On the Interpolation of Analytic Maps

TL;DR

This paper proves that an analytic map between certain Banach space interpolation spaces maintains boundedness properties within intermediate spaces, with explicit bounds derived from endpoint estimates.

Contribution

It establishes interpolation results for analytic maps between Banach spaces, providing explicit bounds for the map's behavior in intermediate spaces.

Findings

Analytic maps preserve boundedness in interpolated Banach spaces.

Explicit bounds relate endpoint estimates to intermediate space behavior.

Results apply to dense, continuous embeddings of Banach spaces.

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_\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 $C_0^{1-\theta}C_1^\theta R/(R-r)$.