用户: Solution/ 习题: 现代分析基础I

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1Fourier Series

The main references of this chapter are [Grafakos, Classical Fourier Analysis (3rd), Chapter 3], and [Rudin, Real and Complex Analysis, Chapter 4].

1.1

1.	Prove the following identities. (i)$$D_N(x)=\frac{\sin (2N+1)\pi x}{\sin \pi x}.$$(ii)$$F_N(x)=\sum _{k=0}^N(1-\frac{k}{N+1})e^{2\pi ikx}=\frac{1}{N+1}[\frac{\sin (N+1)\pi x}{\sin \pi x}{]}^2.$$(iii)$$P_r(x)=\frac{1-r^2}{1-2r\cos (2\pi x)+r^2}.$$
2.	Let $1\le p\le \infty$. Suppose that $f\in L^p({\mathbb{R}}^n)$ and $g\in L^1({\mathbb{R}}^n)$. Then $f\ast g$ exists a.e., and $$\Vert f\ast g{\Vert }_{L^p({\mathbb{R}}^n)}\le \Vert f{\Vert }_{L^p({\mathbb{R}}^n)}\Vert g{\Vert }_{L^1({\mathbb{R}}^n)}.$$
3.	(i) Calculate $\Vert D_N{\Vert }_{L^p(\mathbb{T})}$ for $0<p\le \infty$. (ii) Prove the sharpness of the above result. (iii) Show that $D_N$ is not an approximate identity.

1.2

1.	Suppose that $f\in L^1(\mathbb{T})$, and ${\lim }_{x\to x_0}f(x)$ exists. Show that $F_N\ast f(x_0)\to f(x_0)$ as $N\to \infty$.
2.	Suppose that $f\in C(\mathbb{T})$, and ${\lim }_{N\to \infty }{D}_N\ast f(x_0)$ exists. Prove that ${\lim }_{N\to \infty }{D}_N\ast f(x_0)=f(x_0)$.
3.	Prove the Marcinkiewicz interpolation theorem.
4.	Prove the Vitali covering lemma.

1.3

1.	Let $f(x)={\chi }_{[-1/2,1/2]}(x)$. Show that $\mathcal{M}f\notin L^1$.
2.	Let $1<p<\infty$. Show that $C_0^{\infty }$ is dense in $L^p({\mathbb{R}}^n)$. Show that $C_0^{\infty }$ is not dense in $L^{\infty }({\mathbb{R}}^n)$.

2Fourier tranform

The main references of this chapter are [Grafakos, Classical Fourier Analysis (3rd), Chapter 2, Chapter 4], and [Stein, Weiss, Introduction to Fourier analysis on Euclidean spaces, Chapter VII].

2.1

1.	Prove the following statement: $f\in \mathcal{S}({\mathbb{R}}^n)$ if and only if for any $\alpha \in {\mathbb{N}}^n$ and $N\in \mathbb{N}$, there exists a constant $C_{\alpha ,N}<\infty$ such that $$|{\partial }^{\alpha }f(x)|\le C_{\alpha ,N}(1+|x|{)}^{-N}.$$
2.	Let $f(x)=e^{-\pi x^2}$. Show that $\hat{f}(\xi )=f(\xi )$.

2.2

1.	Suppose that $f_n\to f$ in $L^p$ for some $p\in (0,\infty )$. Show that there exists a subsequence $\{f_{n_j}\}$ such that $f_{n_j}\to f$ a.e.
2.	Let $1<p<2$ and $f\in L^p({\mathbb{R}}^n)$. Show that $f{\chi }_{|f|\ge 1}\in L^1$ and $f{\chi }_{|f|\le 1}\in L^2$.
3.	For $f\in L^p({\mathbb{R}}^n)$ with $1<p<2$, we define $\hat{f}=\widehat{f_1}+\mathcal{F}(f_2)$ with $f_1\in L^1$ and $f_2\in L^2$. Show that this definition is independent of the choice of $f_1$ and $f_2$.
4.	(i) Construct a Schwartz function supported in the unit ball. (ii) Construct a $C_0^{\infty }({\mathbb{R}}^n)$ function equal to $1$ on the annulus $1\le |x|\le 2$, and vanishing off the annulus $1/2\le |x|\le 4$. (iii) Construct a nonnegative nonzero Schwartz function $f$ on ${\mathbb{R}}^n$ such that $\hat{f}$ is nonnegative and compactly supported. (iv) Show that a nonzero Schwartz function $f$ cannot be nonnegative if $\hat{f}(\xi )=1$ for $|\xi |\le 1$.

