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2018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
上课地点 中国科学院大学中关村校区 N313
上课时间 周一、三(10:00-11:40AM), 3月5日--5月9日
课程内容 1 Lp Spaces and Interpolation
1.1 Lp and Weak Lp [Suggested Homework：pp.11-16: 1-6, 10, 11, 15, 16.]
1.2 Convolution and Approximate Identities [Suggested Homework：pp.31-32: 7-9, 12, 13.]
1.3 Interpolation [Suggested Homework：pp.45-48: 2, 3, 6, 9.]
2 Maximal Functions, Fourier Transform, and Distributions
2.1 Maximal Functions [Suggested Homework：pp.99-103: 3, 4, 6, 8, 12.]
2.2 The Schwartz Class and the Fourier Transform [Suggested Homework：pp.116-119：1，2, 4, 6, 8, 13.]
2.3 The Class of Tempered Distributions [Suggested Homework：pp.132-133: 3, 5，7, 9, 11.]
5 Singular Integrals of Convolution Type
5.1 The Hilbert Transform and the Riesz Transforms [Suggested Homework：pp.329-332: 1, 3, 4, 6, 8, 9.]
5.2 Homogeneous Singular Integrals and the Method of Rotations [Suggested Homework：pp.353-355: 3, 4, 5, 6, 9, 10, 13.]
5.3 The Calderon–Zygmund Decomposition and Singular Integrals [Suggested Homework：pp.371-374: 1, 2, 7, 8, 9, 10.]
教材 Loukas Grafakos，Classical Fourier Analysis，GTM 249, 3rd Edition （2014）Springer.
Typos corrections http://faculty.missouri.edu/~grafakosl/FourierAnalysis.html
Some other corrections for Book I:
  1. P. 34, in the 3rd line, Lpj(Xj) should be Lpj(X).
  2. P. 115, in the 5th line, |ξj0|>|ξ|/√n should be |ξj0|⩾, since we can not exclude strictly the case of =, e.g., the cube.
  3. P. 128, in the 5th line from below, "M>2|\alpha|" should be "M>2\max(|\alpha|,n)", since it is necessray to prove the convergence of the integral over the complement of the cube.
  4. P. 129, in 2nd line, it is enough to replace "(1+|x-y|)^M" by "(1+|x-y|)^{M/2}".
  5. P. 130, in 8th line, the "+" symbol between two integrals should be "-".
  6. P. 319, 6th line from below, I think that "Theorem 1.4.19" might be replaced by "Theorem 1.3.2 with p_0=1 and p_1=2, and Remark 1.3.3 since H is linear". This is only a suggestion, since Thm 1.4.19 was not taught as the suggestion in preface (1.1,1.2,1.3,2.1,...) which is in Section 1.4.
  7. P. 321, 6th line from below, "\|H(f)\|_{L^{2p}}<\infty" should be "\|f\|_{L^{2p}}<\infty". By the way, in the inequality just above this line, it is maybe better and more readable to take square for each side.
  8. P. 322, 8th line, "$0
  9. P. 327, in the 2nd line, it is better to omit "|\xi|" in the denominator because it has been assumed to be 1.
  10. P. 344, in the 10th line, \frac{dr}{r} should be dr.
  11. P. 346, in the 10th line, \Omega\in L^1 should be \Omega_j\in L^1.
  12. P. 347, Theorem 5.2.11 should be added the condition "n\geqslant 2" because some estimates are not valid for n=1 in the proof.
  13. P. 349, in the 6th line from below, \Omega() should be its absolute value |\Omega()|.
  14. P. 350, in (5.2.39), \max(p,(p-1)^{-1}) should be \max(p^2,(p-1)^{-2}).
  15. P. 352, in the middle long inequalities, F_j(z) should be G_j(z) or F_j(z/\varepsilon); similarly, in next line F_j(r\theta) should be G_j(r\theta) or F_j(r\theta/\varepsilon).
  16. P. 352, in (5.2.45), \max(p,...) should be \max(p^2,...).
  17. P. 352, Corollary 5.2.12 should be added the condition "n\geqslant 2".

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