Introduction

During the summer of 2000, June 12-22, the II summer school in Analysis and

Mathematical Physics took place in the Institute of Mathematics of the National

Autonomous University of Mexico at Cuernavaca. Following the same idea of the

first school, the purpose was to establish links between standard graduate courses

in mathematics and current research topics in Mathematical Analysis. Thus we

offered seven courses taught by recognized specialists in their working areas. In

this occasion we had representatives of Harmonic and Complex Analysis, Pseudo-

differential Operators, rigorous mathematical theory of Quantum Chaos as well as

Non-linear Analysis.

We had the pleasure of having Cristina Pereyra, Steve Hofmann, Nikolai Vasi-

levski, Yuri V. Egorov, Stephan De Bievre, Pablo Padilla and Jon Jacobsen as

lecturers. The lectures notes can be read in any order, but we recommend to follow

the order in which they appear in this volume.

In the first chapter, Cristina Pereyra exposed the so-called dyadic harmonic

analysis. Her course was thought for students without any experience in harmonic

analysis, hence basic concepts of this field are introduced such as the Hilbert trans-

form. Littlewood-Paley theory, Ap weights and BMO. Then it is explained in

detail the dyadic counterpart of all these notions. This chapter covers concepts

such as Carleson sequences, paraproducts, the use of stopping time techniques and

the nowadays important and very used method of Bellman functions in harmonic

analysis. The paper also includes a proof of the

Tl

theorem for Calderon-Zygmund

operators.

Steve Hofmann devoted his lectures to the exposition of the history and solution

to Kato's "square root problem". The author himself and his collaborators recently

solved this long-standing important problem in mathematics. The students had

the privilege to hear about this problem from one of the main contributors to its

solution and to see strong applications of versions of the concepts described in

Pereyra's lectures.

Nikolai Vasilevski makes a thorough description of the Bergman spaces in the

unit disk, the half space and the corresponding Toeplitz operators. The notes

are thought for advanced students with some knowledge in the topic. The aim

of the author is to explain the structure of various Bergman type spaces and the

representation theory of algebras of Toeplitz operators. The author travels from

representation theory using classical methods like the local principle, all the way to

ix