As it is well-known, to a given plane simple closed curve $\zeta$ with nonvanishing tangent vector, one can associate a conformal welding homeomorphism $\mathbf{w}[\zeta]$ of the unit circle to itself, obtained by composing the restriction to the unit circle of a suitably normalized Riemann map of the domain exterior to $\zeta$ with the restriction to the unit circle of the inverse of a suitably normalized Riemann map of the domain interior to $\zeta$. Now we think the functions $\zeta$ and $\mathbf{w}[\zeta]$ as points in a Schauder function space on the unit circle, and we show that the correspondence $\mathbf{w}$ which takes $\zeta$ to $\mathbf{w}[\zeta]$ is real differentiable for suitable exponents of the Schauder spaces involved. Then we show that $\mathbf{w}$ has a right inverse which is the restriction of a holomorphic nonlinear operator.
Differentiability properties of some nonlinear operators associated to the conformal welding of Jordan curves in Schauder spaces
LANZA DE CRISTOFORIS, MASSIMO;
2003
Abstract
As it is well-known, to a given plane simple closed curve $\zeta$ with nonvanishing tangent vector, one can associate a conformal welding homeomorphism $\mathbf{w}[\zeta]$ of the unit circle to itself, obtained by composing the restriction to the unit circle of a suitably normalized Riemann map of the domain exterior to $\zeta$ with the restriction to the unit circle of the inverse of a suitably normalized Riemann map of the domain interior to $\zeta$. Now we think the functions $\zeta$ and $\mathbf{w}[\zeta]$ as points in a Schauder function space on the unit circle, and we show that the correspondence $\mathbf{w}$ which takes $\zeta$ to $\mathbf{w}[\zeta]$ is real differentiable for suitable exponents of the Schauder spaces involved. Then we show that $\mathbf{w}$ has a right inverse which is the restriction of a holomorphic nonlinear operator.Pubblicazioni consigliate
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