Trace Operator on von Koch’s Snowflake

Abstract

We study properties of the boundary trace operator on the Sobolev space \(W^1_1(\Omega )\). Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222(1), 1-14 2016), we define a surjective operator \(Tr: W^1_1(\Omega _K)\rightarrow X(\Omega _K)\), where \(\Omega _K\) is von Koch’s snowflake and \(X(\Omega _K)\) is a trace space with the quotient norm. Since \(\Omega _K\) 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 \(S: X(\Omega _K) \rightarrow W^1_1(\Omega _K)\) such that \(Tr \circ S= Id_{X(\Omega _K)}\). 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 \(\ell _1\). 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 \(\Omega \) with regular boundary, which explains Banach space geometry cause for this phenomenon.