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 Sierpiński carpet, the Sierpiński gasket, and the von Koch snowflake.