querent

Let $𝜇$ be a non-negative Borel measure on ${ℝ}^{𝑛}$ such that $𝜇(𝐴)<\infty$ for each bounded Borel subset $𝐴\subseteq {ℝ}^{𝑛}$.

(a) Setting , prove that for each $𝜌>0$, is an upper semi-continuous function on ${ℝ}^{𝑛}$. (A real-valued function $𝜃$ on ${ℝ}^{𝑛}$ is upper semi-continuous if for all $𝑥\in {ℝ}^{𝑛}$, $𝜃(𝑥)\ge {lim\; sup}_{𝑦\to 𝑥}𝜃(𝑦)$.)

(b) Give an example of a Borel measure $𝜇$ as above and $𝜌>0$ such that is not continuous.

Recall that $𝑆(ℝ)$ is the space of Schwartz functions on $ℝ$, and ${𝑆}^{\prime }(ℝ)$ the space of tempered distributions. Let ${𝜑}_{𝑛}\in 𝑆(ℝ)$, $𝑛\ge 1$. Suppose that $𝑢\in {𝑆}^{\prime }(ℝ)$, and the distributions corresponding to ${𝜑}_{𝑛}$, namely ${𝑢}_{𝑛}(𝜓)=\int {𝜑}_{𝑛}𝜓$ for $𝜓\in 𝑆(ℝ)$, converge to $𝑢$ in ${𝑆}^{\prime }(ℝ)$.

(a) Suppose that there exists $𝐶>0$ such that ${\Vert {𝜑}_{𝑛}\Vert }_{{𝐿}^2}<𝐶$ for all $𝑛$. Show that there exists $𝜑\in {𝐿}^2$ such that $𝑢(𝜓)=\int 𝜑𝜓$, $𝜓\in 𝑆(ℝ)$ and ${𝜑}_{𝑛}$ converge weakly to $𝜑$ in ${𝐿}^2$.

(b) Show that the analogous statement is not true if ${𝐿}^2$ is replaced by ${𝐿}^1$, namely show that there exist ${𝜑}_{𝑛}$ and $𝑢$ with ${\Vert {𝜑}_{𝑛}\Vert }_{{𝐿}^1}<𝐶$ such that $𝑢$ is not a distribution given by an ${𝐿}^1$ function $𝜑$.

Consider the Hilbert space $𝐻={𝐿}^2(ℝ,{𝑒}^{-{𝑥}^2}d𝑥)$, and let ${𝜑}_{𝑛}(𝑥)={𝑥}^{𝑛}$ for $𝑛\ge 0$ integer, so ${𝜑}_{𝑛}\in 𝐻$.

(a) Let ${𝑒}_{𝜉}(𝑥)={𝑒}^{𝑖𝜉𝑥}$ for $𝑥\in ℝ$. Prove that ${\sum }_{𝑛=0}^{𝑘}\frac{{(𝑖𝜉)}^{𝑛}}{𝑛!}{𝜑}_{𝑛}$ converges to ${𝑒}_{𝜉}\in 𝐻$ in the norm topology as $𝑘\to \infty$.

(b) Using (a) or otherwise show that if $𝜑\in 𝐻$ and for all $𝑛$ then $𝜑=0$.

Hint: Show that the Fourier transform of ${𝑒}^{-{𝑥}^2}𝜑$ vanishes!

(c) Show that there is an orthonormal basis ${\{{𝜓}_{𝑛}\}}_{𝑛=0}^{\infty }$ of $𝐻$ such that for all $𝑛$, $Span\{{𝜑}_0,{𝜑}_1,\dots ,{𝜑}_{𝑛}\}=Span\{{𝜓}_0,{𝜓}_1,\dots ,{𝜓}_{𝑛}\}$.

Suppose $𝑋\subseteq 𝑍$, $𝑌\subseteq 𝑉$, with $𝑋,𝑌,𝑍,𝑉$ Banach spaces, and with both inclusions continuous with respect to their respective norms. Suppose that $𝑃:𝑍\to 𝑉$ is continuous linear and has the property that $𝑢\in 𝑍$, $𝑃𝑢\in 𝑌$ implies $𝑢\in 𝑋$. Show that there exists $𝐶>0$ such that for all $𝑢\in 𝑍$ satisfying $𝑃𝑢\in 𝑌$, one has

