Abstract
We prove a Tb theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound μ(B(x,r))≤Cr d. Our spaces are only assumed to satisfy the geometric doubling property: every ball of radius r can be covered by at most N balls of radius r/2. A key ingredient is the construction of random systems of dyadic cubes in such spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bramanti, M.: Singular integrals in nonhomogeneous spaces: L 2 and L p continuity from Hölder estimates. Rev. Mat. Iberoam. 26(1), 347–366 (2010)
Christ, M.: A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60/61(2), 601–628 (1990)
Coifman, R.R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogènes. In: Étude de certaines intégrales singulières. Lecture Notes in Mathematics, vol. 242. Springer, Berlin (1971)
David, G., Journé, J.-L., Semmes, S.: Opérateurs de Calderón-Zygmund, fonctions para-accretives et interpolation. Rev. Mat. Iberoam. 1(4), 1–56 (1985)
García-Cuerva, J., Gatto, A.E.: Lipschitz spaces and Calderón-Zygmund operators associated to non-doubling measures. Publ. Mat. 49(2), 285–296 (2005)
Hytönen, T.: A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa. Publ. Mat. 54(2), 485–504 (2010)
Luukkainen, J., Saksman, E.: Every complete doubling metric space carries a doubling measure. Proc. Am. Math. Soc. 126(2), 531–534 (1998)
McIntosh, A., Meyer, Y.: Algèbres d’opérateurs définis par des intégrales singulières. C. R. Acad. Sci. Paris Sér. I Math. 301(8), 395–397 (1985)
Macías, R.A., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33(3), 257–270 (1979)
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)
Tchoundja, E.: Carleson measures for the generalized Bergman spaces via a T(1)-type theorem. Ark. Mat. 46(2), 377–406 (2008)
Volberg, A., Wick, B.D.: Bergman-type singular operators and the characterization of Carleson measures for Besov–Sobolev spaces on the complex ball. Am. J. Math. (in press). Preprint. arXiv:0910.1142
Verbitsky, I.E., Wheeden, R.L.: Weighted norm inequalities for integral operators. Trans. Am. Math. Soc. 350(8), 3371–3391 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Michael Lacey.
The authors are supported by the Academy of Finland through projects 130166 and 218148 (“L p methods in harmonic analysis”) and 133264 (“Stochastic and harmonic analysis, interactions and applications”).
Rights and permissions
About this article
Cite this article
Hytönen, T., Martikainen, H. Non-homogeneous Tb Theorem and Random Dyadic Cubes on Metric Measure Spaces. J Geom Anal 22, 1071–1107 (2012). https://doi.org/10.1007/s12220-011-9230-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-011-9230-z