4

M. CWIKEL AND P. G. NILSSON

0. INTRODUCTION.

(a) The problem of characterizing interpolation spaces

The theory of interpolation spaces is usually considered to have begun with the

classical theorems of Riesz-Thorin and Marcinkiewicz which, in their simplest versions,

state that a linear operator which maps the space Lp boundedly into itself (or into

weak Lp) for two distinct values of p must also be bounded on IP for all intermediate

values of p.

In the early '60s the ideas of these two theorems were refined and generalized by

a number of authors, in particular Calderon [Cal] and Lions-Peetre [LP], to give two

constructions, respectively the complex and real methods, for generating spaces with

an analogous "interpolation of operators" property from any given couple of suitably

compatible Banach spaces. At about the same time Aronszajn and Gagliardo [AG]

began the study of interpolation spaces from a more general and abstract point of

view. Subsequent developments of the theory included the discovery of a variety of

other methods for generating interpolation spaces. Let us note that Janson [Ja] has

given a unified description of many of these methods and also the real and complex

methods, in terms of the notions of Aronszajn and Gagliardo.

Given the proliferation of different ways of constructing interpolation spaces it is

very natural to ask how much variety can they actually give us when applied "in prac-

tice" , and to seek some way of characterizing or describing all interpolation spaces in

reasonably concrete terms. Over the past few decades this has been done for a consid-

erable number of specific couples of Banach spaces by several authors. We shall now

attempt to briefly survey these results, since this paper is directed towards extending

and offering a unified perspective of them.

Our starting point is the work of Mityagin [Mi] (1965) and Calderon [Ca2] (1966)

who independently discovered characterizations of the class of all interpolation spaces

with respect to

L1

and L°°. One version of their results can be re-expressed in terms

of Peetre's if-functional for the couple

(L1,!/00).

It states that a space X is an inter-