Math Problems Vault

Problem 4A.1

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $h: X \rightarrow \mathbb {R}$ is an $\mathcal{S}$ -measurable function. Prove that

\mu (\left\{ x \in X:\left|h(x)\right| \geq c\right\} ) \leq \frac{1}{c^{p}} \int \left|h\right|^{p} d \mu

for all positive numbers $c$ and $p$ .

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Problem 4A.2

Suppose $(X, \mathcal{S}, \mu )$ is a measure space with $\mu (X) = 1$ and $h \in \mathcal{L}^{1}(\mu )$ . Prove that

\mu \left(\left\{ x \in X:\left|h(x)-\int h d \mu \right| \geq c\right\} \right) \leq \frac{1}{c^{2}}\left(\int h^{2} d \mu -\left(\int h d \mu \right)^{2}\right)

for all $c > 0$ .

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Remark.

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The result above is called Chebyshev's inequality; it plays an important role in probability theory. Pafnuty Chebyshev (1821--1894) was Markov's thesis advisor.

Problem 4A.3

Suppose $(X, \mathcal{S}, \mu )$ is a measure space. Suppose $h \in \mathcal{L}^{1}(\mu )$ and $\left\| h\right\|_{1} > 0$ . Prove that there is at most one number $c \in (0, \infty )$ such that

\mu (\left\{ x \in X:\left|h(x)\right| \geq c\right\} ) = \frac{1}{c}\left\| h\right\| _{1}.

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Problem 4A.4

Show that the constant 3 in the Vitali Covering Lemma (4.4) cannot be replaced by a smaller positive constant.

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Remark.

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4.4 (Vitali Covering Lemma): Suppose $I_{1}, \ldots , I_{n}$ is a list of bounded nonempty open intervals of $\mathbb {R}$ . Then there exists a disjoint sublist $I_{k_{1}}, \ldots , I_{k_{m}}$ such that $I_{1} \cup \cdots \cup I_{n} \subseteq (3 * I_{k_{1}}) \cup \cdots \cup (3 * I_{k_{m}})$ , where $3 * I$ denotes the open interval with the same center as $I$ and three times the length.

Problem 4A.5

Prove the assertion left as an exercise in the last sentence of the proof of the Vitali Covering Lemma (4.4).

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Remark.

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The proof of the Vitali Covering Lemma (4.4) constructs, from a list $I_{1}, \ldots , I_{n}$ of bounded nonempty open intervals of $\mathbb {R}$ , a disjoint sublist $I_{k_{1}}, \ldots , I_{k_{m}}$ by repeatedly choosing the largest remaining interval disjoint from those already chosen. For $j \in \left\{ 1, \ldots , n\right\}$ not among $k_{1}, \ldots , k_{m}$ , let $I_{k_{L}}$ be the first interval on this sublist not disjoint from $I_{j}$ ; thus $I_{j}$ is disjoint from $I_{k_{1}}, \ldots , I_{k_{L-1}}$ . Because $j$ was not chosen in step $L$ , $\left|I_{k_{L}}\right| \geq \left|I_{j}\right|$ . The assertion to be proved is that because $I_{k_{L}} \cap I_{j} \neq \varnothing$ , this last inequality implies $I_{j} \subseteq 3 * I_{k_{L}}$ (where $3 * I$ denotes the open interval with the same center as $I$ and three times the length).

Problem 4A.6

Verify the formula in Example 4.7 for the Hardy-Littlewood maximal function of $\chi_{[0,1]}$ .

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Remark.

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Example 4.7: for the characteristic function $\chi_{[0,1]}$ of the interval $[0,1]$ ,

\left(\chi _{[0,1]}\right)^{*}(b) = \begin{cases} \frac{1}{2(1-b)} & \text{if } b \leq 0, \\ 1 & \text{if } 0 < b < 1, \\ \frac{1}{2b} & \text{if } b \geq 1.\end{cases}

Problem 4A.7

Find a formula for the Hardy-Littlewood maximal function of the characteristic function of $[0,1] \cup [2,3]$ .

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Problem 4A.8

Find a formula for the Hardy-Littlewood maximal function of the function $h: \mathbb {R} \rightarrow [0, \infty )$ defined by

h(x) = \begin{cases} x & \text{ if } 0 \leq x \leq 1, \\ 0 & \text{ otherwise }.\end{cases}

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Problem 4A.9

Suppose $h: \mathbb {R} \rightarrow \mathbb {R}$ is Lebesgue measurable. Prove that

\left\{ b \in \mathbb {R}: h^{*}(b) > c\right\}

is an open subset of $\mathbb {R}$ for every $c \in \mathbb {R}$ .

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Problem 4A.10

Prove or give a counterexample: If $h: \mathbb {R} \rightarrow [0, \infty )$ is an increasing function, then $h^{*}$ is an increasing function.

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Problem 4A.11

Give an example of a Borel measurable function $h: \mathbb {R} \rightarrow [0, \infty )$ such that $h^{*}(b) < \infty$ for all $b \in \mathbb {R}$ but $\sup \left\{ h^{*}(b): b \in \mathbb {R}\right\} = \infty$ .

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Problem 4A.12

Show that $\left|\left\{ b \in \mathbb {R}: h^{*}(b) = \infty \right\} \right| = 0$ for every $h \in \mathcal{L}^{1}(\mathbb {R})$ .

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Problem 4A.13

Show that there exists $h \in \mathcal{L}^{1}(\mathbb {R})$ such that $h^{*}(b) = \infty$ for every $b \in \mathbf{Q}$ .

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Problem 4A.14

Suppose $h \in \mathcal{L}^{1}(\mathbb {R})$ . Prove that

\left|\left\{ b \in \mathbb {R}: h^{*}(b) \geq c\right\} \right| \leq \frac{3}{c}\left\| h\right\| _{1}

for every $c > 0$ .

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Remark.

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This result slightly strengthens the Hardy-Littlewood maximal inequality (4.8) because the set on the left side above includes those $b \in \mathbb {R}$ such that $h^{*}(b) = c$ . A much deeper strengthening comes from replacing the constant 3 in the Hardy-Littlewood maximal inequality with a smaller constant. In 2003, Antonios Melas answered what had been an open question about the best constant. He proved that the smallest constant that can replace 3 in the Hardy-Littlewood maximal inequality is $(11+\sqrt{61}) / 12 \approx 1.56752$ ; see Annals of Mathematics 157 (2003), 647--688.