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

Abstract

Let $\hat{I}\text{c◯}$ 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 $L^2(\hat{I}\text{c◯})$ with the Dirichlet boundary condition, and the heat semigroup generated by $L$ have the Gaussian property $(G_{\mathrm{diam}(\hat{I}\text{c◯})})$ with the regularity of their kernels measured by $\hat{I}¼\in (0,1]$, where $\mathrm{diam}(\hat{I}\text{c◯})$ denotes the diameter of $\hat{I}\text{c◯}$. Let $\hat{I}¦$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty )$ of upper type 1 and of strictly critical lower type $p_{\hat{I}}¦\in (n/(n+\hat{I}¼),1]$. In this paper, the authors introduce the Orlicz-Hardy space $H_{\hat{I}¦,r}(\hat{I}\text{c◯})$ by restricting arbitrary elements of the Orlicz-Hardy space $H_{\hat{I}}¦({\mathbb{R}}^n)$ to $\boz$ 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 $H_{Φ,\,r}(\boz)$ in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $L$.

65 pages, Rev. Mat. Iberoam. (to appear)