A functional decomposition theorem for the conformal representation (Q1127015)

The author studies the nonlinear operator which associates to each injective nonsingular self-map \(\varphi\) of the complex unit disc \(\mathbb{D}\) of class \(C^{m,\alpha}\) the unique holomorphic map \(g[\varphi]: \mathbb{D}\to \varphi(\mathbb{D})\), normalized by the conditions \(g[\varphi](0)= \varphi(0)\) and \(0< g'[\varphi](0)< \infty\). The crucial idea is that this operator may be decomposed as \(\varphi\mapsto \varphi\circ S[\varphi]^{- 1}\), with \(S\) being some analytic operator. In particular, a recent result due to \(S\). \textit{J. Wu} [Commun. Pure Appl. Math. 46, No. 10, 1303-1326 (1993; Zbl 0818.30005)] may be viewed as a special consequence of Theorem 3.10 of this paper.