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{$Tr:W_1^1({\Omega }_K)\to X({\Omega }_K)$}, 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 that there exists a right inverse to $Tr$, i.e. a linear operator $S:X({\Omega }_K)\to W_1^1({\Omega }_K)$ such that $Tr\circ S=I{d}_{X({\Omega }_K)}$. 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.

Introduction, abstract was changed. We added important references