Besov-type spaces of variable smoothness on rough domains

TL;DR

This paper introduces new Besov spaces of variable smoothness on rough domains, characterizes their traces, and constructs extension operators, advancing the understanding of function spaces on irregular geometries.

Contribution

It defines new Besov-type spaces on rough domains, establishes their trace properties, and constructs extension operators, expanding the theory of variable smoothness spaces.

Findings

Spaces are traces of Besov spaces on lat domains.

Extension operator or Besov spaces is linear.

Extension operator or weighted Besov spaces is nonlinear.

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.