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

Abstract

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

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