2.3

1.	The Dirichlet means does not converge in $L^1(\mathbb{T})$.
2.	Let $1<p<\infty$ and $\frac{1}{p}+\frac{1}{p^{\prime }}=1$. Prove (i) $m\in {\mathcal{M}}_p({\mathbb{R}}^n)$ if and only if $m\in {\mathcal{M}}_{p^{\prime }}({\mathbb{R}}^n)$. (ii) If $m\in {\mathcal{M}}_p({\mathbb{R}}^n)$, then $m\in {\mathcal{M}}_2({\mathbb{R}}^n)$. (iii) $m\in {\mathcal{M}}_2({\mathbb{R}}^n)$ if and only if $m\in L^{\infty }({\mathbb{R}}^n)$.
3.	Let $1<p<\infty$. Then there exists a nonnegative function $\eta \in C_0^{\infty }({\mathbb{R}}^n)$ such that $\eta (0)=1$ and ${\sum }_{k\in {\mathbb{Z}}^n}|\eta (x+k){|}^p=1$ for $x\in {\mathbb{R}}^n$.
4.	Suppose $m(\xi )\in {\mathcal{M}}_p({\mathbb{R}}^n)$. Then (i) $m(\lambda \xi )\in {\mathcal{M}}_p({\mathbb{R}}^n)$ for $\lambda >0$; (ii) $m(\xi +y)\in {\mathcal{M}}_p({\mathbb{R}}^n)$ for $y\in {\mathbb{R}}^n$.
5.	Suppose that $m(\xi ,\eta )\in {\mathcal{M}}_p({\mathbb{R}}^{n+m})$, where $1<p<\infty$. Then for almost every $\xi \in {\mathbb{R}}^n$, the function $\eta \mapsto m(\xi ,\eta )$ is in ${\mathcal{M}}_p({\mathbb{R}}^m)$ with $$\Vert m(\xi ,\cdot ){\Vert }_{{\mathcal{M}}_p({\mathbb{R}}^m)}\le \Vert m{\Vert }_{{\mathcal{M}}_p({\mathbb{R}}^{n+m})}.$$

2.4

1.	Let $\beta ,\delta >0$, and $|m|\le R$. Show that $$(1-\frac{|m{|}^2}{R^2}{)}^{\beta +\delta }=C_{\beta ,\delta }{R}^{-2\beta -2\delta }{\int }_{|m|}^R(R^2-t^2{)}^{\beta -1}{t}^{2\delta +1}(1-\frac{|m{|}^2}{t^2}{)}^{\delta }dt.$$
2.	Let $1<p<\infty$, and $\lambda \ge 0$. Prove that $(1-|\xi {|}^2{)}_{+}^{\lambda }\in {\mathcal{M}}_p({\mathbb{R}}^n)$ if and only if $(1-|\xi |{)}_{+}^{\lambda }\in {\mathcal{M}}_p({\mathbb{R}}^n)$.
3.	Let $k\in \mathbb{Z}$. Suppose $F\in L^1({\mathbb{R}}^2)$ is of the form $$F(r{e}^{i\theta })=f(r)e^{ik\theta }.$$We define the Hankel transform of order $k$ of $f$ by $$H_k(f)(\rho )={\int }_0^{\infty }f(r)J_k(\rho r)rdr,$$where $J_k(r)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{e}^{ir\sin \theta -ik\theta }d\theta$ is the Bessel function of order $k$. Show that $$\hat{F}(\rho e^{i\phi })=2\pi (-i{)}^k{H}_k(f)(\rho )e^{ik\phi }.$$Hint: Use polar coordinates.
4.	Fo $f\in L^2({\mathbb{R}}_{+},rdr)$, we have $$H_k(H_k(f))=f,$$and $${\int }_0^{\infty }|f(r){|}^{2}rdr={\int }_0^{\infty }|H_k(f)(\rho ){|}^2\rho d\rho .$$

