**Graduate Studies in Mathematics**

Volume: 29;
2001;
222 pp;
Hardcover

MSC: Primary 42;

Print ISBN: 978-0-8218-2172-5

Product Code: GSM/29

List Price: $47.00

AMS Member Price: $37.60

MAA Member Price: $42.30

**Electronic ISBN: 978-1-4704-1145-9
Product Code: GSM/29.E**

List Price: $44.00

AMS Member Price: $35.20

MAA Member Price: $39.60

#### Supplemental Materials

# Fourier Analysis

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*Javier Duoandikoetxea*

Fourier analysis encompasses a variety of
perspectives and techniques. This volume presents the real variable
methods of Fourier analysis introduced by Calderón and
Zygmund. The text was born from a graduate course taught at the
Universidad Autónoma de Madrid and incorporates lecture notes
from a course taught by José Luis Rubio de Francia at the same
university.

Motivated by the study of Fourier series and integrals, classical
topics are introduced, such as the Hardy-Littlewood maximal function
and the Hilbert transform. The remaining portions of the text are
devoted to the study of singular integral operators and
multipliers. Both classical aspects of the theory and more recent
developments, such as weighted inequalities, \(H^1\),
\(BMO\) spaces, and the \(T1\) theorem, are
discussed.

Chapter 1 presents a review of Fourier series and
integrals; Chapters 2 and 3 introduce two operators that are basic to
the field: the Hardy-Littlewood maximal function and the Hilbert
transform. Chapters 4 and 5 discuss singular integrals, including
modern generalizations. Chapter 6 studies the relationship between
\(H^1\), \(BMO\), and singular integrals; Chapter 7
presents the elementary theory of weighted norm inequalities. Chapter
8 discusses Littlewood-Paley theory, which had developments that
resulted in a number of applications. The final chapter concludes with
an important result, the \(T1\) theorem, which has been of
crucial importance in the field.

This volume has been updated and translated from the Spanish
edition that was published in 1995. Minor changes have been made to
the core of the book; however, the sections, “Notes and Further
Results” have been considerably expanded and incorporate new
topics, results, and references. It is geared toward graduate students
seeking a concise introduction to the main aspects of the classical
theory of singular operators and multipliers. Prerequisites include
basic knowledge in Lebesgue integrals and functional analysis.

#### Readership

Graduate students and research mathematicians interested in Fourier analysis.

#### Reviews & Endorsements

This is a great introductory book to Fourier analysis on Euclidean spaces and can serve as a textbook in an introductory graduate course on the subject … The chapters on the Hardy-Littlewood maximal function and the Hilbert transform are extremely well written … this is a great book and is highly recommended as an introductory textbook to Fourier analysis. The students will have a lot to benefit from in the simple and quick presentation of the book.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Fourier Analysis

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents ix10 free
- Preface xiii14 free
- Preliminaries xvii18 free
- Chapter 1. Fourier Series and Integrals 120 free
- §1. Fourier coefficients and series 120
- §2. Criteria for pointwise convergence 221
- §3. Fourier series of continuous functions 625
- §4. Convergence in norm 827
- §5. Summability methods 928
- §6. The Fourier transform of L[sup(1)] functions 1130
- §7. The Schwartz class and tempered distributions 1231
- §8. The Fourier transform on L[sup(p)], 1 < p ≤ 2 1534
- §9. The convergence and summability of Fourier integrals 1736
- §10. Notes and further results 1938

- Chapter 2. The Hardy-Littlewood Maximal Function 2544
- §1. Approximations of the identity 2544
- §2. Weak-type inequalities and almost everywhere convergence 2645
- §3. The Marcinkiewicz interpolation theorem 2847
- §4. The Hardy-Littlewood maximal function 3049
- §5. The dyadic maximal function 3251
- §6. The weak (1,1) inequality for the maximal function 3554
- §7. A weighted norm inequality 3756
- §8. Notes and further results 3857

- Chapter 3. The Hilbert Transform 4968
- Chapter 4. Singular Integrals (I) 6988
- Chapter 5. Singular Integrals (II) 91110
- Chapter 6. H[sup(1)] and BMO 115134
- Chapter 7. Weighted Inequalities 133152
- Chapter 8. Littlewood-Paley Theory and Multipliers 157176
- §1. Some vector-valued inequalities 157176
- §2. Littlewood-Paley theory 159178
- §3. The Hörmander multiplier theorem 163182
- §4. The Marcinkiewicz multiplier theorem 166185
- §5. Bochner-Riesz multipliers 168187
- §6. Return to singular integrals 172191
- §7. The maximal function and the Hilbert transform along a parabola 178197
- §8. Notes and further results 184203

- Chapter 9. The T1 Theorem 195214
- Bibliography 217236
- Index 219238
- Back Cover Back Cover1242