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European Mathematical Society Publishing House
2016-09-19 17:05:47
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
29
2013
1
Real-variable characterizations of Orlicz–Hardy spaces on strongly Lipschitz domains of $\mathbb{R}^n$
Dachun
Yang
Beijing Normal University, BEIJING, CHINA
Sibei
Yang
Beijing Normal University, BEIJING, CHINA
Orlicz–Hardy space, divergence form elliptic operator, strongly Lipschitz domain, Dirichlet boundary condition, Gaussian property, nontangential maximal function, Lusin area function, atom
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 $L$.
Fourier analysis
Partial differential equations
Operator theory
General
237
292
10.4171/RMI/719
http://www.ems-ph.org/doi/10.4171/RMI/719