JournalsrmiVol. 29, No. 1pp. 237–292

Real-variable characterizations of Orlicz–Hardy spaces on strongly Lipschitz domains of Rn\mathbb{R}^n

  • Dachun Yang

    Beijing Normal University, China
  • Sibei Yang

    Beijing Normal University, China
Real-variable characterizations of Orlicz–Hardy spaces on strongly Lipschitz domains of $\mathbb{R}^n$ cover
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Abstract

Let Ω\Omega be a strongly Lipschitz domain of Rn\mathbb{R}^n, whose complement in Rn\mathbb{R}^n is unbounded. Let LL be a second order divergence form elliptic operator on L2(Ω)L^2 (\Omega) with the Dirichlet boundary condition, and the heat semigroup generated by LL having the Gaussian property (Gdiam(Ω))(G_{\mathrm{diam}(\Omega)}) with the regularity of its kernels measured by μ(0,1]\mu\in(0,1], where diam(Ω)\mathrm{diam}(\Omega) denotes the diameter of Ω\Omega. Let Φ\Phi be a continuous, strictly increasing, subadditive and positive function on (0,)(0,\infty) of upper type 1 and of strictly critical lower type pΦ(n/(n+μ),1]p_{\Phi}\in(n/(n+\mu),1]. In this paper, the authors introduce the Orlicz–Hardy space HΦ,r(Ω)H_{\Phi,\,r}(\Omega) by restricting arbitrary elements of the Orlicz–Hardy space HΦ(Rn)H_{\Phi}(\mathbb{R}^n) to Ω\Omega and establish its atomic decomposition by means of the Lusin area function associated with {etL}t0\{e^{-tL}\}_{t\ge0}. Applying this, the authors obtain two equivalent characterizations of HΦ,r(Ω)H_{\Phi,\,r}(\Omega) in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by LL.

Cite this article

Dachun Yang, Sibei Yang, Real-variable characterizations of Orlicz–Hardy spaces on strongly Lipschitz domains of Rn\mathbb{R}^n. Rev. Mat. Iberoam. 29 (2013), no. 1, pp. 237–292

DOI 10.4171/RMI/719