Besov-type spaces of variable smoothness on rough domains

Abstract

The paper puts forward new Besov spaces of variable smoothness $B^{\varphi_{0}}_{p,q}(G,\{t_{k}\})$ and $\widetilde{B}^{l}_{p,q,r}(\Omega,\{t_{k}\})$ 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~$\Omega$ is an $(\varepsilon,\delta)$-domain. These spaces are shown to be the traces of the spaces $B^{\varphi_{0}}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ and $\widetilde{B}^{l}_{p,q,r}(\mathbb{R}^{n},\{t_{k}\})$ on domains $G$ and~$\Omega$, respectively. The extension operator $\operatorname{Ext}_{1}:B^{\varphi_{0}}_{p,q}(G,\{t_{k}\}) \to B^{\varphi_{0}}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ is linear, the operator $\operatorname{Ext}_{2}:\widetilde{B}^{l}_{p,q,r}(\Omega,\{t_{k}\}) \to \widetilde{B}^{l}_{p,q,r}(\mathbb{R}^{n},\{t_{k}\})$ 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.