Trace Operator on von Koch’s Snowflake

Abstract We study properties of the boundary trace operator on the Sobolev space W¹₁ () 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 (K) X (K) T r: W 1 1 (Ω K) → X (Ω K), where K Ω K is von Koch’s snowflake and X (K) X (Ω K) is a trace space with the quotient norm. Since 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 (K) W¹₁ (K) S: X (Ω K) → W 1 1 (Ω K) such that Tr S= Idₗ (₊) 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 with regular boundary, which explains Banach space geometry cause for this phenomenon.