Orlicz-Hardy Spaces Associated with Divergence Operators on Unbounded Strongly Lipschitz Domains of $\mathbb{R}^n$

Let $Ω$ be either $\mathbb{R}^n$ or an unbounded strongly Lipschitz domain of $\mathbb{R}^n$, and $Φ$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty)$ of upper type 1 and of strictly critical lower type $p_Φ\in(n/(n+1),1]$. Let $L$ be a divergence form elliptic operator on $L^2 (Ω)$ with the Neumann boundary condition and the heat semigroup generated by $L$ have the Gaussian property $(G_{\infty})$. In this paper, the authors introduce the Orlicz-Hardy space $H_{Φ,\,L}(Ω)$ via the nontangential maximal function associated with $\{e^{-t\sqrt{L}}\}_{t\ge0}$, and establish its equivalent characterization in terms of the Lusin area function associated with $\{e^{-t\sqrt{L}}\}_{t\ge0}$. The authors also introduce the "geometrical" Orlicz-Hardy space $H_{Φ,\,z}(Ω)$ via the classical Orlicz-Hardy space $H_Φ(\mathbb{R}^n)$, and prove that the spaces $H_{Φ,\,L}(Ω)$ and $H_{Φ,\,z}(Ω)$ coincide with equivalent norms, from which, characterizations of $H_{Φ,\,L}(Ω)$, including the vertical and the nontangential maximal function characterizations associated with $\{e^{-tL}\}_{t\ge0}$, and the Lusin area function characterization associated with $\{e^{-tL}\}_{t\ge0}$, are deduced. All the above results generalize the well-known results of P. Auscher and E. Russ by taking $Φ(t)\equiv t$ for all $t\in(0,\infty)$.

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