h1h^1, bmo, blo and Littlewood-Paley gg-functions with non-doubling measures

  • Guoen Hu

    Zhengzhou Information Science and Technology Institute, China
  • Dachun Yang

    Beijing Normal University, China
  • Dongyong Yang

    Beijing Normal University, China


Let μ\mu be a nonnegative Radon measure on Rd{\mathbb R}^d which satisfies the growth condition that there exist constants C0>0C_0 > 0 and n(0,d]n\in(0,d] such that for all xRdx\in{\mathbb R}^d and r>0r > 0, μ(B(x,r))C0rn\mu(B(x,\,r)) \le C_0 r^n, where B(x,r)B(x,r) is the open ball centered at xx and having radius rr. In this paper, we introduce a local atomic Hardy space hatb1,(μ){h_{\rm atb}^{1,\infty}(\mu)}, a local BMO-type space rbmo(μ){\mathop\mathrm{rbmo}(\mu)} and a local BLO-type space rblo(μ){\mathop\mathrm{rblo}(\mu)} in the spirit of Goldberg and establish some useful characterizations for these spaces. Especially, we prove that the space rbmo(μ){\mathop\mathrm{rbmo}(\mu)} satisfies a John-Nirenberg inequality and its predual is hatb1,(μ){h_{\rm atb}^{1,\infty}(\mu)}. We also establish some useful properties of RBLO(μ){\mathop\mathrm{RBLO}\,(\mu)} and improve the known characterization theorems of RBLO(μ){\mathop\mathrm{RBLO}(\mu)} in terms of the natural maximal function by removing the assumption on the regularity condition. Moreover, the relations of these local spaces with known corresponding function spaces are also presented. As applications, we prove that the inhomogeneous Littlewood-Paley gg-function g(f)g(f) of Tolsa is bounded from hatb1,(μ){h_{\rm atb}^{1,\infty}(\mu)} to L1(μ){L^1(\mu)}, and that [g(f)]2[g(f)]^2 is bounded from rbmo(μ){\mathop\mathrm{rbmo}(\mu)} to rblo(μ){\mathop\mathrm{rblo}(\mu)}.

Cite this article

Guoen Hu, Dachun Yang, Dongyong Yang, h1h^1, bmo, blo and Littlewood-Paley gg-functions with non-doubling measures. Rev. Mat. Iberoam. 25 (2009), no. 2, pp. 595–667

DOI 10.4171/RMI/577