Course No.： 011D9102Z*

**Textbook: **[Book I] Loukas Grafakos，Classical Fourier Analysis，GTM 249, 3rd Edition （2014）Springer. **Grading:** Grading for this course has two components: homework sets, whose average counts 40% for the final grade, and a final exam that counts 60%.

The homework sets will be posted in this course's website in SEP system every two weeks. The students are supposed to solve the problems and write down the solutions individually and turn them in, by the indicated deadline, online in SEP system.**Late homework assignments will not be accepted**. You will be evaluated both on the mathematical rigor of your solutions as well as clarity of exposition, so please pay attention to details when preparing your answers. Of course, discussions about the problems, among the students, are a healthy and recommended practice, but each student should write down the solutions in his own words, showing a clear grasp of the material being used. Besides, the deep understanding of the subject is crucial as a form of continuous buildup of knowledge for the final exam. Final exam would be held on Jun.19. **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

Course Hours：80

Course Points：4

Course Title：Harmonic Analysis I, II (调和分析I, II，数学所研究生核心基础课)

Time： Monday, 13:30-15:10; Wednesday, 13:30-15:10 & 15:20-16:10 （2019 Spring, Feb.25-Jun.19）

Place：N401, Teaching Building, Zhongguancun Campus

[Book II] Loukas Grafakos，Modern Fourier Analysis，GTM 250, 3rd Edition （2014）Springer.

The homework sets will be posted in this course's website in SEP system every two weeks. The students are supposed to solve the problems and write down the solutions individually and turn them in, by the indicated deadline, online in SEP system.

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 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 $M_p(\Bbb{R}^n)$

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

**Typos corrections：http://faculty.missouri.edu/~grafakosl/FourierAnalysis.html **

Some other corrections for Book I:

P. 34, in the 3rd line, $L^{p_j}(X_j)$ should be $L^{p_j}(X)$.

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.

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.

P. 129, in 2nd line, it is enough to replace "$(1+|x-y|)^M$" by "$(1+|x-y|)^{M/2}$".

P. 130, in 8th line, the "$+$" symbol between two integrals should be "$-$".

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.

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.

P. 322, 8th line, "$0<x<\pi/2$" should include $\pi/2$, i.e., "$0<x\leqslant\pi/2$", since the case of $x=\pi/2$ is used later.

P. 327, in the 2nd line, it is better to omit "$|\xi|$" in the denominator because it has been assumed to be $1$.

P. 344, in the 10th line, $\frac{dr}{r}$ should be $dr$.

P. 346, in the 10th line, $\Omega\in L^1$ should be $\Omega_j\in L^1$.

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.

P. 349, in the 6th line from below, $\Omega()$ should be its absolute value $|\Omega()|$.

P. 350, in (5.2.39), $\max(p,(p-1)^{-1})$ should be $\max(p^2,(p-1)^{-2})$.

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 $F_j(r\theta/\varepsilon)$.

P. 352, in (5.2.45), $\max(p,...)$ should be $\max(p^2,...)$.

P. 352, Corollary 5.2.12 should be added the condition "$n\geqslant 2$".

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

Some other corrections for Book I:

P. 34, in the 3rd line, $L^{p_j}(X_j)$ should be $L^{p_j}(X)$.

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.

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.

P. 129, in 2nd line, it is enough to replace "$(1+|x-y|)^M$" by "$(1+|x-y|)^{M/2}$".

P. 130, in 8th line, the "$+$" symbol between two integrals should be "$-$".

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.

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.

P. 322, 8th line, "$0<x<\pi/2$" should include $\pi/2$, i.e., "$0<x\leqslant\pi/2$", since the case of $x=\pi/2$ is used later.

P. 327, in the 2nd line, it is better to omit "$|\xi|$" in the denominator because it has been assumed to be $1$.

P. 344, in the 10th line, $\frac{dr}{r}$ should be $dr$.

P. 346, in the 10th line, $\Omega\in L^1$ should be $\Omega_j\in L^1$.

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.

P. 349, in the 6th line from below, $\Omega()$ should be its absolute value $|\Omega()|$.

P. 350, in (5.2.39), $\max(p,(p-1)^{-1})$ should be $\max(p^2,(p-1)^{-2})$.

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 $F_j(r\theta/\varepsilon)$.

P. 352, in (5.2.45), $\max(p,...)$ should be $\max(p^2,...)$.

P. 352, Corollary 5.2.12 should be added the condition "$n\geqslant 2$".