Traces and extensions of certain weighted Sobolev spaces on $$\mathbb {R}^n$$ R n and Besov functions on Ahlfors regular compact subsets of $$\mathbb {R}^n$$ R n
2021
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.
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