Fourier Analysis and Hausdorff Dimension

Modern text examining the interplay between measure theory and Fourier analysis.

Pertti Mattila (Author)

9781107107359, Cambridge University Press

Hardback, published 22 July 2015

452 pages, 5 b/w illus.
23.5 x 15.7 x 2.9 cm, 0.76 kg

'In addition to a clear, direct writing style, one of the main virtues of this book is the bibliography. (There is a three-page two-column index of authors cited.) Though the book was published in 2015, the author has managed to incorporate references and techniques from many articles that were published as late as 2014. Thus this book is still up to date a few years after its publication. This is an excellent place to begin a study of the interplay between dimension and Fourier transforms.' Benjamin Steinhurst, MathSciNet

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.

Preface
Acknowledgements
1. Introduction
2. Measure theoretic preliminaries
3. Fourier transforms
4. Hausdorff dimension of projections and distance sets
5. Exceptional projections and Sobolev dimension
6. Slices of measures and intersections with planes
7. Intersections of general sets and measures
8. Cantor measures
9. Bernoulli convolutions
10. Projections of the four-corner Cantor set
11. Besicovitch sets
12. Brownian motion
13. Riesz products
14. Oscillatory integrals (stationary phase) and surface measures
15. Spherical averages and distance sets
16. Proof of the Wolff–Erdo?an Theorem
17. Sobolev spaces, Schrödinger equation and spherical averages
18. Generalized projections of Peres and Schlag
19. Restriction problems
20. Stationary phase and restriction
21. Fourier multipliers
22. Kakeya problems
23. Dimension of Besicovitch sets and Kakeya maximal inequalities
24. (n, k) Besicovitch sets
25. Bilinear restriction
References
List of basic notation
Author index
Subject index.

Subject Areas: Geometry [PBM], Complex analysis, complex variables [PBKD], Real analysis, real variables [PBKB], Calculus & mathematical analysis [PBK], Mathematics [PB]