Math Problems Vault

Problem 8.3.1

(a)(b)(c)(d)

Let $f, g \in L^{p}$ , where $0<p<1$ . We know that $f+g \in L^{p}$ by the Minkowski-like Inequality result given earlier.

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(a)

We have $(a+b)^{p} \leqslant a^{p}+b^{p}$ for every $0<a, b<\infty$ .

(b)

From (a), we have $\int \left|f+g\right|^{p} \leqslant \int \left|f\right|^{p}+\int \left|g\right|^{p}$ .

(c)

If we write $\left\| f-g\right\|_{p}^{p}$ for the distance between $f$ and $g$ , then this distance function is truly a metric, if we identify functions equal $\mu -$ a. e. on $\Omega$ .

(d)

Writing $\left\| f-g\right\|_{p}$ for the distance between $f$ and $g$ does not define a metric on $L^{p}$ .

Problem 8.3.2

(a)(b)(c)(d)

Consider the space $\Omega =(0,1)$ and let $0<p<1$ . Write B for the Borel subsets of $(0,1)$ , and write $\lambda$ for Lebesgue measure restricted to B. We will show that there exists no norm $\left\| \cdot \right\|$ on $L^{p}$ such that $\lim_{k}\left\| f_{k}\right\|_{p}=0$ forces $\lim_{k}\left\| f_{k}\right\| =0$ .

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(a)

Suppose that such a norm $\left\| \cdot \right\|$ exists. Then there is $C \geqslant 0$ such that $\left\| f \right\| \leqslant C\left\| f\right\|_{p}$ for each $f \in L^{p}$ .

Pick the minimal such $C$ from (a).

(b)

There is $0<c<1$ with $\int_{0}^{c}\left|f\right|^{p}=\int_{c}^{1}\left|f\right|^{p}=\frac{1}{2} \int_{0}^{1}\left|f\right|^{p}$ .

(c)

Let $g=f \; \mathbb {1}_{(0, c]}$ and $h=f \; \mathbb {1}_{(c, 1]}$ , so that $f=g+h$ . Then $\left\| g\right\|_{p}=\left\| h\right\|_{p}= 2^{-1 / p}\left\| f\right\|_{p}$ and $\left\| f\right\| \leqslant \left\| g\right\| +\left\| h\right\| \leqslant C\left\| g\right\|_{p}+C\left\| h\right\|_{p}=C \times 2^{1-1 / p}\left\| f\right\|_{p}$ . Use the minimality of $C$ to obtain $C \leqslant C 2^{1-1 / p}$ , and deduce $C=0$ .

(d)

Conclude that $\left\| f\right\| =0$ for every $f \in L^{p}$ , and hence the assumption in (a) entails a contradiction.

Problem 8.3.3

(a)(b)(c)

Let $0<p_{0}<\infty$ and let $f \in L^{p_{0}}$ be nonnegative. Let $E_{1}=\left\{ \omega \in \Omega : f(\omega )=0\right\} , E_{2}=\left\{ \omega \in \Omega : 0<f(\omega ) \leqslant 1\right\}$ , and $E_{3}=\left\{ \omega \in \Omega : f(\omega )>1\right\}$ .

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(a)

$\lim_{p \rightarrow 0^{+}} \int_{E_{2}}\left|f\right|^{p}=\mu \left(E_{2}\right)$ .

(b)

$\lim_{p \rightarrow 0^{+}} \int_{E_{3}}\left|f\right|^{p}=\mu \left(E_{3}\right)$ .

(c)

$\lim_{p \rightarrow 0^{+}} \int \left|f\right|^{p}=\mu (\left\{ \omega \in \Omega : f(\omega ) \neq 0\right\} )$ .

Problem 8.3.4

Say $\mu (\Omega )=1$ , and let $f \in L^{1}$ be nonnegative. Write $\log 0=-\infty$ .

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(a)

$\int \log f \leqslant \log \int f$ if $\log f \in L^{1}$ .

(b)

If $\log f \notin L^{1}$ , then $\int (\log f)^{+}<\infty$ under the assumption $f \in L^{1}$ , so it must be the case that $\int (\log f)^{-}=\infty$ . Conclude that even if $\log f \notin L^{1}, \int \log f$ still exists and equals $-\infty$ , giving the inequality in (a).

(c)

$\left(f^{r}-1\right) / r$ decreases to $\log f$ as $r \rightarrow 0^{+}$ , hence $\lim_{r \rightarrow 0^{+}}\left(\int f^{r}-1\right) / r=\int \log f$ .

(d)

Verify the inequalities

\frac{1}{r}\left[\int f^{r}-1\right] \geqslant \frac{1}{r} \log \int f^{r} \geqslant \frac{1}{r} \int \log f^{r}=\int \log f

(e)

Conclude that $\lim_{r \rightarrow 0^{+}}\left\| f\right\|_{r}$ exists and equals $\exp \left(\int \log f\right)$ . If $\log f \notin L^{1}$ , this is interpreted as $\lim_{r \rightarrow 0^{+}}\left\| f\right\|_{r}=0$ .

Problem 8.3.5

Find conditions for equality in the Hölder-like Inequality.

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Remark.

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Hölder-like Inequality: Let $0<p<1$ . Let $f \in L^{p}$ be nonnegative, and let $g$ denote a nonnegative $\mathcal{F} / \mathscr {B}^{*}$ -measurable function such that $\int g^{p^{\prime }}$ exists and $0<\int g^{p^{\prime }}<+\infty$ . Then $\int f g \geqslant \left\| f\right\|_{p}\left(\int \left|g\right|^{p^{\prime }}\right)^{1 / p^{\prime }}$ .

Problem 8.3.6

Find conditions for equality in the Minkowski-like Inequality.

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Remark.

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Minkowski-like Inequality: Let $0<p<1$ , and let $f, g \in L^{p}$ be nonnegative. Then $f+g \in L^{p}$ and $\left\| f+g\right\|_{p} \geqslant \left\| f\right\|_{p}+\left\| g\right\|_{p}$ .

Problem 8.3.7

For every $a, b>0$ and $0<p<1$ , we have $a b \geqslant \frac{a^{p}}{p}+\frac{b^{p^{\prime }}}{p^{\prime }}$ . Use this to give a calculus-based proof of the Hölder-like Inequality that is similar to the proof of the original Hölder Inequality.

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Remark.

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Let $f(a, b)=a b-\frac{a^{p}}{p}$ for $a, b>0$ , where $0<p<1$ is fixed. For each $b \geqslant 0$ , we have that $\frac{\partial }{\partial a} f(a, b)=b-a^{p-1} \geqslant 0$ iff $a \geqslant b^{\frac{1}{p-1}}$ [here we use the fact that $p-1<0$ ]. Therefore, we have that $f(a, b) \geqslant f\left(b^{\frac{1}{p-1}}, b\right)$ for every $b>0$ , which gives $a b-\frac{a^{p}}{p} \geqslant \frac{p-1}{p} b^{\frac{p}{p-1}}=\frac{1}{p^{\prime }} b^{p^{\prime }}$ for every $b>0$ .

Problem 8.3.8

Does the Riesz-Fischer Theorem necessarily hold in the case $0<p<1$ ?

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Remark.

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Hint: note the relevance of the original Minkowski inequality in the proof of the Riesz-Fischer Theorem. When $0<p<1$ , the direction of the inequality is reversed.