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.
pdf version
Expository book:
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