2 B. JAWERTH AND M. MILMAN

It has been known for a long time that an operator satisfying the

assumptions of Yano's theorem does not necessarily satisfy a weak type (1,1)

estimate (cf. [34]). In fact, the example in [34] (which is due to Y. Katznelson and

answers a question by A. Zygmund) shows that from a norm estimate like ||Tf|L p

ip(v) ||f||yP , with p(p) - GD , as p -* 1, we cannot extrapolate to a weak type (1,1)

inequality, irrespective of the rate to which p(p) converges to oo. However.

appropriate limiting inequalities exist and can be described in terms of a class of

inequalities between the K and J functional (MKJ inequalities"). These inequalities

can be combined with the Fundamental Lemma of interpolation theory to obtain

intermediate inequalities. In particular, in this way we can derive sharp

rearrangement inequalities, like those of 0'Neil—Weiss and Calderon for the Hilbert

transform (cf. [41], [10]), or the the results of [3] for the strong maximal operator

(cf. §5.2 and §6). Extrapolation theory also provides a new, unified approach for

studying the spaces that appear naturally in the formulation of limiting inequalities

in several areas. In particular, some of the spaces relevant in the implicit function

theorem of Nash-Moser (cf. [26], [59]) as well as Zafran's homogeneous Banach

algebras (cf. [57]), which provide a counterexample to the dichotomy conjecture, are

examples of extrapolation spaces (cf. §5.5 and §5.9). Moreover, the identification of

concrete spaces as extrapolation spaces (eg. L(logL)a, L&gL(&gfogL), Exp La,

Dini, Zygmund—Lipschitz spaces, Wj * j , etc) makes it possible to use our

methods to derive end point inequalities in a simple fashion. For example, the

embedding theorem of M. Weiss, A. P. Calderon, and A. Zygmund, to the effect

that a function satisfying an L Dini condition in (0,1) is in LfogL, follows by the

Fundamental Theorem of Calculus and extrapolation (cf. [11], [20], and compare

with §5.3). Sharp forms of the Sobolev embedding theorem, and other related norm

inequalities important in P.D.E's, can be obtained using extrapolation (cf. [7] and

compare with § 5.3). We may also (partially) improve upon extreme