Stafney's lemma holds for several "classical" interpolation methods

TL;DR

This paper extends Stafney's lemma to various classical interpolation methods, showing that the same norm equivalence holds in broader settings including complex and real interpolation techniques.

Contribution

The paper generalizes Stafney's lemma to multiple well-known interpolation methods, broadening its applicability in functional analysis.

Findings

Stafney's lemma applies to complex interpolation on the annulus.

The lemma holds for the Lions-Peetre real method with various norms.

It also extends to Peetre's 'plus minus' interpolation method.

Abstract

Let (B_0,B_1) be a Banach pair. Stafney showed that one can replace the space F(B_0,B_1) with its subspace G(B_0,B_1) in the definition of the norm in the Calderon complex interpolation method on the strip if the element belongs to the intersection of the spaces B_i. We shall extend this result to a more general setting, which contains well-known interpolation methods: the Calderon complex interpolation method on the annulus, the Lions-Peetre real method (with several different choices of norms), and the Peetre "plus minus" method.