EXTRAPOLATION THEORY 3

forms of Hardy type inequalities due to Moser and Jodeit (cf. [23] and §5.6). We

have also obtained an approach to the extrapolation theorems for weighted norm

inequalities of Rubio de Francia in the context of our theory. The key observation

is that a lattice satisfying a p convexity or concavity condition can be described as

an appropiate extrapolation space of a family of weighted Ir spaces. We refer to a

forthcoming paper [30] for the details. The above examples indicate the potential

breadth of the applications of the theory, and we hope that extrapolation will

become a useful tool for problems in other areas, as well.

The organization of the paper is as follows. In §2 we study the simplest of

extrapolation problems. Given a family of interpolation methods {J«}.

f i

, indexed

by some set 9, when are we able to reconstruct the end points An, A. from

7iA

0

,A A # 6 0 ? Roughly speaking, this means that the family {^n}nfc\ cannot

contain too big holes. Our solution essentially shows that as soon as we can

reconstruct one dimensional spaces we can reconstruct all spaces. This can always

be done using either of the two basic extrapolation methods A and S. We then

address the problem of determining when a given family {A^}^ ~ of Banach spaces

can be obtained by interpolation from a pair A = (A A.) by means of such families

"without holes." In §3 we develop the theory of these extrapolation methods in the

context of the real interpolation method. We show that an operator is bounded on

the whole scale L 0 k l , i f and only if certain inequality involving the S -

method is satisfied. Furthermore, there is a simple transform relating the operator

bound on the intermediate spaces and the particular £ — inequality to which this is

equivalent. It also turns out that these £ — inequalities sometimes can be divided.

Let us consider a simple example: We pick the pair (L jL00). It is well known that

an operator T is bounded from L to L and from L® to L00, both with norm 1, if and

only if