Fourier analysis and Hausdorff dimension

Table of Contents:

Part 1. Preliminaries and some simpler applications of the Fourier transform.
1 Measure theoretic preliminaries
2 Fourier transforms
3 Hausdorff dimension of projections and distance sets
4 Exceptional projections and Sobolev dimension
5 Slices of measures and intersections with planes
6 Intersections of general sets and measures

Part 2. Specific constructions.
7 Cantor measures
8 Bernoulli convolutions
9 Projections of the four-corner Cantor set
10 Besicovitch sets
11 Brownian motion
12 Riesz products
13 Oscillatory integrals (stationary phase) and surface measures

Part 3. Deeper applications of the Fourier transform.
14 Spherical averages and distance sets
15 Proof of the Wolff-Erdoğan Theorem
16 Sobolev spaces, Schrödinger equation and spherical averages
17 Generalized projections of Peres and Schlag

Part 4. Fourier restriction and Kakeya type problems.
18 Restriction problems
19 Stationary phase and restriction
20 Fourier multipliers
21 Kakeya problems
22 Dimension of Besicovitch sets and Kakeya maximal inequalities
23 (n, k) Besicovitch sets
24 Bilinear restriction

During the past two decades there has been active interplay between geometric measure theory and Fourier analysis. This book describes part of that development, concentrating on the relationship between the Fourier transform and Hausdorff dimension. The main topics concern applications of the Fourier transform to geometric problems involving Hausdorff dimension, such as Marstrand type projection theorems and Falconer's distance set problem, and the role of Hausdorff dimension in modern Fourier analysis, especially in Kakeya methods and Fourier restriction phenomena. The discussion includes both classical results and recent developments in the area. The author emphasises partial results of important open problems, for example, Falconer's distance set conjecture, the Kakeya conjecture and the Fourier restriction conjecture. Essentially self-contained, this book is suitable for graduate students and researchers in mathematics.

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