Math Problems Vault

Problem 8.1.1

Pick $\mathfrak {A}, \mathfrak {B} \in \mathfrak {L}^{p}$ . Then $\mathfrak {A}+\mathfrak {B} \in \mathfrak {L}^{p}$ and $\left\| \mathfrak {A}+\mathfrak {B}\right\|_{p} \leqslant \left\| \mathfrak {A}\right\|_{p}+\left\| \mathfrak {B}\right\|_{p}$ .

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Problem 8.1.2

Prove the Cauchy-Schwarz inequality

\left|\sum _{k=1}^{n} a_{k} b_{k}\right| \leqslant \sqrt{\sum _{k=1}^{n} a_{k}^{2}} \sqrt{\sum _{k=1}^{n} b_{k}^{2}} .

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Problem 8.1.3

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

Let $1<p_{1}, \ldots , p_{n}<+\infty$ be such that $1 / p_{1}+\cdots +1 / p_{n}=$ 1 , and pick functions $f_{1} \in L^{p_{1}}, \ldots , f_{n} \in L^{p_{n}}$ . We wish to generalize Hölder's Inequality by showing that $\prod_{i=1}^{n} f_{i} \in L^{1}$ and $\left|\int \prod_{i=1}^{n} f_{i}\right| \leqslant \prod_{i=1}^{n}\left\| f_{i}\right\|_{p_{i}}$ .

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(a)

Show first that $a_{1} \cdots a_{n} \leqslant a_{1}^{p_{1}} / p_{1}+\cdots +a_{n}^{p_{n}} / p_{n}$ by generalizing the calculus result given in the section. [Here $0 \leqslant a_{1}, \ldots , a_{n}<+\infty$ .]

(b)

If $\left\| f_{1}\right\|_{p_{1}}=0$ or $\ldots$ or $\left\| f_{n}\right\|_{p_{n}}=0$ , the claim is trivial.

(c)

Use (a) to prove the claim when $\left\| f_{1}\right\|_{p_{1}}=\cdots =\left\| f_{n}\right\|_{p_{n}}=1$ .

(d)

Prove the claim when $\left\| f_{1}\right\|_{p_{1}}, \ldots ,\left\| f_{n}\right\|_{p_{n}}$ are positive.

Problem 8.1.4

We have equality in Hölder's Inequality iff there are nonnegative numbers $A$ and $B$ , not both zero, with $A\left|f\right|^{p}=B\left|g\right|^{p^{\prime }} \mu$ -a.e. on Ω.

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Problem 8.1.5

Given $f \in L^{p}\left[1<p<+\infty \right]$ , there is $g \in L^{p^{\prime }}$ with $\left\| g\right\|_{p^{\prime }}=1$ and $\int f g=\left\| f\right\|_{p}$ .

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Problem 8.1.6

(a)(b)

We now explore conditions for equality in Minkowski's Inequality. Let $f, g \in L^{p}$ .

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(a)

When $p=1,\left\| f+g\right\|_{p}=\left\| f\right\|_{p}+\left\| g\right\|_{p}$ iff there exists positive $\mathcal{F} / \mathscr {B}^{*}$ measurable $h>0$ defined on $\Omega$ with $f h=g \mu$ -a. e. on $\left\{ \omega \in \Omega : f(\omega ) g(\omega ) \neq 0\right\}$ .

(b)

For $1<p<+\infty$ , equality obtains iff there are nonnegative real numbers $A$ and $B$ , not both zero, such that $A f=B g \mu$ -a. e. on $\Omega$ .

Problem 8.1.7

Let $1 \leqslant p, q, r<+\infty$ be such that $1 / r=1 / p+1 / q$ . Let $f \in L^{p}$ and $g \in L^{q}$ . Then $f g \in L^{r}$ and $\left\| f g\right\|_{r} \leqslant \left\| f\right\|_{p}\left\| g\right\|_{q}$ .

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Problem 8.1.8

If $1 \leqslant p<+\infty$ , Minkowski's Inequality gives $\left|\left\| f\right\|_{p}-\left\| g\right\|_{p}\right| \leqslant \left\| f-g\right\|_{p}$ for every $f, g \in L^{p}$ .

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Problem 8.1.9

Let $\phi$ be a continuous and strictly increasing function defined on $\left[0,+\infty \right)$ with $\phi (0)=0$ and $\lim_{x \rightarrow +\infty } \phi (x)=+\infty$ . Then for each $0 \leqslant a, b<+\infty$ we have

a b \leqslant \int _{0}^{a} \phi (u) \, d u+\int _{0}^{b} \phi ^{-1}(v) \, d v,

with equality iff $\phi (a)=b$ .

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Remark.

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Suggestion: first show that $\int_{0}^{c} \phi (u) \, d u+\int_{0}^{\phi (c)} \phi^{-1}(v) \, d v=c \phi (c)$ for every $0 \leqslant c<+\infty$ . A picture might be helpful. The calculus result in the section may be obtained by setting $\phi (u)=u^{p-1}$ and $\phi^{-1}(v)=v^{\frac{1}{p-1}}$ .

Problem 8.1.10

Prove the following extension of part (c) from Claim 2 of Section 7.5, where $1 \leqslant p<+\infty$ and $\boldsymbol {y} \in \mathbb {R}^{k}$ : If $f \in L^{p}\left(\mathbb {R}^{k}, \mathcal{L}^{k}, \bar\lambda_{k}\right)$ , then $\tau_{\boldsymbol {y}} f \in L^{p}\left(\mathbb {R}^{k}, \mathcal{L}^{k}, \bar\lambda_{k}\right)$ and $\left\| \tau_{\boldsymbol {y}} f\right\|_{p}=\left\| f\right\|_{p}$ .

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Remark.

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Claim 2 of Section 7.5: Let $f: \mathbb {R}^{k} \rightarrow \overline{\mathbb {R}}$ be Lebesgue measurable, and let $\tau_{\boldsymbol {y}}$ denote translation by some $\boldsymbol {y} \in \mathbb {R}^{k}$ : $\tau_{\boldsymbol {y}} f(\boldsymbol {x})=f(\boldsymbol {x}+\boldsymbol {y})$ for every $\boldsymbol {x} \in \mathbb {R}^{k}$ . Then: (a) $\tau_{\boldsymbol {y}} f$ is Lebesgue measurable; (b) if $f$ is nonnegative, then $\int f \, d\bar\lambda_{k}=\int \tau_{\boldsymbol {y}} f \, d\bar\lambda_{k}$ ; (c) if $f \in L^{1}\left(\mathbb {R}^{k}, \mathcal{L}^{k}, \bar\lambda_{k}\right)$ , then $\tau_{\boldsymbol {y}} f \in L^{1}\left(\mathbb {R}^{k}, \mathcal{L}^{k}, \bar\lambda_{k}\right)$ , and again $\int f \, d\bar\lambda_{k}=\int \tau_{\boldsymbol {y}} f \, d\bar\lambda_{k}$ .