Math Problems Vault

Problem VIII.4.1

Let $-\infty <a<b \leqslant \infty$ . Prove that

\int _{(a, b)^{m}} h\left(\max _{1 \leqslant k \leqslant m} x_{k}\right) d x=m \int _{a}^{b}(t-a)^{m-1} h(t) d t

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Problem VIII.4.2

(a)(b)

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(a)

Let

O_{m}(r)=\left\{ \left(x_{1}, \ldots , x_{m}\right) \in \mathbb {R}^{m} \mid \sum _{1 \leqslant k \leqslant m}\left|x_{k}\right| \leqslant r\right\} .

Prove that

\int _{O_{m}(r)} h\left(\sum _{1 \leqslant k \leqslant m}\left|x_{k}\right|\right) d x=\frac{2^{m}}{(m-1)!} \int _{0}^{r} t^{m-1} h(t) d t

(b)

Let $a=\left(a_{1}, \ldots , a_{m}\right) \in \mathbb {R}_{+}^{m}, E=\left\{ x \in \mathbb {R}_{+}^{m} \mid (x, a)>1\right\}$ . Compute the integral $\int_{E} e^{-(a, x)} d x$ .

Problem VIII.4.3

Prove that for $p>0$

\int _{\mathbb {R}^{m}} h\left(\left(\sum _{1 \leqslant k \leqslant m}\left|x_{k}\right|^{p}\right)^{1 / p}\right) d x=m C_{m}(p) \int _{0}^{\infty } t^{m-1} h(t) d t

where $C_{m}(p)=\lambda_{m}\left\{ \left(x_{1}, \ldots , x_{m}\right) \in \mathbb {R}^{m} \mid \left|x_{1}\right|^{p}+\ldots +\left|x_{m}\right|^{p} \leqslant 1\right\}$ (for the value of $C_{m}(p)$ see problem 3.36).

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Problem VIII.4.4

Let $E \subset \mathbb {R}^{m}, g_{E}(t)=\lambda_{m}(E \cap B(0, t))(t>0)$ . Prove that

\int _{E} h(\left\| x\right\| ) d x=\int _{0}^{\infty } h(t) d g_{E}(t)

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Problem VIII.4.5

Let $E \subset \mathbb {R}^{m}, \quad f \in \mathscr {L}^{0}(E)$ , and suppose that for every $t>0 \lambda_{m}\left\{ x \in E \mid \left|f(x)\right|>t\right\} <+\infty$ . Prove that for every nonnegative function $\varphi \in C(\mathbb {R})$ the equality

\int _{E} \varphi (\left|f(x)\right|) d x=-\int _{0}^{+\infty } \varphi (t) d F(t)

holds, where $F$ is the decreasing distribution function of $f$ (see the definition before problem 2.8).

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Problem VIII.4.6

Let $A \subset \mathbb {R}^{2}$ be a convex bounded set, $\rho (x)= =\inf \left\{ \left\| x-y\right\| \mid y \in A\right\}$ the distance from a point $x \in \mathbb {R}^{2}$ to $A$ . Find the integral $\int_{\mathbb {R}^{2}} e^{-\rho (x)} d x$ .

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Problem VIII.4.7

Let $E \subset \mathbb {R}^{m}, f, g$ be nonnegative functions, finite almost everywhere on $E$ , having decreasing distribution functions $F$ and $G$ respectively. Suppose that at the point $t_{0}>0$ the difference $F-G$ changes sign from " $-$ " to " $+$ ", i.e., $F(t) \leqslant G(t)$ for $t \in \left(0, t_{0}\right)$ and $F(t) \geqslant G(t)$ for $t>t_{0}$ . Finally, let $\Delta =\left\{ p>0 \mid f^{p}, g^{p} \in \mathscr {L}(E)\right\}$ . Prove that the function

\varphi (p)=\frac{1}{p t_{0}^{p}} \int _{E}\left(f^{p}(x)-g^{p}(x)\right) d x

is increasing on the set $\Delta$ . In particular, if $\int_{E} f^{p_{0}}(x) d x \geqslant \int_{E} g^{p_{0}}(x) d x$ , then $\int_{E} f^{p}(x) d x \geqslant \int_{E} g^{p}(x) d x$ for all $p>p_{0}$ .

