On the Interpolation of Analytic Maps

Abstract

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_θ. 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 $$\Vert \Phi (x){\Vert }_{H_{\theta }}\le C_{\theta }\Vert x{\Vert }_{H_{\theta }},\forall x\in B_{\theta }(0,R),$$ 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 $C_0^{1-\theta }{C}_1^{\theta }R/(R-r)$.

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• Interpolation of nonlinear maps