內容簡介
《傅裏葉分析(英文版)》講述的是由Calderon和Zygmund引進的傅裏葉分析的實變量方法。這本教材源自馬德裏自治大學的一門研究生課,並吸取瞭JoseLuis Rubiode Francia在同一所大學授課的講義內容。
受傅裏葉級數與積分的研究啓發,《傅裏葉分析(英文版)》引進瞭諸如Hardy-Littlewood大函數和Hilbert變換這些經典論題。全書的其餘部分則緻力於研討奇異積分算子和乘子,討論瞭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。
第一章迴顧瞭傅裏葉級數與積分;第二章和第三章介紹瞭此領域的兩個基本算子:Hardy-Littlewood大函數和Hilbert變換。第四章和第五章討論瞭奇異積分,包括其現代推廣。第六章研討瞭H1、BMO和奇異積分間的關係;第七章講述瞭加權範數不等式。
第八章討論瞭Littlewood-Paley理論,它的發展激發瞭大量應用。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。
《傅裏葉分析(英文版)》的核心部分隻做瞭少量改動,但是在每章的“注釋和進一步的結果”小節中有著相當大的擴充並吸收瞭新的論題、結果和參考文獻。《傅裏葉分析(英文版)》適閤希望找到一本關於奇異算子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。
內頁插圖
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
傅裏葉分析(英文版) [Fourier Analysis] epub pdf mobi txt 電子書 下載 2024
傅裏葉分析(英文版) [Fourier Analysis] 下載 epub mobi pdf txt 電子書