Conformally natural extensions of vector fields and applications (Q2159162)

The authors study an integral operator extending tangent vector fields along the unit circle \(\mathbb S^1\). For an orientation-preserving homeomorphism \(h\) of \(\mathbb S^1\), denote \(\|h\|_{\text{cr}}\) the cross-ratio distortion norm of \(h\). The authors investigate a conformally natural extension of a continuous tangent vector field along \(\mathbb S^1\). Let \(C^0(\mathbb S^1,\mathbb C)\) be the collection of continuous functions from \(\mathbb S^1\) to \(\mathbb C\). Given an element \(V\in C^0(\mathbb S^1,\mathbb C)\) and \(z\in \mathbb D=\{z\in\mathbb C:|z|<1\}\), define \[ L_0(V)(z)=\frac{(1-|z|^2)^3}{2\pi i}\int_{\mathbb S^1}\frac{V(\zeta)d\zeta}{(1-\overline z\zeta)^3(\zeta-z)}. \] The authors prove the following theorems. Theorem 1. The operator \(L_0\) is conformally natural in the following sense: 1. If \(V\in C^0(\mathbb S^1,\mathbb C)\) has a continuous extension \(H\) to the closure \(\overline{\mathbb D}\) of \(\mathbb D\) that is holomorphic in \(\mathbb D\), then \(L_0(V)=H\). 2. For any element \(g\) in \(\text{Möb}(\mathbb S^1)\) and \(V\in C^0(\mathbb S^1,\mathbb C)\), \[ L_0(g^*V)=g^*(L_0(V)), \] where \(g^*(V)=(V\circ g^{-1})/(g^{-1})'.\) Let \(\Lambda(\mathbb S^1)\) be the collection of all Zygmund bounded tangent vector fields along \(\mathbb S^1\). Theorem 2. There is a constant \(C>0\) such that for any \(V\in\Lambda(\mathbb S^1)\), \[ \frac{1}{C}\|V\|_{\text{cr}}\leq\|\overline{\partial} L_0(V)\|_{\infty}\leq C\|V\|_{\text{cr}}. \] There are more characteristics of \(L_0\) and applications to the tangent spaces of the Teichmüller space.