On quasisymmetric homeomorphisms
An orientation preserving homeomorphism \(h\) of the unit circle \(T\) is called quasisymmetric if there exists \(M\geq1\) such that \[ \frac1{M}\leq\frac{h(e^{i(\theta+t)})-h(e^{i\theta})}{h(e^{i\theta})-h(e^{i(\theta-t)})}\leq M \] for all \(\theta, t>0\). Such a homeomorphism is called symmetric if \[ \lim_{t\to 0+}\frac{h(e^{i(\theta+t)})-h(e^{i\theta})}{h(e^{i\theta})-h(e^{i(\theta-t)})}=1 \] uniformly in \(\theta\). Finally, a homeomorphism \(h\) as above is said to belong to the Weil-Petersson class if it has a quasiconformal extension to the unit disc \(\Delta\) with Beltrami coefficient \(\mu\) that satisfies \[ \int_\Delta|\mu(z)|^2(1-|z|^2)^{-2}dxdy<\infty. \] For a given quasisymmetric homeomorphism \(h\) of the unit circle, the authors introduce the following functions and an operator: \[ \phi_h(\zeta,z)=\frac1{2\pi i}\int_T\frac{h(w)}{(1-\zeta w)^2(1-z h(w))}dw, \] \[ \phi_h(z)=\left(\frac1{\pi}\int_\Delta|\phi_h(\zeta,z)|^2d\xi d\eta\right)^{1/2}, \] \[ T_h(\psi(\zeta))=\frac1{\pi}\int_\Delta\phi_h(\zeta, z)\psi(\bar z)dxdy. \] The authors prove that a quasisymmetric homeomorphism \(h\) is symmetric if and only if \(T_h\) is a compact operator, and also if and only if \[ \lim_{|z|\to 1}\phi_h(z)(1-|z|^2)=0. \] As far as the Weil-Petersson class is concerned, they prove that \(h\) belongs to this class if and only if \(T\) is a Hilbert-Schmidt operator, and if and only if \[ \int_\Delta\phi_h^2(z)dxdy<\infty. \]
scientific article; zbMATH DE number 6105489
scientific article