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.