(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
and L°°. One version of their results can be re-expressed in terms
of Peetre's if-functional for the couple
It states that a space X is an inter-
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