## 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.
1. 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.
2. Second homework (due Monday, Feb 12): Notes 6, Q2, 3
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}.
4. 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: