**University Lecture Series**

Volume: 29;
2003;
137 pp;
Softcover

MSC: Primary 42;
Secondary 28

Print ISBN: 978-0-8218-3449-7

Product Code: ULECT/29

List Price: $37.00

Individual Member Price: $29.60

**Electronic ISBN: 978-1-4704-1837-3
Product Code: ULECT/29.E**

List Price: $37.00

Individual Member Price: $29.60

#### Supplemental Materials

# Lectures on Harmonic Analysis

Share this page *Edited by *
*Izabella Łaba; Carol Shubin*

“There were lots of young analysts who flocked to Chicago in those years, but virtually from the start it was clear that Tom had a special brilliance … Eventually, the mathematical door would open a crack as Tom discovered a new technique, usually of astonishing originality. The end would now be in sight, as [he] unleashed his tremendous technical abilities … Time after time, [Wolff] would pick a central problem in an area and solve it. After a few more results, the field would be changed forever … In the mathematical community, the common and rapid response to these breakthroughs was that they were seen, not just as watershed events, but as lightning strikes that permanently altered the landscape.”

—Peter W. Jones, Yale University

“Tom Wolff was not only a deep thinker in mathematics but
also a technical master.”

—Barry Simon, California Institute of Technology

Thomas H. Wolff was a leading analyst and winner of the Salem and Bôcher
Prizes. He made significant contributions to several areas of harmonic
analysis, in particular to geometrical and measure-theoretic questions related
to the Kakeya needle problem. Wolff attacked the problem with awesome power and
originality, using both geometric and combinatorial ideas. This book provides
an inside look at the techniques used and developed by Wolff. It is based on a
graduate course on Fourier analysis he taught at Caltech.

The selection of the material is somewhat unconventional in that it
leads the reader, in Wolff's unique and straightforward way, through
the basics directly to current research topics. The book demonstrates
how harmonic analysis can provide penetrating insights into deep
aspects of modern analysis. It is an introduction to the subject as a
whole and an overview of those branches of harmonic analysis that are
relevant to the Kakeya conjecture.

The first few chapters cover the usual background material: the Fourier
transform, convolution, the inversion theorem, the uncertainty principle, and
the method of stationary phase. However, the choice of topics is highly
selective, with emphasis on those frequently used in research inspired by the
problems discussed in later chapters. These include questions related to the
restriction conjecture and the Kakeya conjecture, distance sets, and Fourier
transforms of singular measures. These problems are diverse, but often
interconnected; they all combine sophisticated Fourier analysis with intriguing
links to other areas of mathematics, and they continue to stimulate first-rate
work.

The book focuses on laying out a solid foundation for further reading and
research. Technicalities are kept to a minimum, and simpler but more basic
methods are often favored over the most recent methods. The clear style of the
exposition and the quick progression from fundamentals to advanced topics
ensure that both graduate students and research mathematicians will benefit
from the book.

#### Table of Contents

# Table of Contents

## Lectures on Harmonic Analysis

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Foreword vii8 free
- Preface ix10 free
- The 𝐿¹ Fourier transform 112 free
- The Schwartz space 718
- Fourier inversion and the Plancherel theorem 1526
- Some specifics, and 𝐿^{𝑝} for 𝑝<2 2334
- The uncertainty principle 3142
- The stationary phase method 3748
- The restriction problem 4556
- Hausdorff measures 5768
- Sets with maximal Fourier dimension and distance sets 6778
- The Kakeya problem 7990
- Recent work connected with the Kakeya problem 91102
- Bibliography for Chapter 11 129140
- Historical notes 133144
- Bibliography 137148
- Back Cover Back Cover1154

#### Readership

Graduate students and research mathematicians interested in harmonic analysis.

#### Reviews

Lovely little book … deft and forceful writing … considerable selectivity … Proofs are tastefully either provided or sketched, so that the reader has a palpable sense of how the subject works … a delightful and satisfying reading experience, leaving the reader with a strong desire to push on and learn more … Both graduate students and experienced mathematicians will learn a great deal from this volume.

-- Mathematical Reviews