Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators |
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Authors: | Renjin Jiang Dachun Yang |
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Institution: | 1. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University, Beijing, 100875, People??s Republic of China
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Abstract: | Let L be a linear operator in L 2(? n ) and generate an analytic semigroup {e ?tL } t??0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by ??(L)??(0,??]. Let ?? on (0,??) be of upper type 1 and of critical lower type $\widetilde{p}_{0}(\omega)\in(n/(n+\theta(L)),1]$ and ??(t)=t ?1/?? ?1(t ?1) for t??(0,??). In this paper, the authors first introduce the VMO-type space VMO ??,L (? n ) and the tent space $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ and characterize the space VMO ??,L (? 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 ??,L (? n ), where L * denotes the adjoint operator of L in L 2(? n ) and B ??,L (? n ) the Banach completion of the Orlicz-Hardy space H ??,L (? n ). These results generalize the known recent results by particularly taking ??(t)=t for t??(0,??). |
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