Classical and modern results on interpolation of operators

Abstract

The main purpose of this project is to study the classical theorems on interpolation of linear operators in order to analyse some modern results on interpolation of multilinear operators. Following the approach of Bennett and Sharpley to the classical interpolation theory of quasilinear operators, we gather all the results that will allow us to tackle the recent developments on multilinear interpolation theory, in particular, the result of Grafakos, Liu, Lu and Zhao. Our goal is to fully understand the different real interpolation techniques presented by the previous authors, so we devote our time and efforts to give detailed, self-contained and complete proofs of the main interpolation results. We focus on the study of real-variable methods and we start with one of the cornerstones of the classical interpolation theory: the Marcinkiewicz interpolation theorem. We continue the study with the K-method of interpolation, which it may be regarded as a lifting of the Marcinkiewicz interpolation theorem from its classical context in spaces of measurable functions to an abstract Banach space setting. Finally, we study multilinear interpolation theory, exposing the proof a version of Marcinkiewicz’s interpolation theorem for bi-sublinear operators.