郝成春的个人主页 | Chengchun Hao's Homepage
博士、研究员、博士生导师,中国科学院数学与系统科学研究院 数学研究所 | Professor, Institute of Mathematics, AMSS, CAS
本讲义是为中科院研究生院研究生及华罗庚班学生上课而准备的讲义,力求精炼、易懂,主要介绍调和分析最基础的内容,将几本经典教材融合在一起并力求前后统一,希望对初学者有所帮助。
Contents
Chapter 1 The Fourier Transform and Tempered Distributions
The $L^1$ theory of the Fourier transform
The $L^2$ theory and the Plancherel theorem
Schwartz spaces
The class of tempered distributions
Characterization of operators commuting with translations
Chapter 2 Interpolation of Operators
Riesz-Thorin's and Stein’s interpolation theorems
The distribution function and weak $L^p$ spaces
The decreasing rearrangement and Lorentz spaces
Marcinkiewicz’ interpolation theorem
Chapter 3 The Maximal Function and Calderón-Zygmund Decomposition
Two covering lemmas
Hardy-Littlewood maximal function
Calderón-Zygmund decomposition
Chapter 4 Singular Integrals
Harmonic functions and Poisson equation
Poisson kernel and Hilbert transform
The Calderón-Zygmund theorem
Truncated integrals
Singular integral operators commuted with dilations
The maximal singular integral operator
Vector-valued analogues
Chapter 5 Riesz Transforms and Spherical Harmonics
The Riesz transforms
Spherical harmonics and higher Riesz transforms
Equivalence between two classes of transforms
Chapter 6 The Littlewood-Paley $g$-function and Multipliers
The Littlewood-Paley $g$-function
Fourier multipliers on $L^p$
The partial sums operators
The dyadic decomposition
The Marcinkiewicz multiplier theorem
Chapter 7 Sobolev and H\"older Spaces
Riesz potentials and fractional integrals
Bessel potentials
Sobolev spaces
H\"older spaces
Chapter 8 Besov and Triebel-Lizorkin Spaces
The dyadic decomposition: the smooth version
Besov spaces and Triebel-Lizorkin spaces
Embedding theorems and Gagliardo-Nirenberg inequalities
Differential-difference norm on Besov spaces
Chapter 9 BMO Spaces
Sharp maximal functions and BMO spaces
Sharp maximal theorem, interpolation between $L^p$ and BMO
C-Z singular integral operator of type ($L^\infty$, BMO)
References
E. M. Stein, “Singular Integrals and Differentiability Properties of Functions”,Princeton University Press,1970.
E. M. Stein, G. Weiss, “Introduction to Fourier Analysis on Euclidean Spaces”,Princeton University Press,1971.
J. Bergh, J. L\"ofstrom, “Interpolation spaces”. An introduction. GMW 223, Springer-Verlag, Berlin, 1976.
J.Duoandikoetxea, Fourier Analysis, AMS, 2001. (Some corrections and additions)
B.X.Wang,Z.H.Huo,Chengchun Hao,Z.H.Guo,“Harmonic Analysis Method for Nonlinear Evolution Equations, I”,World Scientific Publishing Co. Pte. Ltd., 2011.
L. Grafakos, Modern Fourier analysis. Second edition. Graduate Texts in Mathematics, 250.Springer, New York, 2009.
周民强, 《调和分析讲义(实变方法)》,北京大学出版社,1999年.