MATH 247A : Fourier analysis

  • Course description: Introduction to harmonic analysis and Fourier analysis methods, such as Calderon-Zygmund theory, Littlewood-Paley theory, and the theory of various function spaces, in particular Sobolev spaces. Some selected applications to ergodic theory, complex analysis, and geometric measure theory will be given. However, the focus will be on estimates and how to prove them. Fourier analysis per se will be present, but will not be a dominant feature of the course until Math 247B.


        (Nov 29) Some errata to the final has been posted below.

        (Nov 27) The final is now available here. It is due Monday Dec 11. My Thu Nov 30 office hours will be cancelled.

        (Nov 16) My Thu Nov 23 office hours will be cancelled.

        (Nov 9) Next Wednesday’s class (Nov 15) will end early at 4pm due to the Probability seminar.

        (Nov 1) Today’s class will end early at 4pm due to the Probability seminar.

        (Oct 20) My Monday Oct 23 office hours will be cancelled.

        (Oct 9) I may not be able to make the first half Th 10-10:30 of my office hour this week.

        (Oct 4) I have added a second office hour, Th 10-11. Also, starting next week (i.e. Oct 9 onwards), we will shift the class forward by five minutes, to 3:05-4:25, in order to reduce the conflict with Bill Duke’s class.

        (Sep 12) The first class will be on Oct 2.

        Instructor: Terence Tao,, x64844, MS 5622

        Lecture: MW 3:05-4:25, MS5137

        Quiz section: None

        Office Hours: M 11-12, Th 10-11

        Textbook: I will very loosely follow Wolff’s “Lecture notes on harmonic analysis” and Stein’s “Singular integrals” but rely primarily on my own notes (see below)

        Prerequisite: Math 245AB is highly recommended, as is some exposure to undergraduate Fourier analysis (such as Math 133 or equivalent).

        Grading: Homework (50%), Final (50%).

        Exams: The final exam will be a take-home exam, consisting of roughly eight questions, to be released Monday Nov 27 and due Monday Dec 11. The final can be found here (Updated, Dec 5). The errata is as follows:

1.      In Q2, in the scaling condition, n should be 1.

2.      In Q4, the equivalence of the L^infty norm of f^# and the BMO norm of f is of course utterly trivial and this part of the question can be ignored. The implied constants are allowed to depend on d.

3.      In Q5, the good lambda inequality should require f^# to be less than eps lambda, rather than greater than eps lambda. Also, K needs to be sufficiently large (not just larger than 1). Finally, “Lemma 3.4” should be “Lemma 3.7”. It is possible to deduce Lemma 3.7 for all 0 < p,q < infty by using Holder’s inequality, but I will accept a derivation of Lemma 3.7 restricted to the range 1 < p,q < infty.

4.      In Q6, one needs the xi_n to be disjoint; also, there is a summation in n missing in the Fourier series sum_n c_n exp( 2pi i xi_n x ). Finally, since Orlicz spaces were not covered in the course, I am deleting the portions of the question relating to these spaces.

        Reading Assignment: It is strongly recommended that you read the notes concurrently with the course, and try some or all of the problems. The books are optional, but still recommended, simply because they are excellent books; for further reading I also recommend Stein’s “Harmonic analysis”, or Katznelson’s “An introduction to harmonic analysis”. The list of problems for the 2004-2005 incarnation of this course (as taught by Rowan Killip) may also be consulted, although our emphasis here shall differ in many respects from Rowan’s (in particular, we shall emphasise estimation and de-emphasise Fourier analysis).

        Homework: There will be four homework assignments, assigned from the notes. A 10% penalty is imposed for each day past the due date that the homework is turned in.

1.      First homework (Due Monday, Oct 15): Notes 1, Q 1,2,3. Some remarks on the questions can be found here.

2.      Second homework (Due Monday, Oct 29): Notes 1, Q 4; Notes 2, Q 1,5,6. Some remarks on the questions can be found here.

3.      Third homework (Due Monday, Nov 13): Notes 2, Q 14, 15; Notes 3, Q 1. Some remarks on the questions can be found here.

4.      Fourth homework (Due Monday, Nov 27): Notes 3, Q 4, 5, 11. Some remarks on the questions can be found here.


        Notes 0: Overview of harmonic analysis.

        Notes 1: Introduction, estimates, L^p theory, interpolation. (updated, Nov 1. Erratum, Mar 5 2017: On page 5, Y \lesssim X should be Y \gtrsim X. In Problem 5.3, the condition 1 < p < \infty should be added.)

        Notes 2: Interpolation, Schur’s test, Young’s inequality, Hausdorff-Young, Christ-Kiselev. (updated, Dec 6. Erratum, Dec 23 2018: In the definition of f (and its Fourier transform) on page 21, a factor of exp( -pi i n^2 |v|^2) should be added.)

        Notes 3: The Hardy-Littlewood maximal inequality and applications. (updated, Dec 6)

        Notes 4: Calderon-Zygmund theory (updated, Dec 6).

Erratum: in Q13, |xi|/2^j should be |xi|/t (two occurrences). Thanks to Casey Jao for the correction.

        Notes 5: Fefferman-Stein inequality, pseudodifferential operators, Sobolev spaces (updated, Dec 6)

The notes contain both problems and exercises. Homework will be drawn primarily from the exercises. The problems are to test your knowledge, and are recommended, but solutions to these problems will not be collected or graded.