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
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.
(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.
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.
A panorama of harmonic analysis by S. Krantz. The Carus Mathematical Monographs 27, AMS 1999.
Grades: The final grade will be determined by homework , and/or a team project.
Americans with Disabilities Act: Qualified students with disabilities needing appropriate academic adjustments should contact me as soon as possible to ensure your needs are met in a timely manner. Handouts are available in alternative accessible formats upon request.
Return to: Department of Mathematics and Statistics, University of New Mexico
Last updated: January 21, 2008