Real-variable Characterizations of Orlicz-Hardy Spaces on Strongly Lipschitz Domains of ${\mathbb{R}}^n$

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