REAL ANALYSIS GENERAL EXAM AUGUST 2024

Solve as many problems as you can. Full solutions on a smaller number of problems will be worth more than partial solutions on several problems.

Problem 1

Let $A \subset \mathbb{R}$ be a Lebesgue measurable set of finite measure. For $r \in \mathbb{R} \backslash\{0\}$, let $r A:=\left\{x \in \mathbb{R}: r^{-1} x \in A\right\}$ and let $A \triangle r A:=(A \backslash r A) \cup(r A \backslash A)$. Show that

$$\lim _{r \rightarrow 1} m(A \triangle r A)=0$$

Problem 2

Let $\mu$ be a finite Borel measure on $\mathbb{R}$. For $\xi \in \mathbb{R}$, define

$$\widehat{\mu}(\xi):=\int_{\mathbb{R}} e^{-2 \pi i x \xi} d \mu(x)$$

Suppose that

$$\lim _{\xi \rightarrow 0} \frac{\widehat{\mu}(\xi)-\widehat{\mu}(0)}{\xi^{2}}=0$$

(a) Show that

$$\int_{\mathbb{R}} x^{2} d \mu(x)=0$$

(b) Deduce that for any open interval $(a, b) \subseteq \mathbb{R}$, we have that

$$\mu((a, b))= \begin{cases}\mu(\mathbb{R}) & \text { if } 0 \in(a, b), \\ 0 & \text { if } 0 \notin(a, b) .\end{cases}$$

(The measure $\mu$ is a scalar multiple of the Dirac delta mass, though you do not need to prove this fact.)

Problem 3

Fix $\alpha \in(0,1]$. The space $C^{0, \alpha}([0,1])$ of Hölder continuous functions consists of functions $f:[0,1] \rightarrow \mathbb{C}$ for which

$$\|f\|:=|f(0)|+\sup _{\substack{x, y \in[0,1] \\ x \neq y}} \frac{|f(x)-f(y)|}{|x-y|^{\alpha}}$$

is finite. This is a vector space (though you do not need to prove this fact).
(a) Show that $\|\cdot\|$ is a norm on $C^{0, \alpha}([0,1])$.
(b) Show that $\left(C^{0, \alpha}([0,1]),\|\cdot\|\right)$ is a Banach space.

Problem 4

Let ( $X, \mathcal{M}, \mu$ ) be a $\sigma$-finite measure space. Let $p \in[1, \infty)$ and $f \in L^{p}(X, \mathcal{M}, \mu)$. In what follows, you may freely use without proof the fact that the sets $\{x \in X:|f(x)|>\alpha\}$ and $\{(x, \alpha) \in X \times(0, \infty):|f(x)|>\alpha\}$ are measurable.
(a) Show that

$$\int_{X}|f(x)|^{p} d \mu(x)=\int_{0}^{\infty} \alpha^{p-1} \mu(\{x \in X:|f(x)|>\alpha\}) d \alpha$$

(b) Show that for all $\alpha>0$,

$$\mu(\{x \in X:|f(x)|>\alpha\}) \leq \frac{\|f\|_{p}^{p}}{\alpha^{p}} .$$

Problem 5

Let $\phi \in \mathscr{S}\left(\mathbb{R}^{n}\right)$ be a fixed Schwartz function. The Fourier multiplier $M$ acts on functions $f \in L^{2}\left(\mathbb{R}^{n}\right)$ according to the formula

$$\widehat{M f}(\xi):=\phi(\xi) \widehat{f}(\xi) .$$

That is, $M$ is the operation of taking the Fourier transform of $f$, multiplying by the function $\phi(\xi)$, and then taking the inverse Fourier transform. Show that $M$ is a bounded linear map from $L^{2}\left(\mathbb{R}^{n}\right)$ to $L^{2}\left(\mathbb{R}^{n}\right)$.