Trace Operator on von Koch’s Snowflake

We study properties of the boundary trace operator on the Sobolev space \(W^1_1(\Omega )\) $W_1^1(\mathrm{\Omega })$ . Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222(1), 1-14 2016), we define a surjective operator \(Tr: W^1_1(\Omega _K)\rightarrow X(\Omega _K)\) $Tr:W_1^1({\mathrm{\Omega }}_K)\to X({\mathrm{\Omega }}_K)$ , where \(\Omega _K\) ${\mathrm{\Omega }}_K$ is von Koch’s snowflake and \(X(\Omega _K)\) $X({\mathrm{\Omega }}_K)$ is a trace space with the quotient norm. Since \(\Omega _K\) ${\mathrm{\Omega }}_K$ 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 \(S: X(\Omega _K) \rightarrow W^1_1(\Omega _K)\) $S:X({\mathrm{\Omega }}_K)\to W_1^1({\mathrm{\Omega }}_K)$ such that \(Tr \circ S= Id_{X(\Omega _K)}\) $Tr\circ S=I{d}_{X({\mathrm{\Omega }}_K)}$ . 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 \(\ell _1\) ${\ell }_1$ . 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 \(\Omega \) $\mathrm{\Omega }$ with regular boundary, which explains Banach space geometry cause for this phenomenon.