| 课程内容 |
1 $L^p$ Spaces and Interpolation
1.1 $L^p$ and Weak $L^p$ [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.]
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| 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.
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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.
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P. 129, in 2nd line, it is enough to replace "$(1+|x-y|)^M$" by "$(1+|x-y|)^{M/2}$".
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P. 130, in 8th line, the "+" symbol between two integrals should be "-".
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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.
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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.
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P. 322, 8th line, "$0
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P. 327, in the 2nd line, it is better to omit "$|\xi|$" in the denominator because it has been assumed to be 1.
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P. 344, in the 10th line, $\frac{dr}{r}$ should be $dr$.
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P. 346, in the 10th line, $\Omega\in L^1$ should be $\Omega_j\in L^1$.
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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.
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P. 349, in the 6th line from below, $\Omega()$ should be its absolute value $|\Omega()|$.
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P. 350, in (5.2.39), $\max(p,(p-1)^{-1})$ should be $\max(p^2,(p-1)^{-2})$.
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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 $F_j(r\theta/\varepsilon)$.
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P. 352, in (5.2.45), $\max(p,...)$ should be $\max(p^2,...)$.
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P. 352, Corollary 5.2.12 should be added the condition "$n\geqslant 2$".
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2025-04-27
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