2006/07/31 by Pavel Kargaev, Kargaev, Pavel, Evgeny Korotyaev +1
Mathematics · #(34B30) #30C20 #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.CV #math.SP #msc:30C20
paper · pdf · doi:10.48550/arxiv.math/0607814
arxiv created 2006/07/31 · openalex publication_date 2006/07/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the properties of a conformal mapping z(k) from the plane without vertical slits \Gn=[un-ihn, un+ihn], n∈\Z and h=(hn)n∈\Z∈ ℓ2, onto the complex plane without horizontal slits \gnß\R, n∈\Z, with the asymptotics z(iv)=iv+ o(1), v→\iy. Here un+1-un≥ 1, n∈ \Z. Introduce the sequences l=(|\gn|)n∈\Z. % where Jn≥ 0,Jn2=∫\Gn|\Im z(k,h)||dk|/π. We obtain a priori two-sided estimates for ‖h‖p,ø, ‖l‖p,ø, where %‖h‖_øp is the norm of the Banach space %the extension of i)-ii) for the case h∈ℓ_øp, where %ℓ_øp,1≤ p≤ 2 with % the norm ‖h‖p,øp=∑ øn|hn|p, 1≤ p≤ 2 with any weight øn≥ 1, n∈ \Z. Moreover, we determine other estimates.