Descriptions of traces of weighted Sobolev spaces to Ahlfors--David regular sets in the case $p=1$

Abstract

Given $n \in \mathbb{N}$, an Ahlors--David $n$-regular set $S \subset \mathbb{R}^{n+1}$, and a weight $γ$ satisfying the local Muckenhoupt $A_{1}$-condition, we present a complete intrinsic description of the trace-space $W_{1}^{1}(\mathbb{R}^{n+1},γ)|_{S}$ of the weighted first-order Sobolev space $W_{1}^{1}(\mathbb{R}^{n+1},γ)$ to $S$. Furthermore, we construct a new family of nonlinear bounded extension operators acting from $W_{1}^{1}(\mathbb{R}^{n+1},γ)|_{S}$ to $W_{1}^{1}(\mathbb{R}^{n+1},γ)$. Finally, we find conditions on $γ$ that sufficient for the existence of a bounded linear extension operator from $W_{1}^{1}(\mathbb{R}^{n+1},γ)|_{S}$ to $W_{1}^{1}(\mathbb{R}^{n+1},γ)$.