Chapter 4

Differentiation

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Section 4AHardy–Littlewood Maximal Function[14/0%]

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 $p$ .

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 $c > 0$ .

Remark.

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}.

Problem 4A.4

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

Remark.

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).

Remark.

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 $\chi_{[0,1]}$ .

Remark.

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 $[0,1] \cup [2,3]$ .

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}

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 $c \in \mathbb {R}$ .

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.

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 $\sup \left\{ h^{*}(b): b \in \mathbb {R}\right\} = \infty$ .

Problem 4A.12

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

Problem 4A.13

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

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 $c > 0$ .

Remark.

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.

Section 4BDerivatives of Integrals[10/0%]

Problem 4B.1

For $f \in \mathcal{L}^{1}(\mathbb {R})$ and $I$ an interval of $\mathbb {R}$ with $0 < \left|I\right| < \infty$ , let $f_{I}$ denote the average of $f$ on $I$ . In other words, $f_{I} = \frac{1}{\left|I\right|} \int_{I} f$ .

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

\lim _{t \downarrow 0} \frac{1}{2 t} \int _{b-t}^{b+t}\left|f-f_{[b-t, b+t]}\right| = 0

for almost every $b \in \mathbb {R}$ .

Problem 4B.2

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

\lim _{t \downarrow 0} \sup \left\{ \frac{1}{\left|I\right|} \int _{I}\left|f-f_{I}\right|: I \text{ is an interval of length } t \text{ containing } b\right\} = 0

for almost every $b \in \mathbb {R}$ .

Problem 4B.3

Suppose $f: \mathbb {R} \rightarrow \mathbb {R}$ is a Lebesgue measurable function such that $f^{2} \in \mathcal{L}^{1}(\mathbb {R})$ . Prove that

\lim _{t \downarrow 0} \frac{1}{2 t} \int _{b-t}^{b+t}\left|f-f(b)\right|^{2} = 0

for almost every $b \in \mathbb {R}$ .

Problem 4B.4

Prove that the Lebesgue Differentiation Theorem (4.19) still holds if the hypothesis that $\int_{-\infty }^{\infty }\left|f\right| < \infty$ is weakened to the requirement that $\int_{-\infty }^{x}\left|f\right| < \infty$ for all $x \in \mathbb {R}$ .

Remark.

4.19 (Lebesgue Differentiation Theorem, second version): Suppose $f \in \mathcal{L}^{1}(\mathbb {R})$ . Define $g: \mathbb {R} \rightarrow \mathbb {R}$ by $g(x) = \int_{-\infty }^{x} f$ . Then $g^{\prime }(b) = f(b)$ for almost every $b \in \mathbb {R}$ .

Problem 4B.5

Suppose $f: \mathbb {R} \rightarrow \mathbb {R}$ is a Lebesgue measurable function. Prove that

\left|f(b)\right| \leq f^{*}(b)

for almost every $b \in \mathbb {R}$ .

Problem 4B.6

Prove that if $h \in \mathcal{L}^{1}(\mathbb {R})$ and $\int_{-\infty }^{s} h = 0$ for all $s \in \mathbb {R}$ , then $h(s) = 0$ for almost every $s \in \mathbb {R}$ .

Problem 4B.7

Give an example of a Borel subset of $\mathbb {R}$ whose density at 0 is not defined.

Problem 4B.8

Give an example of a Borel subset of $\mathbb {R}$ whose density at 0 is $\frac{1}{3}$ .

Problem 4B.9

Prove that if $t \in [0,1]$ , then there exists a Borel set $E \subseteq \mathbb {R}$ such that the density of $E$ at 0 is $t$ .

Problem 4B.10

Suppose $E$ is a Lebesgue measurable subset of $\mathbb {R}$ such that the density of $E$ equals 1 at every element of $E$ and equals 0 at every element of $\mathbb {R} \backslash E$ . Prove that $E = \varnothing$ or $E = \mathbb {R}$ .