LECTURE NOTES ON DYADIC HARMONIC ANALYSIS 3

Bellman function technique by proving Buckley's characterization of

A)()

weights

by summation conditions.

In the last lecture we use the weight lemma introduced in the third lecture to

prove the boundedness of some non-constant Haar multipliers. From their bounded-

ness one can deduce boundedness on weighted

LP

for our dyadic operators: constant

Haar multipliers, paraproducts and square function. The last section is a survey

on weighted inequalities. Much progress has been made in the last 5 years. Sev-

eral longstanding problems have been solved like the single matrix-valued weight,

and the two-weights problem for the Hilbert transform; as well as the study of

sharp constants for the boundedness of the dyadic square function and the Hilbert

transform on weighted spaces.

Some years ago we had a plan to write a book on this subject with Nets Katz,

there is an unpublished manuscript by Katz that has been very inspiring,

[Ka2].

There are many books in harmonic analysis that contain much more than what

is here, including excellent expository books like

[Duo]

and

[Kr],

or concise and

juicy surveys like

[Ch]

and

[Da2],

or the well known new and old testaments

[Stl],

[St2],

which are of an encyclopedic nature and are compulsory reading for anybody

interested in modern harmonic analysis.

We were very lucky to have Steve Hofmann teaching simultaneously a beautiful

course on his very recent proof of the Kato Problem (a 40 years old longstanding

conjecture), see his Lecture Notes in this volume

[Ho~.

His proof is very classi-

cal and utilizes all these techniques: Littlewood-Paley analysis (square functions),

maximal functions, Carleson's measures, sophisticated versions of the

T(b)

theo-

rem for corresponding singular integral operators adapted to the heat kernel. It

was delightful to see all the classical techniques joining forces to produce such an

astonishing and long overdue result. It does say something about the strength of

the basic techniques.

Last but not least, I would like to warmly thank the organizers, Salvador Perez-

Esteva and Carlos Villegas, for inviting me to teach in the school. I would also like

to thank all the participants, students, colleagues from throughout Mexico and

abroad who attended the course and provided comments during and afterward. In

particular Martha Guzman-Partida, Lucero de Teresa, Magaly Folch, Stephan De

Vievre and Steve Hofmann. You all made this experience a very rewarding one,

socially and mathematically. Finally I would like to thank Kees Onneweer who

volunteered to proofread the manuscript under a tight time schedule.

Disclaimer: All other mistake are my son Nicolas' fault! He was in the making

while these lectures were delivered, and he was born before I had time to finish

them. I thought it was going to be easy to complete this project, little did I know!!

1. Main Characters

1.1.

The Hilbert Transform.

The Hilbert transform is the prototypical ex-

ample of a singular integral operator. It is given formally by the principal value

integral:

Hf(x)

=

p.v.~

J

f(y) dy

:=lim~

1

f(y) dy.

7f X-

y

E----0 7f lx-yiE X-

y