Differential processes generated by two interpolators

TL;DR

This paper investigates the properties of differential processes generated by pairs of interpolators, focusing on their symmetries, commutator theorems, and generalizations of known results in interpolation theory.

Contribution

It introduces a flexible framework that generalizes existing commutator and translation operator results without compatibility assumptions.

Findings

Generalizes known commutator theorems for differential methods

Extends stability and singularity results to broader differential methods

Provides new incomparability results in interpolation theory

Abstract

We study couples of interpolators, the differentials they generate and their associated commutator theorems. An essential part of our analysis is the study of the intrinsic symmetries of the process. Since we work without any compatibility or categorical assumption, our results are flexible enough to generalize most known results for commutators or translation operators, in particular those of Cwikel, Kalton, Milman, Rochberg \cite{ckmr} for differential methods and those of Carro, Cerd\`a and Soria \cite{caceso} for compatible interpolators. We also generalize stability and singularity results in \cite{cfg,ccfg,correa} from the complex method to general differential methods and obtain new incomparability results.