Boundary functions on a bounded balanced domain

Abstract

We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂Ω with u(z) = u(λ z) for |λ| = 1, z ∈ ∂Ω we construct a holomorphic function f ∈ $\mathbb{O}$(Ω) such that $u(z) = \int_{\mathbb{D}z} {|f|^p d\mathfrak{L}_{\mathbb{D}z}^2 }$ for z ∈ ∂Ω, where $\mathbb{D}$= {λ ∈ ℂ: |λ| < 1}.