2019春季学期：调和分析 I，II


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 N401
上课时间 周一(13:30-15:10)、三(13:30-15:10 & 15:20-16:10), 2月25日--6月19日
课程内容 Book I: Classical Fourier Analysis
1 $L^p$ Spaces and Interpolation
1.1 $L^p$ and Weak $L^p$
1.2 Convolution and Approximate Identities
1.3 Interpolation
2 Maximal Functions, Fourier Transform, and Distributions
2.1 Maximal Functions
2.2 The Schwartz Class and the Fourier Transform
2.3 The Class of Tempered Distributions
2.4 More about Distributions and the Fourier Transform
5 Singular Integrals of Convolution Type
5.1 The Hilbert Transform and the Riesz Transforms
5.2 Homogeneous Singular Integrals and the Method of Rotations
5.3 The Calderon–Zygmund Decomposition and Singular Integrals
5.5 Vector-Valued Inequalities
5.6 Vector-Valued Singular Integrals
2.5.5 The space of Fourier Multipliers $M_p(R^n)$
6 Littlewood-Paley Theory and Multipliers
6.1 Littlewood-Paley Theory
6.2 Two Multiplier Theorems
Book II: Modern Fourier Analysis
1 Smoothness and Function Spaces
1.1 Smooth Functions and Tempered Distributions
1.2 Laplacian, Riesz Potentials, and Bessel Potentials
1.3 Sobolev Spaces
2.1 Hardy Spaces
2.1.1 Definition of Hardy Spaces
2.1.2 Quasi-norm Equivalence of Several Maximal Functions
3 BMO Spaces
3.1 Functions of Bounded Mean Oscillation
3.2 Duality between $H^1$ and BMO
教材 [Book I] Loukas Grafakos，Classical Fourier Analysis，GTM 249, 3rd Edition （2014）Springer.
[Book II] Loukas Grafakos，Modern Fourier Analysis，GTM 250, 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, $L^{p_j}(X_j)$ should be $L^{p_j}(X)$.
  2. P. 115, in the 5th line, $|\xi_{j_0}|>|\xi|/\sqrt{n}$ should be $|\xi_{j_0}| \geqslant |\xi|/\sqrt{n}$, 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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