傅里葉分析包括了各種不同的觀點和技巧。賈維爾著的這本《傅里葉分析（英文版）》講述的是由Calderon和Zygmund引進的傅里葉分析的實變數方法。這本教材源自馬德里自治大學的一門研究生課，並吸取了Jose Luis Rubio de Francia在同一所大學授課的講義內容。 受傅里葉級數與積分的研究啟發，本書引進了諸如Hardy-Littlewood極大函數和Hilbert變換這些經典論題。全書的其餘部分則致力於研討奇異積分運算元和乘子，討論了該理論的經典內容和近期發展，諸如加權不等式、H1、BMO空間以及T1定理。 第一章回顧了傅里葉級數與積分；第二章和第三章介紹了此領域的兩個基本運算元：Hardy-Littlewood極大函數和Hilbert變換。第四章和第五章討論了奇異積分，包括其現代推廣。第六章研討了H1、BMO和奇異積分間的關係；第七章講述了加權范數不等式。第八章討論了Littlewood-Paley理論，它的發展激發了大量應用。最後一章以一個重要結果即T1定理結尾，它在此領域具有關鍵性的作用。 本書是1995年西班牙文版的更新和翻譯版本。書的核心部分只做了少量改動，但是在每章的「註釋和進一步的結果」小節中有著相當大的擴充並吸收了新的論題、結果和參考文獻。本書適合希望找到一本關於奇異運算元和乘子的經典理論簡明教材的研究生閱讀，預備知識包括勒貝格積分和泛函分析的基本知識。
作者介紹
(西)賈維爾
目錄
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
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