Lectures: MWF 12-1, MATX 1118.

Office hours: Mon 2-3, Wed 1-2, and by appointment.

This course is a continuation of MATH 541 and will cover more advanced topics in harmonic analysis on Euclidean spaces.

Alternatively, you may elect to give a 1 hour in-class presentation on a topic related to the course material. You would also have to prepare a written exposition to be handed out in class. A presentation would be fairly substantial (a research paper rather than just a section of a textbook) and would replace 3 of the homeworks. Here are a few possible topics:

- Dvir's proof of the finite field Kakeya conjecture (Z. Dvir,
*On the size of Kakeya sets in finite fields*J. Amer. Math. Soc. 22 (2009), 1093-1097) (Rhoda Sollazzo) - Bourgain's circular maximal theorem (J. Bourgain,
*Averages in the plane over convex curves and maximal operators*, J. Anal. Math. 47 (1986) 69-85. This would replace all 4 homeworks if anyone wants to do it)a (Marc Carnovale) - Salem's construction of Salem sets (R. Salem,
*On singular monotonic functions whose spectrum has a given Hausdorff dimension*, Ark. Mat. 1, (1951). 353-365) (Tatchai Titichetrakun) - Fefferman's counterexample to the ball multiplier problem and an introduction
to the Bochner-Riesz conjecture (C. Fefferman,
*The multiplier problem for the ball*, Ann. Math. 94 (1971), 330-336, plus some expository reference on Bochner-Riesz) (Vince Chan) - Maximal functions over a Cantor set of directions (M. Bateman, N. Katz,
*Kakeya sets in Cantor directions*, Math. Res. Lett. 15 (2008), 73-81 (Ed Kroc)

- Homework 1, due on
**Wednesday, January 26** - Homework 2, due on
**Friday, February 11** - Homework 3, due on
**Friday, March 11** - Homework 4, due on
**Friday, April 1**

- Maximal functions and differentiation theorems
- The Hardy-Littlewood maximal function
- The spherical maximal function in higher dimensions
- Kakeya sets and the Kakeya maximal function

- The restriction problem for the sphere
- The Tomas-Stein theorem
- The restriction conjecture and the Kakeya problem

- Fourier transforms of singular measures
- Hausdorff measure and Hausdorff dimension
- The energy inequality and Fourier dimension
- Salem sets
- Applications to geometric measure theory: convolutions, projections, distances

*Fourier Analysis*, J. Duoandikoetxea, American Mathematical Society, 2001*Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals*, E.M. Stein, Princeton Univ. Press, 1993*Real Analysis*, E.M. Stein and R. Shakarchi, Princeton Univ. Press, 2003*Lectures on Harmonic Analysis*, T. Wolff, American Mathematical Society, 2003