3Disc multiplier

The main references of this chapter are [de Guzman, Differentiation of integrals in ${\mathbb{R}}^n$, Chapter V], [Stein, Harmonic Analysis, Chapter X], [Wolff, Lectures on Harmonic Analysis, Chapter 11], and [Grafakos, Modern Fourier Analysis (3rd), Chapter 5].

3.1

1.	Show that$$|{\Phi }_{\alpha }(△ABC)|=2(1-\alpha {)}^2|△ABC|.$$
2.	Show that for any Kakeya set $E$, and $\epsilon ,\delta >0$, there exists a Kakeya set $F\subset E(\epsilon )$ such that $|F|<\delta$, where $E(\epsilon )$ is the $\epsilon$–neighborhood of $E$.

3.2

1.	Let $T$ be a linear operator bounded on $L^p({\mathbb{R}}^n)$. Prove that $$\Vert (\sum _{j=1}^M|T(f_j){|}^2{)}^{1/2}{\Vert }_{L^p}\le C\Vert (\sum _{j=1}^M|f_j{|}^2{)}^{1/2}{\Vert }_{L^p}.$$
2.	For a collection $\{v_j{\}}_{j=1}^M$ of unit vectors in ${\mathbb{R}}^2$, we define $H_j=\{\xi \in {\mathbb{R}}^n:\xi \cdot v_j\ge 0\}$, where $v_j\in {\mathbb{R}}^2$ is a direction. Let $S_j$ be the linear operator whose multiplier is ${\chi }_{H_j}$. Prove that if ${\chi }_{B(0,1)}\in {\mathcal{M}}_p({\mathbb{R}}^2)$, then $$\Vert (\sum _{j=1}^M|S_j(f_j){|}^2{)}^{1/2}{\Vert }_{L^p}\le C\Vert (\sum _{j=1}^M|f_j{|}^2{)}^{1/2}{\Vert }_{L^p}.$$
3.	Let $H$ be the Hilbert transform. Show that, for $y\in [2,3]$, we have $$\frac{I+iH}{2}({\chi }_{[0,1]})(y)=\frac{i}{2}\ln (1+\frac{1}{y-1}).$$

4The Uncertainty Principle

The main references of this chapter are [Wolff, Lectures on Harmonic Analysis, Chapter 5] and [Muscalu, Schlag, Classical and multilinear harmonic analysis, I, Chapter 10].

