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.
Author(s): Javier Duoandikoetxea (writ.), David Cruz-Uribe (tr.)
Series: Graduate Studies in Mathematics
Publisher: American Mathematical Society
Year: 2000
Language: English
Pages: 237
Cover......Page 1
Fourier Analysis......Page 2
Copyright - ISBN: 0821821725......Page 3
Contents......Page 6
Preface......Page 10
Preliminaries......Page 14
§1. Fourier coefficients and series......Page 16
§2. Criteria for pointwise convergence......Page 17
§3. Fourier series of continuous functions......Page 21
§4. Convergence in norm......Page 23
§5. Summability methods......Page 24
§6. The Fourier transform of L^1 functions......Page 26
§7. The Schwartz class and tempered distributions......Page 27
§8. The Fourier transform on L^p, 1 < p <= 2......Page 30
§9. The convergence and summability of Fourier integrals......Page 32
§10. Notes and further results......Page 34
§1. Approximations of the identity......Page 40
§2. Weak-type inequalities and almost everywhere convergence......Page 41
§3. The Marcinkiewicz interpolation theorem......Page 43
§4. The Hardy-Littlewood maximal function......Page 45
§5. The dyadic maximal function......Page 47
§6. The weak (1,1) inequality for the maximal function......Page 50
§7. A weighted norm inequality......Page 52
§8. Notes and further results......Page 53
§1. The conjugate Poisson kernel......Page 64
§2. The principal value of 1/x......Page 65
§3. The theorems of M. Riesz and Kolmogorov......Page 66
§4. Truncated integrals and pointwise convergence......Page 70
§5. Multipliers......Page 73
§6. Notes and further results......Page 76
§1. Definition and examples......Page 84
§2. The Fourier transform of the kernel......Page 85
§3. The method of rotations......Page 88
§4. Singular integrals with even kernel......Page 92
§5. An operator algebra......Page 95
§6. Singular integrals with variable kernel......Page 98
§7. Notes and further results......Page 100
§1. The Calderon-Zygmund theorem......Page 106
§2. Truncated integrals and the principal value......Page 109
§3. Generalized Calderon-Zygmund operators......Page 113
§4. Calderon-Zygmund singular integrals......Page 116
§5. A vector-valued extension......Page 120
§6. Notes and further results......Page 122
§1. The space atomic H^1......Page 130
§2. The space BMO......Page 132
§3. An interpolation result......Page 136
§4. The John-Nirenberg inequality......Page 138
§5. Notes and further results......Page 141
§1. The A_p condition......Page 148
§2. Strong-type inequalities with weights......Page 152
§3. A_1 weights and an extrapolation theorem......Page 155
§4. Weighted inequalities for singular integrals......Page 158
§5. Notes and further results......Page 162
§1. Some vector-valued inequalities......Page 172
§2. Littlewood-Paley theory......Page 174
§3. The Hormander multiplier theorem......Page 178
§4. The Marcinkiewicz multiplier theorem......Page 181
§5. Bochner-Riesz multipliers......Page 183
§6. Return to singular integrals......Page 187
§7. The maximal function and the Hilbert transform along a parabola......Page 193
§8. Notes and further results......Page 199
§1. Cotlar's lemma......Page 210
§2. Carleson measures......Page 212
§3. Statement and applications of the T1 theorem......Page 216
§4. Proof of the T1 theorem......Page 220
§5. Notes and further results......Page 227
Bibliography......Page 232
Index......Page 234
