Besov-type spaces of variable smoothness on rough domains

Abstract

The paper puts forward new Besov spaces of variable smoothness $B_{p,q}^{{\phi }_0}(G,\{t_k\})$ and ${\tilde{B}}_{p,q,r}^l(\Omega ,\{t_k\})$ 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 $(\epsilon ,\delta )$-domain. These spaces are shown to be the traces of the spaces $B_{p,q}^{{\phi }_0}({\mathbb{R}}^n,\{t_k\})$ and ${\tilde{B}}_{p,q,r}^l({\mathbb{R}}^n,\{t_k\})$ on domains $G$ and~$\Omega$, respectively. The extension operator ${\mathrm{Ext}}_1:B_{p,q}^{{\phi }_0}(G,\{t_k\})\to B_{p,q}^{{\phi }_0}({\mathbb{R}}^n,\{t_k\})$ is linear, the operator ${\mathrm{Ext}}_2:{\tilde{B}}_{p,q,r}^l(\Omega ,\{t_k\})\to {\tilde{B}}_{p,q,r}^l({\mathbb{R}}^n,\{t_k\})$ 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.

Submitted in Nonlinear Analysis: Theory, Methods and Applications