Journal of Fourier Analysis and Applications

, Volume 17, Issue 1, pp 1–35 | Cite as

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators



Let L be a linear operator in L 2(ℝ n ) and generate an analytic semigroup {e tL }t≥0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by θ(L)∈(0,∞]. Let ω on (0,∞) be of upper type 1 and of critical lower type \(\widetilde{p}_{0}(\omega)\in(n/(n+\theta(L)),1]\) and ρ(t)=t −1/ω −1(t −1) for t∈(0,∞). In this paper, the authors first introduce the VMO-type space VMOρ,L(ℝ n ) and the tent space \(T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})\) and characterize the space VMOρ,L(ℝ n ) via the space \(T^{\infty}_{\omega,\mathrm{v}}({{\mathbb{R}}}^{n+1}_{+})\). Let \(\widetilde{T}_{\omega}({{\mathbb{R}}}^{n+1}_{+})\) be the Banach completion of the tent space \(T_{\omega}({\mathbb{R}}^{n+1}_{+})\). The authors then prove that \(\widetilde{T}_{\omega}({\mathbb{R}}^{n+1}_{+})\) is the dual space of \(T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})\). As an application of this, the authors finally show that the dual space of \(\mathrm{VMO}_{\rho,L^{\ast}}({\mathbb{R}}^{n})\) is the space B ω,L(ℝ n ), where L * denotes the adjoint operator of L in L 2(ℝ n ) and B ω,L(ℝ n ) the Banach completion of the Orlicz-Hardy space H ω,L(ℝ n ). These results generalize the known recent results by particularly taking ω(t)=t for t∈(0,∞).


Operator Orlicz function Orlicz-Hardy space VMO Predual space Banach completion Tent space Molecule 

Mathematics Subject Classification (2000)

42B35 42B30 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of EducationBeijing Normal UniversityBeijingPeople’s Republic of China

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