Abstract
One defines a non-homogeneous space (X,μ) as a metric space equipped with a non-doubling measure μ so that the volume of the ball with center x, radius r has an upper bound of the form r n for some n>0. The aim of this paper is to study the boundedness of Calderón–Zygmund singular integral operators T on various function spaces on (X,μ) such as the Hardy spaces, the L p spaces, and the regularized BMO spaces. This article thus extends the work of X. Tolsa (Math. Ann. 319:89–149, 2011) on the non-homogeneous space (ℝn,μ) to the setting of a general non-homogeneous space (X,μ). Our framework of the non-homogeneous space (X,μ) is similar to that of Hytönen (2011) and we are able to obtain quite a few properties similar to those of Calderón–Zygmund operators on doubling spaces such as the weak type (1,1) estimate, boundedness from Hardy space into L 1, boundedness from L ∞ into the regularized BMO, and an interpolation theorem. Furthermore, we prove that the dual space of the Hardy space is the regularized BMO space, obtain a Calderón–Zygmund decomposition on the non-homogeneous space (X,μ), and use this decomposition to show the boundedness of the maximal operators in the form of a Cotlar inequality as well as the boundedness of commutators of Calderón–Zygmund operators and BMO functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bui, T.A., Duong, X.T.: Endpoint estimates for maximal operator and boundedness of maximal commutators. Preprint
Coifman, R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)
Di Fazio, G., Gutiérrez, C.E., Lanconelli, E.: Covering theorems, inequalities on metric spaces and applications to PDE’s. Math. Ann. 341(2), 255–291 (2008)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. Springer, New York (2001)
Hofmann, S., Mayboroda, S.: Hardy and BMO spaces associated to divergence form elliptic operators. Math. Ann. 344(1), 37–116 (2009)
Hytönen, T.: A framework for non-homogenous analysis on metric spaces, and RBMO spaces of Tolsa. Preprint (2011)
Hytönen, T., Martikainen, H.: Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces. Preprint
Hytönen, T., Yang, D., Yang, D.: The Hardy space H 1 on non-homogeneous metric spaces. Preprint
Journé, J.-L.: Calderón–Zygmund Operators, Pseudo-differential Operators and the Cauchy Integral of Calderón. Lecture Notes in Mathematics, vol. 994. Springer, Berlin (1983)
Nazarov, F., Treil, S., Volberg, A.: Cauchy integral and Calderón–Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 15, 703–726 (1997)
Nazarov, F., Treil, S., Volberg, A.: Weak type estimates and Cotlar inequalities for Calderón–Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 9, 463–487 (1998)
Nazarov, F., Treil, S., Volberg, A.: The Tb-theorem on non-homogeneous spaces. Acta Math. 190(2), 151–239 (2003)
Mateu, J., Mattila, P., Nicolau, A., Orobitg, J.: BMO for non doubling measures. Duke Math. J. 102, 533–565 (2000)
Tchoundja, E.: Carleson measures for the generalized Bergman spaces via a T(1)-type theorem. Ark. Mat. 46(2), 377–406 (2008)
Tolsa, X.: BMO, H 1, and Calderón–Zygmund operators for non doubling measures. Math. Ann. 319, 89–149 (2001)
Tolsa, X.: A proof of the weak (1,1) inequality for singular integrals with non doubling measures based on a Calderón–Zygmund decomposition. Publ. Mat. 45, 163–174 (2001)
Verdera, J.: On the T(1) theorem for the Cauchy integral. Ark. Mat. 38, 183–199 (2000)
Volberg, A., Wick, B.D.: Bergman-type singular operators and the characterization of Carleson measures for Besov–Sobolev spaces on the complex ball. Preprint arXiv:0910.1142 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Loukas Grafakos.
The Anh Bui was supported by a Macquarie University scholarship.
Xuan Thinh Duong was supported by research grants from Australian Research Council and Macquarie University.
This article is a slightly revised version of our paper arXiv:1009.1274v2. Later, the paper arXiv:1011.2937 of Hytönen et al. is an independent work and contains some results similar to ours in Sect. 6.3 by using a different approach.
Rights and permissions
About this article
Cite this article
Bui, T.A., Duong, X.T. Hardy Spaces, Regularized BMO Spaces and the Boundedness of Calderón–Zygmund Operators on Non-homogeneous Spaces. J Geom Anal 23, 895–932 (2013). https://doi.org/10.1007/s12220-011-9268-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-011-9268-y