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
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