Traces and extensions of certain weighted Sobolev spaces on $\mathbb{R}^n$ and Besov functions on Ahlfors regular compact subsets of $\mathbb{R}^n$

Abstract

The focus of this paper is on Ahlfors $Q$-regular compact sets $E\subset\mathbb{R}^n$ such that, for each $Q-2<\alpha\le 0$, the weighted measure $\mu_{\alpha}$ given by integrating the density $\omega(x)=\text{dist}(x, E)^\alpha$ yields a Muckenhoupt $\mathcal{A}_p$-weight in a ball $B$ containing $E$. For such sets $E$ we show the existence of a bounded linear trace operator acting from $W^{1,p}(B,\mu_\alpha)$ to $B^\theta_{p,p}(E, \mathcal{H}^Q\vert_E)$ when $0<\theta<1-\tfrac{\alpha+n-Q}{p}$, and the existence of a bounded linear extension operator from $B^\theta_{p,p}(E, \mathcal{H}^Q\vert_E)$ to $W^{1,p}(B, \mu_\alpha)$ when $1-\tfrac{\alpha+n-Q}{p}\le \theta<1$. We illustrate these results with $E$ as the Sierpi\'nski carpet, the Sierpi\'nski gasket, and the von Koch snowflake.