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The Hardy space H1 on non-homogeneous metric spaces


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).

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[2]R. R. Coifman and G. Weiss Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes in Math. 242 (Springer-Verlag, 1971).

[3]R. R. Coifman and G. Weiss Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc. 83 (1977), 569645.

[4]J. Heinonen Lectures on Analysis on Metric Spaces (Springer-Verlag, 2001).

[6]T. Hytönen A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa. Publ. Mat. 54 (2010), 485504.

[8]J.-L. Journé Calderón–Zygmund operators, pseudodifferential operators and the Cauchy integral of Calderón. Lecture Notes in Math. 994 (Springer-Verlag, 1983).

[9]L. Liu , Da. Yang and Do. Yang Atomic Hardy-type spaces between H1 and L1 on metric spaces with non-doubling measures. Acta Math. Sin. (Engl. Ser.) 27 (2011), 24452468.

[10]J. Luukkainen and E. Saksman Every complete doubling metric space carries a doubling measure. Proc. Amer. Math. Soc. 126 (1998), 531534.

[11]S. Meda , P. Sjögren and M. Vallarino On the H1-L1 boundedness of operators. Proc. Amer. Math. Soc. 136 (2008), 29212931.

[12]F. Nazarov , S. Treil and A. Volberg The Tb-theorem on non-homogeneous spaces. Acta Math. 190 (2003), 151239.

[13]E. M. Stein and G. Weiss On the theory of harmonic functions of several variables. I. The theory of Hp-spaces. Acta Math. 103 (1960), 2562.

[14]X. Tolsa BMO, H1, and Calderón–Zygmund operators for non doubling measures. Math. Ann. 319 (2001), 89149.

[15]X. Tolsa Littlewood–Paley theory and the T(1) theorem with non-doubling measures. Adv. Math. 164 (2001), 57116.

[16]X. Tolsa Painlevé's problem and the semiadditivity of analytic capacity. Acta Math. 190 (2003), 105149.

[17]X. Tolsa The space H1 for nondoubling measures in terms of a grand maximal operator. Trans. Amer. Math. Soc. 355 (2003), 315348.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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