2025/11/19 by Caetano, António, Chandler-Wilde, Simon N., Hewett, David P. · 1 citation
Engineering · Mathematics · #28A80 #46B70 #46E35 #65N30 #65R20 #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Dynamics and Fractals #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.2511.15213
openalex publication_date 2025/11/19 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28
We study fractal sets Γ⊂ ℝn with non-empty interior Ω, that are attractors of iterated function systems (IFSs) of contracting similarities satisfying the open set condition. Examples for n=2 are the closures of the Koch snowflake domain and the Gosper island domain. Our first result is that Ω is thick in the sense of Triebel. A consequence is that C0^∞(Ω) is dense in the Sobolev space HsΓ:= \ϕ∈ Hs(ℝn): supp(ϕ)⊂ Γ\ for all s∈ℝ. Our second result, accompanied by results on pointwise multiplication by characteristic functions and uniform extension operators, is that the spaces \Hs(Ω)\s∈ ℝ, where Hs(Ω):=\ϕ|Ω: u∈ Hs(ℝn)\, form an interpolation scale. This is established as a special case of new extension and interpolation results for Besov and Triebel-Lizorkin spaces, applying to large classes of domains Ω that are thick and have boundary with Assouad dimension