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

Abstract

Let $\Omega$ 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 $L^2 (\Omega)$ with the Dirichlet boundary condition, and the heat semigroup generated by $L$ having the Gaussian property $(G_{\mathrm{diam}(\Omega)})$ with the regularity of its kernels measured by $\mu\in(0,1]$, where $\mathrm{diam}(\Omega)$ denotes the diameter of $\Omega$. Let $\Phi$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty)$ of upper type 1 and of strictly critical lower type $p_{\Phi}\in(n/(n+\mu),1]$. In this paper, the authors introduce the Orlicz–Hardy space $H_{\Phi,\,r}(\Omega)$ by restricting arbitrary elements of the Orlicz–Hardy space $H_{\Phi}(\mathbb{R}^n)$ to $\Omega$ and establish its atomic decomposition by means of the Lusin area function associated with $\{e^{-tL}\}_{t\ge0}$. Applying this, the authors obtain two equivalent characterizations of $H_{\Phi,\,r}(\Omega)$ in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $L$.