Math Problems Vault

Problem 8.5.1

(a)(b)

Consider the measure space $(\mathbb {R}, \mathfrak {B}, \lambda )$ . Let $1 \leqslant p<q< \infty$ , and let $r$ be such that $1 / q<r<1 / p$ .

?

(a)

Define $f$ on $\mathbb {R}$ by writing $f(x)=x^{-r} \; \mathbb {1}_{(0,1)}(x)$ for each $x \in \mathbb {R}$ . Then $f \in L^{p}$ but $f \notin L^{q}$ . Therefore, we do not in general have $L^{p} \subseteq L^{q}$ when $p<q$ .

(b)

Let $g(x)=x^{-r} \; \mathbb {1}_{(1, \infty )}(x)$ for every $x \in \mathbb {R}$ . Then $g \in L^{q}$ but $g \notin L^{p}$ . Therefore, we do not in general have $L^{q} \subseteq L^{p}$ when $p<q$ .

Problem 8.5.2

Let $0<p<r<\infty$ . Then $L^{p} \cap L^{\infty } \subseteq L^{r}$ , and for any $f \in L^{p} \cap L^{\infty }$ we have $\left\| f\right\|_{r} \leqslant \left\| f\right\|_{p}^{p / r}\left\| f\right\|_{\infty }^{1-p / r}$ .

?

Problem 8.5.3

For any $0<p<\infty$ and $0<M<\infty$ we have

\left(\int \left|f\right|^{p}\right)^{1 / p} \geqslant M \mu (\left\{ \omega \in \Omega :\left|f(\omega )\right| \geqslant M\right\} )^{1 / p} .

?

Problem 8.5.4

(a)(b)(c)

Let $0<r<\infty$ and assume that $f \in L^{r} \cap L^{\infty }$ , so that $f \in L^{p}$ for every $r<p<\infty$ by Exercise 8.5.2. We wish to show that $\lim_{p \rightarrow \infty }\left\| f\right\|_{p}=\left\| f\right\|_{\infty }$ . Follow this outline:

?

(a)

Ignoring the trivial case where $f=0 \mu$ -a.e. on $\Omega$ , let $f \neq 0$ on a set of positive measure, so that $\left\| f\right\|_{\infty }>0$ . Show that $\liminf_{p \rightarrow \infty }\left\| f\right\|_{p} \geqslant \left\| f\right\|_{\infty }$ .

(b)

Show that $\limsup_{p \rightarrow \infty }\left\| f\right\|_{p} \leqslant \left\| f\right\|_{\infty }$ .

(c)

Put (a) and (b) together to prove the claim.

Problem 8.5.5

Let $\mu (\Omega )=1$ and $1 \leqslant p \leqslant q \leqslant \infty$ . Show for arbitrary $f$ that

\int \left|f\right| \leqslant \left(\int \left|f\right|^{p}\right)^{1 / p}=\left(\int \left|f\right|^{q}\right)^{1 / q} \leqslant \operatorname {ess} \sup f

so that $\left\| f\right\|_{1} \leqslant \left\| f\right\|_{p} \leqslant \left\| f\right\|_{q} \leqslant \left\| f\right\|_{\infty }$ .

?

Problem 8.5.6

We know that $l^{p} \subseteq l^{q} \subseteq l^{\infty }$ when $0<p<q<+\infty$ by Claim 5. Show further that

\operatorname {ess} \sup f \leqslant \left(\sum _{\omega \in \Omega }\left|f(\omega )\right|^{q}\right)^{1 / q} \leqslant \left(\sum _{\omega \in \Omega }\left|f(\omega )\right|^{p}\right)^{1 / p}

for arbitrary $f \in l^{p}$ , so that $\left\| f\right\|_{\infty } \leqslant \left\| f\right\|_{q} \leqslant \left\| f\right\|_{p}$ for every $f \in l^{p}$ .

?

Remark.

?

Claim 5: For every $0<p<q<+\infty$ we have $l^{p} \subseteq l^{q} \subseteq l^{\infty }$ .

Problem 8.5.7

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

Consider $\left(\mathbb {R}, \mathfrak {B}, \lambda \right)$ .

?

(a)

Find $f$ such that $f \in L^{p}$ but $f \notin L^{q}$ for every $p<q<+\infty$ .

(b)

Find $f$ such that $f \in L^{q}$ for every $0<q<p$ , but $f \notin L^{p}$ .

(c)

Find a nonnegative finite-valued function $f$ with $f \notin L^{p}$ for every $0<p<+\infty$ .

(d)

Find $f$ such that $f \in L^{p}$ but $f \notin L^{q}$ for every $q \in (0, \infty )-\left\{ p\right\}$ .

Problem 8.5.8

Let $(\Omega , \mathcal{F}, \mu )$ be such that $\mu (A)>0$ forces $\mu (A) \geqslant 1$ for each $A \in \mathcal{F}$ . Let $1 \leqslant p \leqslant q<+\infty$ . Then $L^{p} \subseteq L^{q} \subseteq L^{\infty }$ and $\left\| f\right\|_{\infty } \leqslant \left\| f\right\|_{q} \leqslant \left\| f\right\|_{p}$ .

?