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