Analyticity of a Nonlinear Operator Associated to the Conformal Representation in Schauder Spaces. An Integral Equation Approach

We consider the Riemann map g_ζ,w of the complex unit disk to the plane domain 𝕀ζ] enclosed by the Jordan curve ζ and normalized by the conditions g_ζ,w(0) = w, g′_ζ,w(0) > 0, where w is a point of 𝕀ζ], and we present a nonlinear singular integral equation approach to prove that the nonlinear operator which takes the pair (ζ, w) to the map g^(–1)_ζ,w ○ ζ is real analytic in Schauder spaces.