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 $θ\in [0,1]$ denote by $B_θ(0,R)$ the ball of radius R centered at zero in the interpolation spaces E_θ. Assume that an analytic map $Φ$ maps the ball B_0(0,R) into H_0, $Φ$ maps B_1(0,R) into H_1 and for $θ=0,1$ the estimates $\|Φ(x)\|_{H_θ} \le C_θ\|x\|_{H_θ}, \forall\ x\in B_θ(0,R),$ hold. Then for all $θ\in(0, 1)$ and r<R $Φ$ maps the ball $B_θ(0,r)$ into $H_θ$ and the same estimate holds for $x\in B_θ(0,r)$ if the constant $C_θ$ is replaced by $C_0^{1-θ}C_1^θR/(R-r)$.

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