Fourier transform and Hausdorff dimension, spring 2014

Lecturer

Pertti Mattila
 

Scope

10 sp.

Type

Advanced studies

Prerequisites

Basic real analysis, knowledge of some Fourier analysis would be useful but not necessary

Lectures

Weeks 3-9 and 11-18, Thursday 14-16 and Friday 10-12 in room C123. The first lecture will be on January 16.

Course description

This is a course on modern Fourier analysis with emphasis on questions  related to Hausdorff dimension. Work of Cordoba, Fefferman, Bourgain, Wolff and Tao will be discussed. It is mainly based on a book manuscript I am preparing. A version of it will be avalable when the course starts. Wolff's book (below) is also closely related. Probable topics are

basics of Fourier transform in R^n

oscillating integrals and Fourier transforms of surface measures

restriction problems; mapping properties of the restriction of the Fourier transform to the sphere

Fourier multipliers; Fefferman's ball example and Bochner-Riesz multipliers

Kakeya maximal functions (maximal functions with narrow tubes) and their relations to restriction problems

Besicovitch sets (sets of measure zero containing line segments in all directions), their Hausdorff dimension and relations to restriction problems

Bilinear restriction problems

Bibliography

P. Mattila: Fourier transform and Hausdorff dimension

E.M. Stein: Harmonic Analysis, Princeton University Press, 1993

T.H. Wolff: Lectures on Harmonic Analysis, AMS, 2003

Good for preliminaries on Fourier transform:

J. Duoandikoetxea: Fourier Analysis, AMS

Exercises

Exercises 1

Exercises 2

Week by week

16-17.1. Sections 3.1-3 of Fourier transform and Hausdorff dimension

23-24.1. Chapter 14: Oscillating integrals

30-31.1. Part of Chapter 19 on restriction, Section 2.8; Khintchine's inequality

6-7.2. Restriction conjecture 19.3, Stationary phase and restriction Chapter 20

13-14.2. Chapter 17:  Sobolev spaces and Schrödinger equation

20-21.2. Chapter 21: Fourier multipliers

27-28.2. Chapter 11: Besicovitch sets

13-14.3. Chapter 22: Kakeya maximal function

20-21.3. End of Chapter 22, beginning of bilinear restriction, Chapter 25

27-28.3. End of Chapter 25

3-4.4. Chapter 23: Hausdorff dimension of Besicovitch sets

10-11.4. end of Chapter 23 and Chapter 24: (n,k) Besicovitch sets

24.4. Laura Venieri: Spherical maximal function

25.4. Joni Teräväinen: Uncertainty principles in Fourier analysis

2.5. Jesse Jääsaari: Hilbert transform