Stein 3장 연습문제

여기서도 4k+2만 푼다. (+ Prob 8)

숙제 문제는 6,7,9,17,19,24,32 + Problem 8.

 

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2. (Kernel의 조건이 바뀌면 어떻게 될까?)

Suppose $\{K_{\delta }\}$ is a family of kernels that satisfies:

1. $|K_{\delta }(x)|\le A{\delta }^{-d}$ for all $\delta >0$
2. $|K_{\delta }(x)|\le A\delta /|x{|}^{d+1}$ for all $\delta >0$.
3. $\int K_{\delta }=0$ (Not 1!!)

Show that if $f$ is integrable on ${\mathbb{R}}^d$, then

$$(f\ast K_{\delta })(x)\to 0\text{ for a.e. }x\text{, as }\delta \to 0\text{.}$$

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6. (숙제문제)

In one dimension there is a version of the basic inequality

$$m(\{x\in \mathbb{R}:f^{\ast }>\alpha \})\le \frac{A}{\alpha }\Vert f{\Vert }_1$$

in the form of an identity. We define the "one-sided" maximal function

 

$$f_{+}^{\ast }(x)=\underset{h>0}{sup}\frac{1}{h}{\int }_x^{x+h}|f(y)|dy.$$

 

If $E_{\alpha }^{+}=\{x\in \mathbb{R}:f_{+}^{\ast }>\alpha \}$, then

 

$$m(E_{\alpha }^{+})=\frac{1}{\alpha }{\int }_{E_{\alpha }^{+}}|f(y)|dy.$$

 

[Hint: Apply Lemma 3.5 to $F(x)={\int }_0^x|f(y)|dy-\alpha x$. Then $E_{\alpha }^{+}$ is the union of disjoint intervals $(a_k,b_k)$ with ${\int }_{a_k}^{b_k}|f(y)|dy=\alpha (a_k-b_k)$.]

Proof

 

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10.

Construct an increasing function on $\mathbb{R}$ whose set of discontinuities is precisely $\mathbb{Q}$.

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14.

The following measurability issues arose in the discussion of differentiability of functions.

1. Suppose $F$ is continuous on $[a,b]$. Show that $$D^{+}(F)(x)=\underset{h\to 0+}{lim sup}\frac{F(x+h)-F(x)}{h}$$ is measurable.
2. Suppose $J(x)=\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n{j}_n(x)$ is a jump function as in Section 3.3. Show that $$\underset{h\to 0}{lim sup}\frac{J(x+h)-J(x)}{h}$$ is measurable.

[Hint: For (a), the continuity of $F$ allows one to restrict to countably many $h$ in takeing the limsup. For (b), given $k>m$, let $F_{k,m}^N=\underset{1/k\le |h|\le 1/m}{sup}\left|\frac{J_N(x+h)-J_N(x)}{h}\right|$, where $J_N(x)=\sum _{n=1}^N{\alpha }_n{j}_n(x)$. Note that each $F_{k,m}^N$ is measurable. Then, successively, let $N\to \mathrm{\infty },k\to \mathrm{\infty }$, and finally $m\to \mathrm{\infty }$.]

 

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18.
Verify the agreement between the two definitions given for the Cantor-Lebesgue function in Exercise 2. Chapter 1 and in Section 3.1 of this chapter.

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22.

Suppose that $F$ and $G$ are absolutely continuous in $[-\pi ,\pi ]$. Show that their product $FG$ is also absolutely continuous. This has the following consequences.

1. $${\int }_{-\pi }^{\pi }{F}^{\prime }(x)G(x)dx=-{\int }_{-\pi }^{\pi }F(x)G^{\prime }(x)dx+[F(x)G(x){]}_{-\pi }^{\pi }$$
2. Let $F(\pi )=F(-\pi )$. Show that if $$a_n=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }F(x)e^{-inx}dx,$$ such that $F(x)\sim \sum a_n{e}^{inx}$, then $$F^{\prime }(x)\sim \sum in{a}_n{e}^{inx}.$$
3. What happens if $F(-\pi )\ne F(\pi )$? [Hint: Consider $F(x)=x$.]

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26.

외측도를 Cube가 아니라 Ball로 정의하면 어떻게 될까? 그래도 똑같은데, 원래 정의를 $m_{\ast }$로, Ball로 만든 정의를 $m_{\ast }^{\mathcal{B}}$라고 하면 $m_{\ast }(E)\le m_{\ast }^{\mathcal{B}}(E)$는 자명하고, Reverse Inequality를 보여야 한다. 다음을 보여라. $$\sum _{j}m(B_j)\le m_{\ast }(E)+ϵ$$

Note that for any preassigned $\delta$, we can choose the balls to have diameter $<\delta$. 

[Hint: $E$를 열린집합으로 싸매고 적당히 근사되게 finite ball 고르고, 나머지 부분은 cube로 덮고 그걸 ball로 바꾼다.]

 

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30.

$F$가 bounded이고 임의의 Bounded suinterval $[a,b]$에서 유계변동이고 $\underset{a,b}{sup}{T}_F(a,b)<\mathrm{\infty }$이면, $\mathbb{R}$에서 유계변동이라고 한다. 다음을 보여라.

• ${\int }_{\mathbb{R}}|F(x_h)-F(x)|dx\le A|h|$, for some constant $A$ and all $h\in \mathbb{R}$.
• $|{\int }_{\mathbb{R}}F(x){\phi }^{\prime }(x)dx|\le A,$ where $\phi$ ranges over all $C^1$ functions of bounded support with $sup|\phi (x)|\le 1$.

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Prob 8.  (숙제문제)

Let $\mathcal{R}$ denote the set of all rectangles in ${\mathbb{R}}^2$ that contain the origin and with sides parallel ot the coordinate axis. Conseider the maximal operator associated to this family, namely

$$f_{\mathcal{R}}^{\ast }=\underset{R\in \mathcal{R}}{sup}\frac{1}{m(R)}{\int }_R|f(x-y)|dy.$$

1. Then, $f\mapsto f_{\mathcal{R}}^{\ast }$ does not satisfy the weak type inequality $$m(\{x:f_{\mathcal{R}}^{\ast }(x)>\alpha \})\le \frac{A}{\alpha }\Vert f{\Vert }_1$$
2. Using this, one can show that there exists $f\in L^1(\mathbb{R})$ so that for $R\in \mathcal{R}$ $$\underset{\text{diam}(R)\to 0}{lim sup}\frac{1}{m(R)}{\int }_{R}f(x-y)dy=\mathrm{\infty }\text{ for a.e. }x\text{.}$$

[Hint: For part (a), let $B$ be the unit ball, and consider the function $\varphi (x)={\chi }_B(x)/m(B)$. For $\delta >0$, let ${\varphi }_{\delta }(x)={\delta }^{-2}\varphi (x/\delta )$. Then $$({\varphi }_{\delta }{)}_{\mathcal{R}}^{\ast }(x)\to \frac{1}{|x_1||x_2|}\text{ as }\delta \to 0\text{,}$$ for every $(x_1,x_2)$, with $x_1{x}_2\ne 0.$ If the weak type inequality held, then we would have $$m(\{|x|\le 1:|x_1{x}_2{|}^{-1}>\alpha \})\le \frac{A}{\alpha }.$$ This is a contradiction since the left-hand side is of the order of $(\log \alpha )\alpha$ as $\alpha$ tends to infinity.]