Fourier Analysis and Hausdorff Dimension (Cambridge Studies in Advanced Mathematics)
Cambridge University Press, 7/22/2015
EAN 9781107107359, ISBN10: 1107107350
Hardcover, 452 pages, 23.5 x 15.7 x 2.9 cm
Language: English
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–Erdoğan 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.