Regularity Properties of a Nonlinear Operator Associated to the Conformal Welding

As it is well known, given a plane simple closed curve $\zeta$ with nonvanishing tangent vector, there exists a pair of suitably normalized Riemann maps $(F,G)$, where $F$ maps the open unit disk $\mathbb{D}$ of $\mathbb{C}$ onto the domain $\mathbb{I}[\zeta]$ interior to $\zeta$, and where $G$ maps the exterior $\mathbb{C} \setminus \mathrm{cl}\,\mathbb{D}$ of $\mathrm{cl}\,\mathbb{D}$ onto the domain $\mathbb{E}[\zeta]$ exterior to $\zeta$. It is also well known that $F$ and $G$ can be extended to boundary homeomorphisms. Thus one can consider the conformal welding homeomorphism $F^{(-1)}\circ G_{|\partial \mathbb{D}}$ of $\partial \mathbb{D}$ to itself, which we denote by $\mathbf{w}[\zeta]$. Now we think both the set of simple closed curves $\zeta$ and the set of welding homeomorphisms as subsets of the Schauder space $C^{m,\alpha}_{*}(\partial \mathbb{D},\mathbb{C} )$ of the $m$ times continuously differentiable complex-valued functions on $\partial\mathbb{D}$ which have $m$-th order $\alpha$-H\"{o}lder continuous derivative, with $\alpha\in ]0,1[$, $m\geq 1$. Then we present some differentiability Theorems for the dependence of $\mathbf{w}[\zeta]$ upon $\zeta$, and a complex analyticity result for a right inverse of $\mathbf{w}[\cdot]$.