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
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