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 $B_{p,p}^{\theta }(E,{\mathcal{H}}^Q{|}_E)$ when $0<\theta <1-\frac{\alpha +n-Q}{p}$, and the existence of a bounded linear extension operator from $B_{p,p}^{\theta }(E,{\mathcal{H}}^Q{|}_E)$ to $W^{1,p}(B,{\mu }_{\alpha })$ when $1-\frac{\alpha +n-Q}{p}\le \theta <1$. We illustrate these results with $E$ as the Sierpiński carpet, the Sierpiński gasket, and the von Koch snowflake.

20 pages