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, we define a surjective operator \mbox{$Tr: W^1_1(\Omega_K)\rightarrow X(\Omega_K)$}, 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. Mal\'y 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 about non-existence of the right inverse for domain $\Omega$ with regular boundary, which explains Banach space geometry cause for this phenomenon.