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