傅里叶分析(英文版)(精)/美国数学会经典影印系列

傅里叶分析包括了各种不同的观点和技巧。贾维 尔著的这本《傅里叶分析（英文版）》讲述的是由 Calderon和Zygmund引进的傅里叶分析的实变量方法 。这本教材源自马德里自治大学的一门研究生课，并 吸取了Jose Luis Rubio de Francia在同一所大学 授课的讲义内容。 受傅里叶级数与积分的研究启发，本书引进了诸 如Hardy-Littlewood极大函数和Hilbert变换这些经 典论题。全书的其余部分则致力于研讨奇异积分算子 和乘子，讨论了该理论的经典内容和近期发展，诸如 加权不等式、H1、BMO空间以及T1定理。 第一章回顾了傅里叶级数与积分；第二章和第三 章介绍了此领域的两个基本算子：Hardy-Littlewood 极大函数和Hilbert变换。第四章和第五章讨论了奇 异积分，包括其现代推广。第六章研讨了H1、BMO和 奇异积分间的关系；第七章讲述了加权范数不等式。 第八章讨论了Littlewood-Paley理论，它的发展激发 了大量应用。最后一章以一个重要结果即T1定理结尾 ，它在此领域具有关键性的作用。 本书是1995年西班牙文版的更新和翻译版本。书 的核心部分只做了少量改动，但是在每章的“注释和 进一步的结果”小节中有着相当大的扩充并吸收了新 的论题、结果和参考文献。本书适合希望找到一本关 于奇异算子和乘子的经典理论简明教材的研究生阅读 ，预备知识包括勒贝格积分和泛函分析的基本知识。

Preface Preliminaries Chapter 1. Fourier Series and Integrals §1. Fourier coefficients and series §2. Criteria for pointwise convergence §3. Fourier series of continuous functions §4. Convergence in norm §5. Summability methods §6. The Fourier transform of L1 functions §7. The Schwartz class and tempered distributions §8. The Fourier transform on Lp, 1 < p < 2 §9. The convergence and summability of Fourier integrals §10. Notes and further results Chapter 2. The Hardy-Littlewood Maximal Function §1. Approximations of the identity §2. Weak-type inequalities and almost everywhere convergence §3. The Marcinkiewicz interpolation theorem §4. The Hardy-Littlewood maximal function §5. The dyadic maximal function §6. The weak (1, 1) inequality for the maximal function §7. A weighted norm inequality §8. Notes and further results Chapter 3. The Hilbert Transform §1. The conjugate Poisson kernel §2. The principal value of 1/x §3. The theorems of M. Riesz and Kolmogorov §4. Truncated integrals and pointwise convergence §5. Multipliers §6. Notes and further results Chapter 4. Singular Integrals (I) §1. Definition and examples §2. The Fourier transform of the kernel §3. The method of rotations §4. Singular integrals with even kernel §5. An operator algebra §6. Singular integrals with variable kernel §7. Notes and further results Chapter 5. Singular Integrals (II) §1. The Calderon-Zygmund theorem §2. Truncated integrals and the principal value §3. Generalized Calderon-Zygmund operators §4. CalderSn-Zygmund singular integrals §5. A vector-valued extension §6. Notes and further results Chapter 6. H1 and BMO §1. The space atomic H1 §2. The space BMO §3. An interpolation result §4. The John-Nirenberg inequality §5. Notes and further results Chapter 7. Weighted Inequalities §1. The Ap condition §2. Strong-type inequalities with weights §3. A1 weights and an extrapolation theorem §4. Weighted inequalities for singular integrals §5. Notes and further results Chapter 8. Littlewood-Paley Theory and Multipliers §1. Some vector-valued inequalities §2. Littlewood-Paley theory §3. The HSrmander multiplier theorem §4. The Marcinkiewicz multiplier theorem §5. Bochner-Riesz multipliers §6. Return to singular integrals §7. The maximal function and the Hilbert transform along a parabola §8. Notes and further results Chapter 9. The T1 Theorem §1. Cotlar's lemma §2. Carleson measures §3. Statement and applications of the T1 theorem §4. Proof of the T1 theorem §5. Notes and further results Bibliography Index