Math 601 topics

Math 601 (Topics in Analysis) topics, Winter/Spring 2015

This list will be updated as we proceed with the course material. The references for any background needed, or for proofs omitted from the lecture, will be posted here.

Hausdorff and Fourier dimension

Jan 8: Hausdorff measure and dimension
Jan 13: Frostman's lemma and the dimensionality of Cantor sets; energy integrals. The proof of the "hard" part of Frostman's lemma (omitted in class) can be found e.g. in [W], Chapter 8. More general constructions of Cantor-type sets and measures supported on them can be found in many places, e.g. [Falconer], Example 4.6 and Proposition 4.1, or [LP2], Lemma 2.1.
Jan 15: Fourier transform and the Fourier-analytic form of energy integrals. This is covered in [Mattila], Chapter 12, or [W], Chapter 8. The Fourier transform of |x|^-a (we used this in converting the energy integral to its Fourier form) is computed in [W], Chapter 4.
Jan 20: Fourier transforms of Cantor measures. I have used Pertti Mattila's unpublished lecture notes; they are currently not available online as far as I know, but should be published as a book in the near future.
Jan 22: The "middle-(1-2d)" Cantor set supports a measure \mu with Fourier transform decaying pointwise at infinity if and only if 1/d is not a Pisot number. I used Mattila's notes again, but this can also be found in [KS], Theorem V.7.IV. The proof that Pisot numbers admit the diophantine characterization stated in class can be found in Chapter 6 of Algebraic Numbers and Harmonic Analysis by Y. Meyer, North-Holland and Elsevier, 1972.
Jan 27: Salem sets. The construction presented in class is taken from [LP1], Section 6, except simpler because we did not try to keep track of the constants. The fact that it suffices to prove decay for the integer Fourier coefficients (k in Z) is in [W], Lemma 9.A.4.
Applications to geometric measure theory: convolutions, projections, distances

Maximal functions and differentiation theorems

The Hardy-Littlewood maximal function
The spherical maximal function
Maximal functions and differentiation theorems for fractal sets

Restriction estimates

The Tomas-Stein theorem
Restriction theorems for fractals

Resources:

[D]Fourier Analysis, J. Duoandikoetxea, American Mathematical Society, 2001
[Falconer]Fractal Geometry: Mathematical Foundations and Applications, K. Falconer, John Wiley & Sons Ltd., 1995.
[KS] J-P. Kahane and R. Salem, Ensembles parfaits et series trigonometriques, Hermann, (1963).
[LP1]I. Laba and M. Pramanik, Arithmetic progressions in sets of fractional dimension", Geom. Funct. Anal. 19 (2009), 429-456. The preprint version is available here.
[LP2]I. Laba and M. Pramanik, Maximal operators and differentiation theorems for sparse sets, Duke Math. J. 158 (2011), 347-411. The preprint version is available here.
{Mattila]Geometry of sets and measures in Euclidean spaces: fractals and rectifiability, P. Mattila, Cambridge Univ. Press, 2003
[Stein1]Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, E.M. Stein, Princeton Univ. Press, 1993
[StSh]Real Analysis, E.M. Stein and R. Shakarchi, Princeton Univ. Press, 2003
[W]Lectures on Harmonic Analysis, T. Wolff, American Mathematical Society, 2003. I have ordered it as a "recommended textbook" so you should be able to get it from the bookstore, but you can also download the PDF from my webpage.