Math 254A, Winter 2001
Harmonic analysis in the phase plane
Class is at MS 5217, MWF 12. My office is MS 5622, my email is tao@math.ucla.edu,
and my phone is x64844. Office hours are MW 23.
There will not be a class on Friday March 16. Have a good spring
break!

The description of this course.

Week 1 notes: Review of Fourier transform; phase space;
uncertainty principle (postscript) (Corrected
Jan 10! Further correction (Apr 2 2021, from Jacob Denson): In the conclusion of Proposition 5.5, all the appearances of I and J should be swapped.)

Week 2/3 notes: LittlewoodPaley theory; Sobolev spaces
(postscript) (Corrected Feb 20, 2003, thanks to Mark Keel). A further correction (Mar 10, 2010, from Veronica Quitalo): On page 5, the inequality 1/p  1/q > 1/n should be 1/p1/q < 1/n. Yet another correction (Mar 16, 2012, from Nefton Pali): in the proof of Lemma 1.1 on page 3, one should replace phi with a function like psi (basically, one needs to smoothly truncate phi near the origin to avoid unnecessary frequency divergences for the second part of the lemma).

Week 4 notes: Product estimates, multilinear estimates
(postscript)

Week 5 notes: Regularity of harmonic maps (postscript)

Week 6/7/8 notes: Walsh wave packets, Walsh Carleson
theorem (postscript) (Corrected Nov 26, thanks
to Dmitry Ryabogin)

Week 8/9 notes: Haar basis, the Cauchy integral
(postscript)

Week 10 notes: The Lewy example (postscript)
Note: if you are on a Windows system and wish to read DVI files, try
this link.
ERRATA for Homework 2:

Question 1: W^{p,s} should be W^{s,p}. Similarly for W^{p,ssigma}.

Question 2: The condition 1/q = 1/p + \theta/n should be 1/p = 1/q + \theta/n.

Question 4(b): The righthand side should contain a f_p as well as
\nabla f_p.
Also, the factor in Bernstein's inequality in Week 1 should be I^{1/p1/q}rather
than I^{1/q1/p}.
ERRATA for Homework 4:

Question 1: One should have 0 < s < 1 rather than just s>0.
ERRATA for Homework 5:

Question 1: One should ignore sets of measure zero in this question (and
indeed, in all questions in this course).
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