We consider the Riemann map $g_{\zeta,w}$ of the complex unit disk to the plane domain $\mathbb I[\zeta]$ enclosed by the Jordan curve $\zeta$ and normalized by the conditions $ g_{\zeta,w}(0) = w$, $ g'_{\zeta,w}(0) > 0$, where $w$ is a point of $\mathbb I[\zeta]$, and we present a nonlinear singular integral equation approach to prove that the nonlinear operator which takes the pair $(\zeta,w)$ to the map $g^{(-1)}_{\zeta,w}\circ\zeta$ is real analytic in Schauder spaces.
Analyticity of a nonlinear operator associated to the conformal representation in Schauder spaces. An integral equation approach
LANZA DE CRISTOFORIS, MASSIMO;
2000
Abstract
We consider the Riemann map $g_{\zeta,w}$ of the complex unit disk to the plane domain $\mathbb I[\zeta]$ enclosed by the Jordan curve $\zeta$ and normalized by the conditions $ g_{\zeta,w}(0) = w$, $ g'_{\zeta,w}(0) > 0$, where $w$ is a point of $\mathbb I[\zeta]$, and we present a nonlinear singular integral equation approach to prove that the nonlinear operator which takes the pair $(\zeta,w)$ to the map $g^{(-1)}_{\zeta,w}\circ\zeta$ is real analytic in Schauder spaces.
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