Abstract
In this chapter, we introduce and study a class of metric measure spaces \((\mathcal{X},d,\nu )\), which include both Euclidean spaces with nonnegative Radon measures satisfying the polynomial growth condition and spaces of homogeneous type as special cases. We also introduce the BMO-type space \(\mathrm{RBMO}\,(\mathcal{X},\,\nu )\) and the atomic Hardy space \({H}^{1}(\mathcal{X},\,\nu )\) in this setting, establish the John–Nirenberg inequality for \(\mathrm{RBMO}\,(\mathcal{X},\,\nu )\) and some equivalent characterizations of \(\mathrm{RBMO}\,(\mathcal{X},\,\nu )\) and \({H}^{1}(\mathcal{X},\,\nu )\), respectively, and show that the dual space of \({H}^{1}(\mathcal{X},\,\nu )\) is \(\mathrm{RBMO}\,(\mathcal{X},\,\nu )\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
P. Auscher, T. Hytönen, Orthonormal bases of regular wavelets in spaces of homogeneous type. Appl. Comput. Harmon. Anal. 34, 266–296 (2013)
T.A. Bui, X.T. Duong, Hardy spaces, regularized BMO spaces and the boundedness of Calderón–Zygmund operators on non-homogeneous spaces. J. Geom. Anal. 23, 895–932 (2013)
M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60/61, 601–628 (1990)
R.R. Coifman, G. Weiss, Analyse Harmonique Non-commutative sur Certains Spaces Homogènes. Lecture Notes in Mathematics, vol. 242 (Springer, Berlin/New York, 1971)
R.R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)
J. Heinonen, Lectures on Analysis on Metric Spaces (Springer, New York, 2001)
G. Hu, Y. Meng, D. Yang, A new characterization of regularized BMO spaces on non-homogeneous spaces and its applications. Ann. Acad. Sci. Fenn. Math. 38, 3–27 (2013)
G. Hu, Da. Yang, Do. Yang, A new characterization of RBMO (μ) by John–Strömberg sharp maximal functions. Czechoslovak Math. J. 59, 159–171 (2009)
T. Hytönen, A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa. Publ. Mat. 54, 485–504 (2010)
T. Hytönen, H. Martikainen, Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces. J. Geom. Anal. 22, 1071–1107 (2012)
T. Hytönen, Da. Yang, Do. Yang, The Hardy space H 1 on non-homogeneous metric spaces. Math. Proc. Camb. Phil. Soc. 153, 9–31 (2012)
A.K. Lerner, On the John–Strömberg characterization of BMO for nondoubling measures. Real Anal. Exchange 28, 649–660 (2002/03)
J. Luukkainen, E. Saksman, Every complete doubling metric space carries a doubling measure. Proc. Am. Math. Soc. 126, 531–534 (1998)
W. Rudin, Functional Analysis (McGraw-Hill Book, New York/Düsseldorf/Johannesburg, 1973)
J.O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces. Indiana Univ. Math. J. 28, 511–544 (1979)
J. Wu, Hausdorff dimension and doubling measures on metric spaces. Proc. Am. Math. Soc. 126, 1453–1459 (1998)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Yang, D., Yang, D., Hu, G. (2013). The Hardy Space \({H}^{1}(\mathcal{X},\,\nu )\) and Its Dual Space \(\mathrm{RBMO}(\mathcal{X},\nu )\) . In: The Hardy Space H1 with Non-doubling Measures and Their Applications. Lecture Notes in Mathematics, vol 2084. Springer, Cham. https://doi.org/10.1007/978-3-319-00825-7_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-00825-7_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-00824-0
Online ISBN: 978-3-319-00825-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)