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Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of \(\mathbb{R}^n\). (English) Zbl 1266.42056

Authors’ 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 \(\left(G_{\text{diam}(\Omega)}\right)\) with the regularity of its kernels measured by \(\mu\in(0,1],\) where 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 \(p_{\Phi}\in\left(n/(n+\mu),1\right].\) In this paper, the authors introduce the Orlicz-Hardy space \(H_{\Phi, r}(\Omega)\) by restricting arbitrary elements of the Orlicz-Hardy space \(H_{\Phi}\left(\mathbb{R}^{n}\right)\) to \(\Omega\) and establish its atomic decomposition by means of the Lusin area function associated with \(\left\{e^{-tL}\right\}_{t\geq 0}.\) 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 \(L\).”

MSC:

42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
35J25 Boundary value problems for second-order elliptic equations
42B37 Harmonic analysis and PDEs
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