On the Interpolation of Analytic Maps

Let (E_0,E_1) and (H_0,H_1) be a pair of Banach spaces with dense and continuous embeddings E_1 into E_0, H_1 into H_0. For $\theta \in [0,1]$ denote by $B_\theta(0,R)$ the ball of radius R centered at zero in the interpolation spaces E_\theta. Assume that an analytic map $\Phi$ maps the ball B_0(0,R) into H_0, $\Phi$ maps B_1(0,R) into H_1 and for $\theta =0,1$ the estimates $$\|\Phi(x)\|_{H_\theta} \le C_\theta\|x\|_{H_\theta}, \forall\ x\in B_\theta(0,R),$$ hold. Then for all $\theta\in(0, 1)$ and r<R $\Phi$ maps the ball $B_\theta (0,r)$ into $H_\theta$ and the same estimate holds for $x\in B_\theta(0,r)$ if the constant $C_\theta$ is replaced by $C_0^{1-\theta}C_1^\theta R/(R-r)$.