2019/03/04 by Krystian Kazaniecki, Kazaniecki, Krystian, Michał Wojciechowski +1
Computer Science · Mathematics · #46B03 #46B45 (Secondary) #46E35 (Primary) 46B25 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1903.01100
openalex publication_date 2019/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study properties of the boundary trace operator on the Sobolev space W11(Ω). Using the density result by Koskela and Zhang, we define a surjective operator Tr: W11(ΩK)→ X(ΩK), where ΩK is von Koch's snowflake and X(ΩK) is a trace space with the quotient norm. Since ΩK 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(ΩK) → W11(ΩK) such that Tr ∘ S= IdX(ΩK). 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 ℓ1. 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 Ω with regular boundary, which explains Banach space geometry cause for this phenomenon.