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The focus of this paper is on Ahlfors Q -regular compact sets $$E\subset \mathbb {R}^n$$ E ⊂ R n such that, for each $$Q-2<\alpha \le 0$$ Q - 2 < α ≤ 0 , the weighted measure $$\mu _{\alpha }$$ μ α given by integrating the density $$\omega (x)=\text {dist}(x, E)^\alpha $$ ω ( x ) = dist ( x , E ) α yields a Muckenhoupt $$\mathcal {A}_p$$ 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 $$W^{1,p}(B,\mu _\alpha )$$ W 1 , p ( B , μ α ) to $$B^\theta _{p,p}(E, \mathcal {H}^Q\vert _E)$$ B p , p θ ( E , H Q | E ) when $$0<\theta <1-\tfrac{\alpha +n-Q}{p}$$ 0 < θ < 1 - α + n - Q p , and the existence of a bounded linear extension operator from $$B^\theta _{p,p}(E, \mathcal {H}^Q\vert _E)$$ B p , p θ ( E , H Q | E ) to $$W^{1,p}(B, \mu _\alpha )$$ W 1 , p ( B , μ α ) when $$1-\tfrac{\alpha +n-Q}{p}\le \theta <1$$ 1 - α + n - Q p ≤ θ < 1 . We illustrate these results with E as the Sierpiński carpet, the Sierpiński gasket, and the von Koch snowflake.

complex analysis and its synergies

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