Problem Sets

Continuous on compact implies uniformly continuous

2010 6.1

2010 6.2

Show that $$\begin{align*} \int_{{\mathbf{R}}^n} {\left\lvert { f} \right\rvert} = \int_0^{\infty } m(A_t)\,dt&& A_t \coloneqq\left\{{x\in {\mathbf{R}}^n {~\mathrel{\Big\vert}~}{\left\lvert {f(x)} \right\rvert} > t}\right\} .\end{align*}$$

2010 6.5

Suppose $F \subseteq {\mathbf{R}}$ with $m(F^c) < \infty$ and let $\delta(x) \coloneqq d(x, F)$ and $$\begin{align*} I_F(x) \coloneqq\int_{\mathbf{R}}{ \delta(y) \over {\left\lvert {x-y} \right\rvert}^2 } \,dy .\end{align*}$$

• Show that $\delta$ is continuous.

• Show that if $x\in F^c$ then $I_F(x) = \infty$.

• Show that $I_F(x) < \infty$ for almost every $x$

2010 7.1

Let $(X, \mathcal{M}, \mu)$ be a measure space and prove the following properties of $L^ \infty (X, \mathcal{M}, \mu)$:

• If $f, g$ are measurable on $X$ then $$\begin{align*} {\left\lVert {fg} \right\rVert}_1 \leq {\left\lVert {f} \right\rVert}_1 {\left\lVert {g} \right\rVert}_{\infty } .\end{align*}$$

• ${\left\lVert {{-}} \right\rVert}_{\infty }$ is a norm on $L^{\infty }$ making it a Banach space.

• ${\left\lVert {f_n - f} \right\rVert}_{\infty } \overset{n\to \infty }\to 0 \iff$ there exists an $E\in \mathcal{M}$ such that $\mu(X\setminus E) = 0$ and $f_n \to f$ uniformly on $E$.

• Simple functions are dense in $L^{\infty }$.

2010 7.2

Show that for $0 < p < q \leq \infty$, ${\left\lVert {a} \right\rVert}_{\ell^q} \leq {\left\lVert {a} \right\rVert}_{\ell^p}$ over ${\mathbf{C}}$, where ${\left\lVert {a} \right\rVert}_{\infty } \coloneqq\sup_j {\left\lvert {a_j} \right\rvert}$.

2010 7.3

Let $f, g$ be non-negative measurable functions on $[0, \infty)$ with $$\begin{align*} A &\coloneqq\int_0^{\infty } f(y) y^{-1/2} \,dy< \infty \\ B &\coloneqq\qty{ \int_0^{\infty } {\left\lvert { g(y) } \right\rvert} }^2 \,dy< \infty .\end{align*}$$

Show that $$\begin{align*} \int_0^{\infty } \qty{ \int_0^{\infty } f(y) \,dy} {g(x) \over x} \,dx\leq AB .\end{align*}$$

2010 7.4

Let $(X, \mathcal{M}, \mu)$ be a measure space and $0 < p < q< \infty$. Prove that if $L^q(X) \subseteq L^p(X)$, then $X$ does not contain sets of arbitrarily large finite measure.

2010 7.5

Suppose $0 < a < b \leq \infty$, and find examples of functions $f \in L^p((0, \infty ))$ if and only if:

• $a < p < b$

• $a \leq p \leq b$

• $p = a$

Hint: consider functions of the following form: $$\begin{align*} f(x) \coloneqq x^{- \alpha} {\left\lvert { \log(x) } \right\rvert}^{ \beta} .\end{align*}$$

2010 7.6

Define $$\begin{align*} F(x) &\coloneqq\qty{ \sin(\pi x) \over \pi x}^2 \\ G(x) &\coloneqq \begin{cases} 1 - {\left\lvert {x} \right\rvert} & {\left\lvert {x} \right\rvert} \leq 1 \\ 0 & \text{else}. \end{cases} \end{align*}$$

• Show that $\widehat{G}(\xi) = F(\xi)$

• Compute $\widehat{F}$.

• Give an example of a function $g\not \in L^1({\mathbf{R}})$ which is the Fourier transform of an $L^1$ function.

Hint: write $\widehat{G}(\xi) = H(\xi) + H(-\xi)$ where $$\begin{align*} H(\xi) \coloneqq e^{2\pi i \xi} \int_0^1 y e^{2\pi i y \xi }\,dy .\end{align*}$$

2010 7.7

Show that for each $\epsilon>0$ the following function is the Fourier transform of an $L^1({\mathbf{R}}^n)$ function: $$\begin{align*} F(\xi) \coloneqq\qty{1 \over 1 + {\left\lvert {\xi} \right\rvert}^2}^{\epsilon} .\end{align*}$$

Hint: show that

$$\begin{align*} K_\delta(x) &\coloneqq\delta^{-n/2} e^{-\pi {\left\lvert {x} \right\rvert}^2 \over \delta} \\ f(x) &\coloneqq\int_0^{\infty } K_{\delta}(x) e^{-\pi \delta} \delta^{\epsilon - 1} \,d \delta \\ \Gamma(s) &\coloneqq\int_0^{\infty } e^{-t} t^{s-1} \,dt\\ \implies \widehat{f}(\xi) &= \int_0^{\infty } e^{- \pi \delta {\left\lvert {\xi} \right\rvert}^2} e^{ -\pi \delta} \delta^{\epsilon - 1} = \pi^{-s} \Gamma(\epsilon) F(\xi) .\end{align*}$$

2010 7 Challenge 1: Generalized Holder

Suppose that $$\begin{align*} 1\leq p_j \leq \infty, && \sum_{j=1}^n {1\over p_j} = {1\over r} \leq 1 .\end{align*}$$

