§1. INTRODUCTION

The interpolation theorems of Riesz—Thorin and Marcinkiewicz are at the

basis of the theory of interpolation spaces which has been developed by Calderon,

Gagliardo, Krein, Lions, Peetre, and many others during the last decades.

Interpolation theory has become an established field with interesting applications to

classical and modern analysis.

The point of departure of this paper is a classical extrapolation theorem

obtained by Yano [55] (although special cases had been considered earlier by

Titchmarsh and Marcinkiewicz (cf. [60])). Suppose that T is a bounded linear

operator on

LP(0,1)

for p 1 with ||T|| = ^((p-l)~

a

), as p - + 1, for some a

Lp-Lp

0. Then the

Lp

estimates can be extrapolated to T:

L(LogL)a

L . There is

also a dual statement for operators T acting on LP(0,1) for p close to m, with

||T|| = ^(p a ), as p - + u, for some a 0; then T: L00 Exp \}la. This

Lp-+Lp

result can be considered a converse of the Marcinkiewicz interpolation theorem.

Similarly, we could ask what end point inequalities does an operator T have to

satisfy to be bounded on with the bound depending on p in some given way.

This would lead to "best possible" Marcinkiewicz type theorems (cf. [23] and [28] for

preliminary results in this direction). More generally, the question of best possible

interpolation theorems can be asked in the context of interpolation scales of spaces,

and we may inquire whether Yano's theorem can be extended to this setting.

In this paper we develop a general theory of extrapolation which is a

converse of interpolation theory. In the process we also obtain sharper versions of

Yano's classical theorem, even in theL p setting.

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