Let $\top =ℝ/(2𝜋\mathbb{Z})$ be the unit circle, and for $𝑚>0$ let ${𝐻}^{𝑚(\top )}$ be the Sobolev space consisting of ${𝐿}^2(\top )$-functions $𝑓$ whose Fourier coefficients $\widehat{𝑓}(𝑛)$ satisfy ${\sum }_{𝑛\in \mathbb{Z}}{\left|\widehat{𝑓}(𝑛)\right|}^2{(1+{𝑛}^2)}^{𝑚}<\infty$. Suppose $𝐴:{𝐻}^{𝑚(\top )}\to {𝐿}^2(\top )$ is a continuous linear map, and there is $𝐵\in \mathrm{ℒ︀}({𝐿}^2(\top ),{𝐻}^{𝑚(\top )})$ such that $𝐴𝐵-𝐼=𝐸$ and $𝐵𝐴-𝐼=𝐹$ are compact on ${𝐿}^2(\top )$, resp. ${𝐻}^{𝑚(\top )}$ (and in fact ${𝐸}^{\ast }\in \mathrm{ℒ︀}({𝐿}^2,{𝐻}^{𝑚})$). Suppose also that when $𝑓,𝑔\in {𝐻}^{𝑚(\top )}$ (and for $𝑓,𝑔\in {𝐿}^2(\top )$). Show that $𝐴-𝜆𝐼:{𝐻}^{𝑚}\to {𝐿}^2$ is invertible when $𝜆\notin ℝ$.

Let ${𝕊}^1=ℝ/\mathbb{Z}$.

(a) Assume that for a given function $𝜑\in {𝐿}^1([0,1])$ there exists an irrational number $𝛼$ such that $𝜑(𝑥)=𝜑(𝑥+𝛼)$ for almost all $𝑥\in [0,1]$, where $+$ is addition modulo $\mathbb{Z}$. Show that $𝜑(𝑥)$ equals a constant for almost all $𝑥\in [0,1]$.

(b) Given an irrational number $𝛼$, consider the equation

for an unknown function $𝑔(𝑥)$, with a given function $𝑝\in {𝐶}^{\infty ({𝕊}^1)}$, such that ${\int }_{{𝕊}^1}𝑝(𝑥)d𝑥=0$. Give a condition on $𝛼$ that would guarantee that $𝑔\in {𝐶}^1({𝕊}^1)$ for any such function $𝑝$.

Let $𝑋$ be a Banach space, and let $𝑆$ denote the unit sphere $𝑆=\{𝑥\in 𝑋:\Vert 𝑥\Vert =1\}$.

(a) Suppose ${𝑦}_{𝑛}\in 𝑆$ for all $𝑛$, and ${𝑦}_{𝑛}\to 𝑦\in 𝑋$ weakly. Show that $\Vert 𝑦\Vert \le 1$.

(b) Suppose that $𝑋$ is a separable infinite dimensional Hilbert space, $𝑦\in 𝑋$ and $\Vert 𝑦\Vert \le 1$. Show that there exists a sequence ${\{{𝑦}_{𝑛}\}}_{𝑛=1}^{\infty }$ with ${𝑦}_{𝑛}\in 𝑆$ for all $𝑛$ such that ${𝑦}_{𝑛}\to 𝑦$ weakly.

Two quick problems.

(a) Show that a subspace of a Banach space is closed in the norm topology if and only if it is closed in the weak topology.

(b) Suppose that $𝐻$ is a Hilbert space and $𝐴\in \mathrm{ℒ︀}(𝐻)$ (bounded linear operator on $𝐻$) is self-adjoint. Show that $\Vert 𝐴\Vert =sup\{|𝜆|:𝜆\in 𝜎(𝐴)\}$ (with $𝜎(𝐴)$ denoting the spectrum).

Let $1<𝑝<\infty$. Suppose that ${𝑓}_{𝑛}\in {𝐿}^{𝑝([0,1])}$ and ${\Vert {𝑓}_{𝑛}\Vert }_{𝑝}\le 1$ for all $𝑛$. Assuming that ${𝑓}_{𝑛}(𝑥)\to 0$ a.e., prove that ${𝑓}_{𝑛}\to 0$ weakly in ${𝐿}^{𝑝}$.

(a) Is it possible to find uncountably many disjoint measurable subsets of $ℝ$ with strictly positive Lebesgue measure? Give an example or show that this is impossible.

(b) Suppose $𝑓\in {𝐿}^1(ℝ)$. Show that for any $𝜆>0$, ${lim}_{𝑛\to \infty }{𝑛}^{-𝜆}𝑓(𝑛𝑥)=0$ for almost all $𝑥\in ℝ$, where $𝑛\in {\mathbb{N}}^{+}$ is understood in the limit.