Interpolation of Nonlinear Maps

Let $(X_0, X_1)$ and $(Y_0, Y_1)$ be complex Banach couples and assume that $X_1\subseteq X_0$ with norms satisfying $\|x\|_{X_0} \le c\|x\|_{X_1}$ for some $c >\nobreak 0$. For any $0<\theta <1$, denote by $X_\theta = [X_0, X_1]_\theta$ and $Y_\theta = [Y_0, Y_1]_\theta$ the complex interpolation spaces and by $B(r, X_\theta)$, $0 \le \theta \le 1$, the open ball of radius $r>0$ in $X_\theta$ centered at zero. Then, for any analytic map $\Phi\colon B(r, X_0) \to Y_0+ Y_1$ such that $\Phi\colon B(r, X_0)\to Y_0$ and $\Phi\colon B(c^{-1}r, X_1)\to Y_1$ are continuous and bounded by constants $M_0$ and $M_1$, respectively, the restriction of $\Phi$ to $B(c^{-\theta}r, X_\theta)$, $0 < \theta <\nobreak 1$, is shown to be a map with values in $Y_\theta$ which is analytic and bounded by $ M_0^{1-\theta} M_1^\theta$.