Overview

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.

Full Product Details

Author:   Pertti Mattila (University of Helsinki)
Publisher:   Cambridge University Press
Imprint:   Cambridge University Press
Volume:   150
Dimensions:   Width: 15.70cm , Height: 2.90cm , Length: 23.50cm
Weight:   0.760kg
ISBN:  

9781107107359

ISBN 10:   1107107350
Pages:   452
Publication Date:   22 July 2015
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

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.

Reviews

'Mattila deserves kudos for having written an excellent text for the community of graduate students and research mathematicians with an analytic bent, one that exposes in considerable detail a particularly rich seam of mathematics at the interface between harmonic analysis and geometric measure theory in Euclidean space ... Libraries should be encouraged to buy their copies in haste.' Tushar Das, MAA Reviews

Author Information

Pertti Mattila is Professor of Mathematics at the University of Helsinki and an expert in geometric measure theory. He has authored the book Geometry of Sets and Measures in Euclidean Spaces as well as more than 80 other scientific publications.

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