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 $\{e^{-tL}{\}}_{t\ge 0}$. 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 $L$.