The theory of complex interpolation of Banach spaces and operators, which was developed by Calderón, Lions and S. G. Krein and extended by us and others, centers on the use of the maximum principle for analytic functions. In this paper we investigate the significance in interpolation theory of estimates for derivatives of analytic functions. We obtain norm estimates for certain linear and non-linear commutators and obtain new classical interpolation theorems.
The paper has four sections. In the first section we show how Thorin’s proof of the Riesz-Thorin theorem can be extended to give an estimate for a non-linear commutator and how analogous computations can be done in families of interpolation spaces. The second section introduces an extension of the complex interpolation theory which is designed to incorporate in a systematic way the type of estimates obtained in the first section. In the third section, the details of the abstract theory of the second section are filled in for several standard examples including $L^p$ spaces and weighted $L^p$ spaces. A brief final section contains some general comments and observations.