2019/12/30 by Igor Chyzhykov, Chyzhykov, Igor, Yulia Kosanyak +1
Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1912.12872
openalex publication_date 2019/12/30 · openalex created_date 2020/01/10 · openalex updated_date 2026/07/28
Let E be an arbitrary closed set on the unit circle ∂ \mathbbD, u be a harmonic function on the unit disk \mathbbD satisfying |u(z)|\lesssim (1-|z|)γρ-q(z) where ρ(z)= \mathop\rm dist(z, E), γ, q are some real constants, γ≤ q. We establish an estimate of the conjugate u of the same type which is sharp in some sense and in the case E=∂ D coincides with known estimates. As an application we describe growth classes defined by the non-radial condition |u(z)|\lesssim ρ-q(z) in terms of smoothness of the Stieltjes measure associated to the harmonic function u.