921–2931.CrossRefGoogle Scholar
[12]Nazarov, F., Treil, S. and Volberg, A.The Tb-theorem on non-homogeneous spaces. Acta Math. 190 (2003), 151–239.CrossRefGoogle Scholar
[13]Stein, E. M. and Weiss, G.On the theory of harmonic functions of several variables. I. The theory of H p-spaces. Acta Math. 103 (1960), 25–62.CrossRefGoogle Scholar
[14]Tolsa, X.BMO, H 1, and Calderón–Zygmund operators for non doubling measures. Math. Ann. 319 (2001), 89–149.CrossRefGoogle Scholar
[15]Tolsa, X.Littlewood–Paley theory and the T(1) theorem with non-doubling measures. Adv. Math. 164 (2001), 57–116.CrossRefGoogle Scholar
[16]Tolsa, X.Painlevé's problem and the semiadditivity of analytic capacity. Acta Math. 190 (2003), 105–149.CrossRefGoogle Scholar
[17]Tolsa, X.The space H 1 for nondoubling measures in terms of a grand maximal operator. Trans. Amer. Math. Soc. 355 (2003), 315–348.CrossRefGoogle Scholar
[18]Wu, J.Hausdorff dimension and doubling measures on metric spaces. Proc. Amer. Math. Soc. 126 (1998), 1453–1459.CrossRefGoogle Scholar
[19]Yosida, K.Functional Analysis (Springer-Verlag, 1995).CrossRefGoogle Scholar