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Problem VIII.4.8

Using the result of the previous problem, prove that $\int_{0}^{+\infty }\left|\frac{\sin x}{x}\right|^{p} d x<\frac{\pi }{\sqrt{2 p}}$ for $p>2$ , while for $1<p<2$ the opposite inequality holds.

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Problem VIII.4.9

Let $\sigma$ be the Cantor function (see problem III.3.14). Compute the integrals

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(a)

$\int_{0}^{1} x d \sigma (x)$ ;

(b)

$\int_{0}^{1} x^{2} d \sigma (x)$ ;

(c)

$\int_{0}^{1} \sigma (x) d x$ ;

(d)

$\int_{0}^{1} \sigma (x) d \sigma (x)$ ;

(e)

$\int_{0}^{1} \sigma (1-x) d \sigma (x)$ .

Problem VIII.4.10

(a)(b)(c)

Let $C$ be the Cantor set, $\sigma$ the Cantor function. For which $p \in \mathbb {R}$ are the following integrals finite:

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(a)

$\int_{0}^{1} \int_{0}^{1} \frac{d \sigma (x) d \sigma (y)}{\left(x^{2}+y^{2}\right)^{p / 2}}$ ;

(b)

$\int_{0}^{1} \int_{0}^{1} \frac{d \sigma (x) d \sigma (y)}{\left((x-u)^{2}+(y-v)^{2}\right)^{p / 2}}$ , where $(u, v) \in C \times C$ ;

(c)

$\int_{0}^{1} \int_{0}^{1} \frac{d \sigma (x) d \sigma (y)}{\left|x-y\right|^{p}}$ ? (For a generalization of the statement in c) see problem 5.25.)

Problem VIII.4.11

(a)(b)

Let $C$ be the Cantor set, $\sigma$ the Cantor function. Define the function $\varphi$ by the equality

\varphi (z)=\int _{0}^{1} \int _{0}^{1} \frac{d \sigma (\xi ) d \sigma (\eta )}{z-(\xi +i \eta )} \quad (z \in \mathbb {C})

Prove that

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(a)

$\varphi \in C(\mathbb {C}), z \varphi (z) \rightarrow 1$ as $z \rightarrow \infty$ ;

(b)

the function $\varphi$ is holomorphic outside the set $C \times C$ (compare with Problem 3.46).

Problem VIII.4.12

Let $A \subset \mathbb {R}^{m}, B \subset \mathbb {R}^{n}$ be convex compact centrally symmetric sets. What fraction of the volume of the set $A \times B$ is the volume of the convex hull of the set $(A \times \left\{ 0\right\} ) \cup (\left\{ 0\right\} \times B)$ ?

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Problem VIII.4.13

Let $p>1, E \subset \mathbb {R}^{m}, f \in \mathscr {L}^{0}(E)$ , and let $f^{*}$ be the nonincreasing rearrangement of the function $\left|f\right|$ . Prove that if the integral

I_{p}(f)=\int _{0}^{+\infty } f^{*}(t) t^{\frac{1}{p}-1} d t, \text{ is finite, then } \int _{E}\left|f(x)\right|^{p} d x<\infty

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Problem VIII.4.14

Keeping the notation of the previous problem, set for $n \in \mathbb {Z}$

\begin{array}{ll} E_{n}=\left\{ x \in E \mid 2^{n}<\left|f(x)\right|\right\} , & S=\sum _{n \in \mathbb {Z}} 2^{n}\left(\lambda _{m}\left(E_{n}\right)\right)^{1 / p} ; \\ e_{n}=\left\{ x \in E \mid 2^{n}<\left|f(x)\right| \leqslant 2^{n+1}\right\} , & s=\sum _{n \in \mathbb {Z}} 2^{n}\left(\lambda _{m}\left(e_{n}\right)\right)^{1 / p} . \end{array}

Prove that $(s<+\infty ) \Rightarrow \left(I_{p}(f)<+\infty \right) \Rightarrow (S<+\infty )$ .

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