Contemporary Mathematics

Volume 289, 2001

Lecture Notes on Dyadic Harmonic Analysis

Maria Cristina Pereyra

CONTENTS

Introduction

1. Main Characters

2. Classical Tools

3.

BMO,

Acxo

and Stopping Times

4. The T(1) Theorem

5. Carleson's Lemma and Bellman Functions

6. Haar multipliers and Weighted Inequalities

References

Introduction

1

3

16

27

36

43

49

57

These Lecture Notes grew out of a series of lectures delivered by the author

at the Analysis Summer School in the Instituto de Matematicas of the Universi-

dad Autonoma de Mexico, Unidad Cuernavaca in June 2000. The lectures were

intended for begining graduate students with a basic knowledge of real and com-

plex analysis, measure theory and functional analysis. There were many exercises

sprinkled throughout the lectures, which hopefully complemented and helped the

reader test his/her understanding of the material presented. I have included those

exercises and more in the lecture notes. Also, while I had hoped to cover topics

related to weights, I did not have sufficient time to present them in Cuernavaca,

but have included them in these notes.

The notes contain what I consider are the main actors and universal tools used

in this area of mathematics. They also contain an overview of the classical problems

that lead mathematicians to study these objects and to develop the tools that are

now considered the abc of harmonic analysis. The modern twist is the connection to

a parallel dyadic world were objects, statements and sometimes proofs are simpler,

1991 Mathematics Subject Classification. Primary 42B25, 42B20, 42A45; Secondary 47B38.

Key words and phrases. dyadic harmonic analysis, Hilbert transform, maximal function,

square function, paraproducts, Haar multipliers,

BMO, Ap

and

RHp

weights, singular inte-

gral operators,

T(1)

Theorem, Carleson's Lemma, Bellman functions, stopping time, weighted

inequalities.

©

2001 American Mathematical Society

http://dx.doi.org/10.1090/conm/289/04874