2024/10/03 by Huaying Wei, Wei, Huaying, Michel Zinsmeister +1
Computer Science · Mathematics · #30C62 #31A05 #31C25 #Advanced Mathematical Modeling in Engineering #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2410.02183
openalex publication_date 2024/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Γ be a rectifiable Jordan curve in the complex plane, Ωi and Ωe respectively the interior and exterior domains of Γ, and p≥ 2. Let E be the vector space of functions defined on Γ consisting of restrictions to Γ of functions in C1(\mathbb C). We define three semi-norms on E: \beginenumerate \item \Vert u‖i=((1)/(2π)\iintΩi|∇ Ui(z)|pλΩi2-p(z) dxdy)1/p, where Ui is the harmonic extension of u∈ E to Ωi and λΩi is the density of hyperbolic metric of domain Ωi, \item ‖u‖e defined similarly for the exterior domain Ωe, \item ‖u‖Bp(Γ) =((1)/(4π2)\iintΓ×Γ(|u(z)-u(ζ)|p)/(|z-ζ|2)|dz| |dζ|)1/p. \endenumerate The equivalences of these three semi-norms are well-known when Γ is the unit circle. We prove that they are equivalent if and only if Γ is a chord-arc curve.