A Unified Theory of Commutator Estimates for a Class of Interpolation Methods

Abstract

A general family of interpolation methods is introduced which includes, as special cases, the real and complex methods and also the so-called ± or G 1 and G 2 methods defined by Peetre and Gustavsson–Peetre. Derivation operators Ω and translation operators R are introduced for all methods of this family. A theorem is proved about the boundedness of the commutators [ T, Ω] and [ T, R ] for operators T which are bounded on the spaces of the pair to which the interpolation method is applied. This extends and unifies results previously known for derivation and translation operators in the contexts of the real and complex methods. Other results deal with higher order commutators and also include an “equivalence theorem,” i.e. it is shown that, as previously known only for real interpolation spaces, all these interpolation spaces have two different equivalent definitions in the style of the “ J method” and “ K method.” Auxiliary results which may also be of independent interest include the equivalence of Lions–Schechter complex interpolation spaces defined using an annulus with the same spaces defined in the usual way, using a strip.