2
MARIA CRISTINA PEREYRA
but yet illuminating enough to guarantee that one can translate them into the non-
dyadic world. This philosophy has been pushed to unexpected limits by Nazarov,
Treil and Volberg, as well as by their students and collaborators. Most of the
material related to Bellman functions I learned from them, either in preprints or in
the Spring School on Analysis held in Paseky1 a couple weeks before the School in
Cuernavaca.
In these lectures we will concentrate on Haar analogues, this is just the tip
of the time/frequency iceberg. A full dyadic model for the phase plane is given
by the Walsh functions. Beautiful results are being obtained now with the more
sophisticated time/frequency tools and very delicate combinatorial arguments. The
pioneering work was done by C. Fefferman in 1972
[Fe~.
A few years ago C. Thiele,
relying heavily on Fefferman's ideas, presented a solution of a famous conjecture
of Calderon for a Walsh model of the bilinear Hilbert transform in his PhD Thesis
[Th].
Joining forces with M. Lacey they were able to prove the full conjecture in a
work that earned them the 1997 Salem Price
[LT].
There is lots of work in progress
along these lines in connection to PDE's, for an overview see the lecture notes of a
course taught by T. Tao during Spring 2001 at UCLA
[Tao].
The main problem analyzed in the following pages is £P boundedness of op-
erators. Namely, we want to know if a given linear (or sub linear) operator T acts
continuously from £P(X) into
Lq(Y),
where (X,
11)
and
(Y, v)
are measure spaces,
for some 1 ::;
p, q::;
oo, i.e. is there a constant
C
0 such that for all
f
E
£P(X),
1 1
IITJIILq(Y) = ([
ITJ(yWdv(y))"::; C
(L
lf(x)IPdf-L(x))
1'
= CIIJII£P(X)?
We try to illustrate in the first lecture why people were interested in such
inequalitiel:l. We do so by revil:liting the most classical operators: the Hilbert trans-
form, Hardy-Littlewood maximal function, square functions and paraproducts; not-
ing their place in history as well as their dyadic counterparts.
In the second lecture we introduce the classical tools used to handle bounded-
ness: Schur's Lemma, Cotlar's Lemma, interpolation and extrapolation, Calder6n-
Zygmund decomposition. We illustrate how to use these tools to prove boundedness
estimates for the classic operators.
The third lecture introduces the space of bounded mean oscillation
(BMO)
and
Aoo
weights, as well as their dyadic counterparts; the "self-improvement" theorems
of John-Nirenberg and Gehring are proved as the first examples of the power of
stopping time techniques. An analogue of the John-Nirenberg Theorem for RHp
weights is presented, the Weight Lemma, and its use is illustrated in proving char-
acterizations of weights by summation conditions.
In the fourth lecture singular integral operators are introduced and the cele-
brated T(1) theorem of David-Journe is proved following the dyadic proof of Coif-
man and Semmes. Some history on the Cauchy integral is provided as well as
the
T(b)
theorem. Finally we present a short survey on recent progress done in
extending these tools to non-homogeneous (non-doubling measures) spaces.
In the fifth lecture the classical Carleson embedding theorem is presented and
its dyadic counterparts. A stopping time proof is presented as well as Nazarov-
Treil-Volberg's proof using Bellman functions. Either proof can be extended to
handle weighted versions of the embedding theorem. We illustrate furthermore the
1School
that I attented thanks to a Travel Grant from AWM/NSF, May 2000
Previous Page Next Page