Trace Operator on von Koch’s Snowflake

We study properties of the boundary trace operator on the Sobolev space W¹₁(Ω ) W 1 1 ( Ω ) . Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222 (1), 1-14 2016), we define a surjective operator Tr: W¹₁(Ω K)→ X(Ω K) T r : W 1 1 ( Ω K ) → X ( Ω K ) , where Ω K Ω K is von Koch’s snowflake and X(Ω K) X ( Ω K ) is a trace space with the quotient norm. Since Ω K Ω K is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý (2017) that there exists a right inverse to Tr , i.e. a linear operator S: X(Ω K) → W¹₁(Ω K) S : X ( Ω K ) → W 1 1 ( Ω K ) such that Tr ∘ S= IdX(Ω K) T r ∘ S = I d X ( Ω K ) . In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch’s snowflake. Moreover we identify the isomorphism class of the trace space as ₁ ℓ 1 . As an additional consequence of our approach we obtain a simple proof of the Peetre’s theorem (Special Issue 2 , 277-282 1979) about non-existence of the right inverse for domain Ω Ω with regular boundary, which explains Banach space geometry cause for this phenomenon.