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Besov-type spaces of variable smoothness on rough domains. (English) Zbl 1365.46029
Summary: The paper puts forward new Besov spaces of variable smoothness \(B_{p, q}^{\varphi_0}(G, \{t_k \})\) and \(\widetilde{B}_{p, q, r}^l(\operatorname{\Omega}, \{t_k \})\) on rough domains. A domain \(G\) is either a bounded Lipschitz domain in \(\mathbb{R}^n\) or the epigraph of a Lipschitz function, a domain \(\operatorname{\Omega}\) is an \((\varepsilon, \delta)\)-domain. These spaces are shown to be the traces of the spaces \(B_{p, q}^{\varphi_0}(\mathbb{R}^n, \{t_k \})\) and \(\widetilde{B}_{p, q, r}^l(\mathbb{R}^n, \{t_k \})\) on domains \(G\) and \(\operatorname{\Omega}\), respectively. The extension operator \(\operatorname{Ext}_1 : B_{p, q}^{\varphi_0}(G, \{t_k \}) \to B_{p, q}^{\varphi_0}(\mathbb{R}^n, \{t_k \})\) is linear, the operator \(\operatorname{Ext}_2 : \widetilde{B}_{p, q, r}^l(\operatorname{\Omega}, \{t_k \}) \to \widetilde{B}_{p, q, r}^l(\mathbb{R}^n, \{t_k \})\) is nonlinear. As a corollary, an exact description of the traces of 2-microlocal Besov-type spaces and weighted Besov-type spaces on rough domains is obtained.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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