Show that if $f_j \in L^{p_j}$ for each $1\leq j \leq n$, then $$\begin{align*} \prod f_j \in L^r, && {\left\lVert { \prod f_j } \right\rVert}_r \leq \prod {\left\lVert {f_j} \right\rVert}_{p_j} .\end{align*}$$

2010 7 Challenge 2: Young’s Inequality

Suppose $1\leq p,q,r \leq \infty$ with $$\begin{align*} {1\over p } + {1 \over q} = 1 + {1 \over r} .\end{align*}$$

Prove that $$\begin{align*} f \in L^p, g\in L^q \implies f \ast g \in L^r \text{ and } {\left\lVert {f \ast g} \right\rVert}_r \leq {\left\lVert {f} \right\rVert}_p {\left\lVert {g} \right\rVert}_q .\end{align*}$$

2010 9.1

Show that the set $\left\{{ u_k(j) \coloneqq\delta_{ij} }\right\} \subseteq \ell^2({\mathbf{Z}})$ and forms an orthonormal system.

2010 9.2

Consider $L^2([0, 1])$ and define $$\begin{align*} e_0(x) &= 1 \\ e_1(x) &= \sqrt{3}(2x-1) .\end{align*}$$

• Show that $\left\{{e_0, e_1}\right\}$ is an orthonormal system.

• Show that the polynomial $p(x)$ where $\deg(p) = 1$ which is closest to $f(x) = x^2$ in $L^2([0, 1])$ is given by $$\begin{align*} h(x) = x - {1\over 6} .\end{align*}$$

Compute ${\left\lVert {f - g} \right\rVert}_2$.

2010 9.3

Let $E \subseteq H$ a Hilbert space.

• Show that $E\perp \subseteq H$ is a closed subspace.

• Show that $(E^\perp)^\perp = { \operatorname{cl}}_H(E)$.

2010 9.5b

Let $f\in L^1((0, 2\pi))$.

• Show that for an $\epsilon>0$ one can write $f = g+h$ where $g\in L^2((0, 2\pi))$ and ${\left\lVert {H} \right\rVert}_1 < \epsilon$.

2010 9.6

Prove that every closed convex $K \subset H$ a Hilbert space has a unique element of minimal norm.

2010 9 Challenge

Let $U$ be a unitary operator on $H$ a Hilbert space, let $M \coloneqq\left\{{x\in H {~\mathrel{\Big\vert}~}Ux = x}\right\}$, let $P$ be the orthogonal projection onto $M$, and define $$\begin{align*} S_N \coloneqq{1\over N} \sum_{n=0}^{N-1} U^n .\end{align*}$$ Show that for all $x\in H$, $$\begin{align*} {\left\lVert { S_N x - Px} \right\rVert}_H \overset{N\to \infty } \to 0 .\end{align*}$$

2010 10.1

Let $\nu, \mu$ be signed measures, and show that $$\begin{align*} \nu \perp \mu \text{ and } \nu \ll {\left\lvert { \mu} \right\rvert} \implies \nu = 0 .\end{align*}$$

2010 10.2

Let $f\in L^1({\mathbf{R}}^n)$ with $f\neq 0$.

• Prove that there exists a $c>0$ such that $$\begin{align*} Hf(x) \geq {c \over (1 + {\left\lvert {x} \right\rvert})^n } .\end{align*}$$

2010 10.3

Consider the function $$\begin{align*} f(x) \coloneqq \begin{cases} {1\over {\left\lvert {x} \right\rvert} \qty{ \log\qty{1\over x}}^2 } & {\left\lvert {x} \right\rvert} \leq {1\over 2} \\ 0 & \text{else}. \end{cases} \end{align*}$$

• Show that $f \in L^1({\mathbf{R}})$.

• Show that there exists a $c>0$ such that for all ${\left\lvert {x} \right\rvert} \leq 1/2$, $$\begin{align*} Hf(x) \geq {c \over {\left\lvert {x} \right\rvert} \log\qty{1\over {\left\lvert {x} \right\rvert}} } .\end{align*}$$ Conclude that $Hf$ is not locally integrable.

2010 10.4

Let $f\in L^1({\mathbf{R}})$ and let $\mathcal{U}\coloneqq\left\{{(x, y) \in {\mathbf{R}}^2 {~\mathrel{\Big\vert}~}y > 0}\right\}$ denote the upper half plane. For $(x, y) \in \mathcal{U}$ define $$\begin{align*} u(x, y) \coloneqq f \ast P_y(x) && \text{where } P_y(x) \coloneqq{1\over \pi}\qty{y \over t^2 + y^2} .\end{align*}$$

• Prove that there exists a constant $C$ independent of $f$ such that for all $x\in {\mathbf{R}}$, $$\begin{align*} \sup_{y > 0} {\left\lvert { u(x, y) } \right\rvert} \leq C\cdot Hf(x) .\end{align*}$$

  Hint: write the following and try to estimate each term: $$\begin{align*} u(x, y) = \int_{{\left\lvert {t} \right\rvert} < y} f(x - t) P_y(t) \,dt+ \sum_{k=0}^{\infty } \int_{A_k} f(x-t) P_y(t)\,dt&& A_k \coloneqq\left\{{2^ky \leq {\left\lvert {t} \right\rvert} < 2^{k+1}y}\right\} .\end{align*}$$

• Following the proof of the Lebesgue differentiation theorem, show that for $f\in L^1({\mathbf{R}})$ and for almost every $x\in {\mathbf{R}}$, $$\begin{align*} u(x, y) \overset{y\to 0} \to f(x) .\end{align*}$$