Analyticity of a nonlinear operator associated to the conformal representation in schauder spaces. An integral equation approach

We consider the Riemann map g(zeta ,w) of the complex unit disk to the plan e domain I[zeta] enclosed by the Jordan curve zeta and normalised by the co nditions g(zeta ,w)(0) = omega, g'(zeta ,w) (0) > 0, where w is a point of I[zeta], and we present a nonlinear singular integral equation approach to prove that the nonlinear operator which takes the pair (zeta ,w) to the map g(zeta ,w)((-1)) o zeta is real analytic in Schauder spaces.