Interpolation of nonlinear maps

Abstract

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 $c > 0$. For any $0<θ<1$, denote by $X_θ= [X_0, X_1]_θ$ and $Y_θ= [Y_0, Y_1]_θ$ the complex interpolation spaces and by $B(r, X_θ)$, $0 \le θ\le 1,$ the open ball of radius $r>0$ in $X_θ$, centered at zero. Then for any analytic map $Φ: B(r, X_0) \to Y_0+ Y_1$ such that $Φ: B(r, X_0)\to Y_0$ and $Φ: B(c^{-1}r, X_1)\to Y_1$ are continuous and bounded by constants $M_0$ and $M_1$, respectively, the restriction of $Φ$ to $B(c^{-θ}r, X_θ)$, $0 < θ< 1,$ is shown to be a map with values in $Y_θ$ which is analytic and bounded by $M_0^{1-θ} M_1^θ$.