Math 565 Syllabus

Math 565: Introduction to Harmonic Analysis

Instructor: Cristina Pereyra

M ATH 565 - INTRODUCTION TO HARMONIC ANALYSIS

SRING 2008
Instructor: Cristina Pereyra
E-mail: crisp AT math . unm . edu
Office: Humanities 459, 277-4147
Schedule: TTh 9:30-10:45pm, HUM 424
Office Hours: TBA, or by appointment

Course content: This is a graduate course on Harmonic Analysis. Often Harmonic analysis is equated with the study of Fourier series and integrals, however the subject has evolved so much that often the most interesting problems are those where one can not use the techniques of Fourier analysis. In this course we will quickly review classical Fourier Analysis, and distribution theory. The chore of the course will be devoted to study the main ideas and basic techniques in modern harmonic analysis (a.k.a. the Calder\'on-Zygmund theory of singular integral operators).

The basic operators to be studied are the maximal function, Hilbert transform (in dimension 1) and Riesz Transforms (in R^n), square functions and paraproducts. We will note their origin in the study of classical problems in Fourier and Complex analysis, and their role as basic models for a large class of operators that appear in many problems in mathematics, the Calderon-Zygmund singular integral operators. One of the main problems will be to understand boundedness of these operators in L^p spaces, as well as in the limiting spaces of BMO (bounded mean oscillation), weak L^1, and the Hardy space H^1 (a beautiful theorem of C. Fefferman says that the dual of H^1 is BMO). In doing so, we will be forced to study some basic techniques in harmonic analysis: Schur's Lemma, almost orthogonality (Cotlar's Lemma), interpolation and extrapolation, stopping time arguments, and the celebrated T(1) Theorem of David and Journ\'e.

We will assume the students are familiar with measure theory (in particular L^p-spaces), and basic functional analysis. It helps also to have been exposed to the classical theory of Fourier series (last semester's Math 472/572 in particular you will find there the lecture notes: Harmonic Analysis: from Fourier to Haar , that I used for this course, which you might want to review.)

Prerequisites: Real analysis, measure theory, and basic functional analysis, or permission from the instructor.

Textbooks:
(Required) - Fourier Analysis by Javier Duoandicoetxea. AMS Graduate Studies in Mathematics Volume 29, 2001.

(Recommended) - Classical and Modern Fourier Analysis by Loukas Grafakos. Prentice Hall 2003. (Note: a second edition is in the works and we will get pdf files from the author, it will consist of two volumes in Springer's GTM series expected to appear in the fall 2008.)

(Recommended) - Harmonic Analysis. Real-variable methods, orthogonality, and oscillatory integrals by E. Stein. Princeton University Series 43, 1993.

(Recommended) - An introduction to harmonic analysis by Y. Katznelson. Dover Publications Inc, 1976.

Lecture Notes/articles:
From harmonic analysis to arithmetic combinatorics by Izabella Laba. Bulletin (New Series) of the AMS, vol. 45, No. 1, January 2008, pp. 77-115. pdf copy

Lecture notes on dyadic harmonic analysis by M. C. Pereyra. Contemporary Mathematics, Volume 289 (2001) p. 1-60.