MATH 247B :
Fourier analysis
- Course description: Continuation of Math 247A.
Topics include the paradifferential calculus, the T(1)
theorem, averaging on surfaces, and Fourier analysis on more general
groups.
Announcements:
- (Mar
8) There will be no class on Mar 14, as I have run out of course
material.
- (Feb
20) I will end class early on Wed Feb 28, so as not to conflict with
Peter Lax’s talk.
- (Jan 19) I
am also canceling the office hour on Tue Jan 23 (I will be in Australia
at the time).
- (Dec 26)
The first class will be on Mon Jan 8. In
addition to the MLK holiday on Mon Jan 15, I will not have classes on
Wed Jan 17 and Wed Jan 24.
- Instructor:
Terence Tao, tao@math.ucla.edu,
x64844, MS 5622
- Lecture:
MW 3:00-4:20, MS5137
- Quiz
section: None
- Office
Hours: M 11-12, Tu 2-3 (note change in OH from last quarter)
- Textbook:
I will rely primarily on my own notes (see below)
- Prerequisite:
Math 247A.
- Grading:
Homework (100%)
- Exams:
None.
- Reading
Assignment: It is strongly recommended that you read the notes
concurrently with the course, and try some or all of the problems.
- 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. Some homework is
assigned from the Math
247A notes.
- First
homework (due Monday, Jan 29): Notes 5, Q5, 6. Errata:
in Q6, the implied constant should depend on delta (or equivalently, on
s). In Q5, there are several approaches to solve
the problem. One is to do the integer s case first
and then interpolate. The other is to exploit the
fractional integral formulation of |nabla|^{-s}. A
third is to use Littlewood-Paley theory. In the
latter case, you may find my other notes on Littlewood-Paley theory
(see here
and also the appendix to this book)
to be useful; see also Stein’s “Singular integrals” for more on Sobolev
spaces. You may also find various PDE texts (e.g. Taylor) to be
useful.
- Second
homework (due Monday, Feb 12): Notes 6, Q2, 3
- Third
homework (due Monday, Feb 26): Notes 6, Q7; Notes 7, Q1. (Hint:
for Notes 7, Q1, use the T(1) theorem.) Errata: in
Q7, 2^{-j \alpha} should be 2^{j\alpha}.
- Fourth homework (due Monday, Mar 12): Notes 8, Q2, Q3.
Errata: in Q2, the disk example given only works when d=2; for d>2
one should use balls instead.
Notes:
- Notes 6: Paradifferential
calculus, fractional chain and Leibnitz rules. (Updated, Feb 5. Erratum (Feb 17 2019): the proof of Lemma 3.3 is not correct as stated. One can proceed by first using the Littlewood-Paley inequality to estimate the L^r norm of pi_lh(f,g) by the L^r norm of the square function of the summands pi_lh(f, psi_j(D) psi_j(D) g), at which point the rest of the argument will work; alternatively one can dualise and then apply the high-high argument. Thanks to Jason Murphy for pointing out the issue.)
- Notes 7: The T(1)
theorem; the Cauchy integral. (Erratum, May 9 2019: In Lemma 1.4, the exponent -d-theta in the first display should be -2d-2theta, and in the proof, K(x_B',y) should be K(x,y_B), and the factor of |B| in the bound for T phi_B should be omitted; fially, "vanishes outside of 5B'" should be "vanishes on 2B'", and (2) should be (3). Thanks to Hans Lindblad for pointing out the issue.)
- Notes 8: Stationary
phase, spherical averages, high-dimensional Hardy-Littlewood maximal
inequality.
- Notes 9: Fourier
analysis on abelian groups. (Updated, Mar 1)
- Notes 10: Fourier
analysis on non-abelian groups.