1.	(Hardy’s inequality) For $n\ge 3$, we have $$-\Delta \ge \frac{(n-2{)}^2}{4|x{|}^2}$$in the sense that, for any $f\in \mathcal{S}({\mathbb{R}}^n)$, $$-⟨\Delta f,f⟩\ge \frac{(n-2{)}^2}{4}⟨|x{|}^{-2}f,f⟩,$$i.e. $$\Vert |x{|}^{-1}f{\Vert }_{L^2({\mathbb{R}}^n)}\le \frac{2}{n-2}\Vert \nabla f{\Vert }_{L^2({\mathbb{R}}^n)}.$$Hint: Let $p:=-i\nabla$, $r:=|x|$, and $Xf(x):=xf(x)$. Use the commutator $[r^{-1}p{r}^{-1},X]$.
2.	Prove Young’s inequality: Let $1\le p\le q\le \infty$. Then $$\Vert f\ast K{\Vert }_{L^q({\mathbb{R}}^n)}\le \Vert K{\Vert }_{L^r({\mathbb{R}}^n)}\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}$$for $\frac{1}{q}+1=\frac{1}{r}+\frac{1}{p}$.
3.	Let $I=[-1/2,1/2]$, and $w_N(x)=(1+|x|{)}^{-N}$ an associated weight with $N\ge 100$. Suppose that $f\in \mathcal{S}(\mathbb{R})$ and $\mathrm{supp}\hat{f}\subset I$. Prove that, for $1\le p\le q\le \infty$, we have $$\Vert f{\Vert }_{L_{avg}^p(w_N)}\lesssim \Vert f{\Vert }_{L_{avg}^q(w_N)}$$and $$\Vert f{\Vert }_{L_{avg}^q(w_N)}\lesssim \Vert f{\Vert }_{L_{avg}^p(w_{N^{\prime }})}$$with $N^{\prime }=Np/q$. Here $\Vert f{\Vert }_{L_{avg}^q(w)}=(\frac{1}{\Vert w\Vert }{\int }_{\mathbb{R}}|f(x){|}^{q}w(x)dx{)}^{\frac{1}{q}}$. Hint: $\mathbb{R}={\cup }_{k\in \mathbb{Z}}{I}_k$ with $I_k=I+k$, hence $w_N(x)\lesssim {\sum }_{k\in \mathbb{Z}}{a}_k{\chi }_{I_k}$ with $a_k=w_N(k)$. Then apply Bernstein’s inequality to each $I_k$.
4.	Let $0<p\le q\le \infty$. Then for any $f\in \mathcal{S}({\mathbb{R}}^n)$ with $\mathrm{supp}\hat{f}\subset B(0,R)$, it holds $$\Vert f{\Vert }_{L^q({\mathbb{R}}^n)}\le C{R}^{n(\frac{1}{p}-\frac{1}{q})}\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}.$$

5Oscillatory Integrals

The main reference of this chapter is [Stein, Harmonic Analysis, Chapter VIII].

1.

Supppose $\mathbf{\psi }\in C_c^{\infty }(\mathbb{R})$. Then, for $\mu >-1$, we have the asymptotic expansion $${\int }_0^{\infty }{e}^{i\lambda x}\mathbf{\psi }(x)x^{\mu }dx\sim \sum _{j=0}^{\infty }{a}_j{\lambda }^{-j-1-\mu },$$where $a_j=i^{j+\mu +1}\Gamma (j+\mu +1)\frac{{\mathbf{\psi }}^{(j)}(0)}{j}$. Hint: $${\int }_0^{\infty }{e}^{i\lambda x}\mathbf{\psi }(x)x^{\mu }dx=\underset{\epsilon \to 0}{\lim }{\int }_0^{\infty }{e}^{i\lambda x}{e}^{-\epsilon x}\mathbf{\psi }(x)x^{\mu }dx.$$

2.

(1) There exists a constant $C>0$ such that for any $\gamma >1,\lambda ,b>1$, it holds $$\left|{\int }_0^b{e}^{i\lambda t^{\gamma }}dt\right|\le C{\lambda }^{-1/\gamma }.$$(2) For any $\gamma \in (0,1)$, there exists a constant $C$ such that $$\left|{\int }_0^1{e}^{i\lambda t^{\gamma }}dt\right|\le C{\gamma }^{-1}{\lambda }^{-1}.$$(3) Show that the result in (2) is sharp in that there is no $\alpha \in (1,\infty )$ such that $$\left|{\int }_0^1{e}^{i\lambda t^{\gamma }}dt\right|\le C{\lambda }^{-\alpha }.$$

6Restriction

The main references of this chapter are [Tao, Restriction theorems and applications (course notes 1-3), 1999. https://www.math.ucla.edu/tao/254b.1.99s/], and [Muscalu, Schlag, Classical and multilinear harmonic analysis, I, Chapter 11].

