2012春季学期：调和分析基础

授课教师	郝成春，邮箱地址 : hcc [at] amss.ac.cn
课程内容	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 Hölder Spaces Riesz potentials and fractional integrals Bessel potentials Sobolev spaces Hölder 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)
参考文献	E.M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970. E.M. Stein and Guido Weiss. Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, N.J., 1971. J. Bergh, J. Löfström, “Interpolation spaces”. An introduction. GMW 223, Springer-Verlag, Berlin, 1976. J. Duoandikoetxea. Fourier Analysis, volume 29 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. B.X. Wang, Z.H. Huo, C.C. Hao, and Z.H. Guo. Harmonic Analysis Method for Nonlinear Evolution Equations, volume I. World Scientific Publishing Co. Pte. Ltd., 2011. L. Grafakos. Modern Fourier Analysis, volume 250 of Graduate Texts in Mathematics. Springer, New York, third edition, 2014. 周民强, 实变方法 (调和分析讲义)，北京大学出版社. 1999.

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2012春季学期：调和分析基础