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BMO-Estimates for Maximal Operators via Approximations of the Identity with Non-Doubling Measures

  • Dachun Yang (a1) and Dongyong Yang (a1)
Abstract

Let μ be a nonnegative Radon measure on ℝ d that satisfies the growth condition that there exist constants C 0 > 0 and n ∈ (0, d] such that for all x ∈ ℝ d and r > 0, μ(B(x, r)) ≤ C 0 rn , where B(x, r) is the open ball centered at x and having radius r. In this paper, the authors prove that if f belongs to the BMO-type space RBMO(μ) of Tolsa, then the homogeneous maximal function S( f ) (when ℝ d is not an initial cube) and the inhomogeneous maximal function ℳ S ( f ) (when ℝ d is an initial cube) associated with a given approximation of the identity S of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, S and ℳ S are bounded from RBMO(μ) to the BLO-type space RBLO(μ). The authors also prove that the inhomogeneous maximal operator ℳ S is bounded from the local BMO-type space rbmo(μ) to the local BLO-type space rblo(μ).

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Dachun Yang is supported by National Science Foundation for Distinguished Young Scholars (No. 10425106) and NCET (No. 04-0142) of Ministry of Education of China.

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References
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[1] Bennett, C., De Vore, R. A., and Sharpley, R., Weak-L1 and B MO. Ann. of Math. (2) 113(1981), no. 13, 601–611. doi:10.2307/2006999
[2] Hu, G., Da. Yang, and Do. Yang, h1, bmo, blo and Littlewood-Paley g-functions with non-doubling measures. Rev. Mat. Iberoam. 25(2009), no. 2, 595–667.
[3] Jiang, Y., Spaces of type BLO for non-doubling measures. Proc. Amer. Math. Soc. 133(2005), no. 7, 2101–2107. doi:10.1090/S0002-9939-05-07795-6
[4] Mateu, J., Mattila, P., Nicolau, A., and Orobitg, J., B MO for nondoubling measures. Duke Math. J. 102(2000), no. 3, 533–565. doi:10.1215/S0012-7094-00-10238-4
[5] Tolsa, X., B MO, H1 and Calderón-Zygmund operators for non doubling measures. Math. Ann. 319(2001), no. 1, 89–149. doi:10.1007/PL00004432
[6] Tolsa, X., Littlewood-Paley theory and the T(1) theorem with non-doubling measures. Adv. Math. 164(2001), no. 1, 57–116. doi:10.1006/aima.2001.2011
[7] Tolsa, X., The space H1 for nondoubling measures in terms of a grand maximal operator. Trans. Amer. Math. Soc. 355(2003), no. 1, 315–348. doi:10.1090/S0002-9947-02-03131-8
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[12] Tolsa, X., Painlevé's problem and analytic capacity. Collect. Math. 2006, Extra, 89–125.
[13] Verdera, J., The fall of the doubling condition in Calderón-Zygmund theory. Publ. Mat. 2002, Extra, 275–292.
[14] Yang, Da. and Yang, Do., Endpoint estimates for homogeneous Littlewood-Paley g-functions with non-doubling measures. J. Funct. Spaces Appl. 7(2009), no. 2, 187–207.
[15] Yang, Da. and Yang, Do., Uniform boundedness for approximations of the identity with nondoubling measures. J. Inequal. Appl. (2007), Art. ID 19574, 25 pp.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
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