6.1

1.	Let $1\le q\le \infty$, and $S$ the unit ball of a hyperplane in ${\mathbb{R}}^n$. Show that $\Vert \hat{f}{\Vert }_{L^q(S)}\le C\Vert f{\Vert }_{L^p}$ if and only if $p=1$.
2.	Let $\sigma$ be the surface measure on ${\mathbb{S}}^{n-1}$. Prove that $$\widehat{d\sigma }(\xi )=c_1{e}^{-2\pi i|\xi |}|\xi {|}^{-\frac{n-1}{2}}+c_2{e}^{2\pi i|\xi |}|\xi {|}^{-\frac{n-1}{2}}+O(|\xi {|}^{-\frac{n+1}{2}}).$$
3.	Let $\phi$ be a Schwartz function. Prove that there exists a constant $C$ such that $$|{\phi }_k\ast (d\sigma )(\xi )|\le C{2}^k$$for all $k\ge 1$, where ${\phi }_k(x)=2^{kn}\phi (2^{k}x)$. Hint: Decompose $\phi$ dyadically into ${\sum }_{j\ge 0}{\mathbf{\psi }}_j$ where ${\mathbf{\psi }}_0$ is supported in the unit ball, and ${\mathbf{\psi }}_j$ is supported in $\{\xi :|\xi |\sim 2^j\}$ for $j\ge 1$.
4.	Let $1<p,q<\infty$. Show that $R(p\to q)$ if and only if $R^{\ast }(q^{\prime }\to p^{\prime })$.

6.2

1.	Suppose that $T(f)$ is a linear operator such that there exists a constant $R>0$ satisfying $\mathrm{supp}T(f)\subset B(x,2R)$ for any $f$ with $\mathrm{supp}f\subset B(x,R)$. Then, for $0<p\le q<\infty$, $$\Vert T(f){\Vert }_{L^q({\mathbb{R}}^n)}\le C\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}$$is equivalent to $$\Vert T(f){\Vert }_{L^q({\mathbb{R}}^n)}\le C\Vert f{\Vert }_{L^p({\mathbb{R}}^n)},\forall \mathrm{supp}f\subset B(x,R)\text{ and }x\in {\mathbb{R}}^n.$$
2.	Let $N\in \mathbb{N}$ be a large number, and ${\varphi }_k(x)=2^{kn}(1+2^k|x|{)}^{-N}$. Show that $$|{\varphi }_k\ast (d\sigma )(x)|\le C{2}^k(1+2^{k}d(x,{\mathbb{S}}^{n-1}){)}^{-N^{\prime }}$$for some large number $N^{\prime }$. Here $d(x,{\mathbb{S}}^{n-1})$ is the distance between $x$ and ${\mathbb{S}}^{n-1}$.
3.	$m_0^{\lambda }(\xi )=(1-|\xi {|}^2{)}_{+}^{\lambda }[1-\phi (2^k(1-|\xi |))]$. Show that $$|({\Delta }^N{m}_0^{\lambda })(\xi )|\le 2^{-k(\lambda -2N)}[1+2^{k}d(\xi ,{\mathbb{S}}^{n-1}){]}^{-(2N-\lambda )}.$$

7Kakeya

The main references of this chapter are [Tao, Restriction theorems and applications (course notes 6-7), 1999. https://www.math.ucla.edu/tao/254b.1.99s/], [Wolff, Lectures on Harmonic Analysis, Chapter 10] and [Mattila, Fourier analysis and Hausdorff dimension, Chapter 22].

1.	(Schur’s test) Let $T(f)(x)={\int }_{{\mathbb{R}}^n}K(x,y)f(y)dy$. Suppose that $$\underset{y}{\sup }{\int }_{{\mathbb{R}}^n}|K(x,y)|dx\le A$$and $$\underset{x}{\sup }{\int }_{{\mathbb{R}}^n}|K(x,y)|dy\le B.$$Prove that$$\Vert T(f){\Vert }_{L^2}\le \sqrt{AB}\Vert f{\Vert }_{L^2}.$$
2.	Use Schur’s test to prove the Kakeya tube conjecture in the plane.
3.	Let $2\le p\le s:=2\frac{n+1}{n-1}$. Show that $$\Vert \widehat{fd\sigma }{\Vert }_{L^p(B_R)}\le C{R}^{\frac{n+1}{2}(\frac{1}{p}-\frac{1}{s})}\Vert f{\Vert }_{L^2({\mathbb{S}}^{n-1})}.$$