2010秋季学期：奇异积分与函数的可微性


授课教师 郝成春，邮箱地址 : hcc [at] amss.ac.cn
课程内容 Chapter 1 Some Fundamental Notions of Real-Variable Theory
1.1 *The maximal function
1.2 Behavior near general points of measurable sets
1.3 Decomposition in cubes of open sets in $R^n$
1.4 *Two interpolation theorems for $L^p$
Chapters 2 Singular Integrals
2.1 Review of certain aspects of harmonic analysis in $R^n$
2.2 Singular integrals: the heart of the matter
2.3 Singular integral: some extensions and variants of the preceding
2.4 Singular integral operators which commute with dilations
2.5 Vector-valued analogues
Chapter 3 Riesz Transforms, Poisson Integrals, and Spherical Harmonics
3.1 The Riesz transforms
3.2 Poisson integrals and approximations to the identity
3.3 Higher Riesz transforms and spherical harmonics
Chapter 4 The Littlewood-Paley Theory and Multipliers
4.1 The Littlewood-Paley $g$-function
4.2 The function $g^*_\lambda$
4.3 *Fourier multipliers on $L^p$
4.4 Application of the partial sums operators
4.5 The Littlewood-Paley decomposition
4.6 The Marcinkiewicz multiplier theorem
4.7 *The smooth Littlewood-Paley decomposition
Chapter 5 *Smoothness and Function Spaces
5.1 Riesz potentials and fractional integrals
5.2 Bessel potentials
5.3 *Sobolev spaces
5.4 Lipschitz continuous function spaces
5.5 *Inhomogeneous Besov and Lizorkin-Triebel Spaces
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