querent

Let $𝐴=\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$. Then $𝐴$ acts on ${\top }^2={ℝ}^2/{\mathbb{Z}}^2$ by matrix multiplication, and thus on $𝐶({\top }^2)$ via pullback: $({\mathrm{\Phi }}_{𝐴}𝑓)(𝑥)=𝑓(𝐴𝑥)$, $𝑓\in 𝐶({\top }^2)$, $𝑥\in {\top }^2$.

(a) Show that the action on $𝐶({\top }^2)$ extends to a weak-* continuous action on ${\mathrm{𝒟︀}}^{\prime }({\top }^2)$, i.e. on distributions on the torus.

(b) Express the Fourier coefficients $\widehat{{\mathrm{\Phi }}_{𝐴}𝑢}(𝑘)$, $𝑘\in {\mathbb{Z}}^2$, of ${\mathrm{\Phi }}_{𝐴}𝑢\in {\mathrm{𝒟︀}}^{\prime }({\top }^2)$ in terms of the Fourier coefficients of $𝑢$. (Recall that the Fourier series of $𝑓$ on ${\top }^2$ is of the form ${\sum }_{𝑘\in {\mathbb{Z}}^2}\widehat{𝑓}(𝑘){𝑒}^{2𝜋𝑖𝑘\cdot 𝑥}$.)

(c) Show that if $𝑓\in {𝐿}^1({\top }^2)$ and ${\mathrm{\Phi }}_{𝐴}𝑓=𝑓$, then $𝑓$ is an a.e. constant function.

(d) Show that the space of invariant distributions, i.e. $𝑢\in {\mathrm{𝒟︀}}^{\prime }({\top }^2)$ such that ${\mathrm{\Phi }}_{𝐴}𝑢=𝑢$, is however infinite dimensional.

Prove that if $𝑓\in {𝐿}^1$ and its distributional derivative ${𝑓}^{\prime }$ is also in ${𝐿}^1$, then $𝑓$ is absolutely continuous.

(a) Prove that there is an uncountable number of disjoint balls of radius $1$ in ${\ell }^{\infty }$, and hence ${\ell }^{\infty }$ is not separable.

(b) Show that on the other hand ${\ell }^1$ is separable.

Let $(𝑋,𝜇)$ be a measure space such that $𝜇(𝑋)=1$. Assume that the functions ${𝑓}_{𝑛(𝑥)}$, $𝑥\in 𝑋$ are measurable and for each $𝑦\in ℝ$ and $𝑛\in \mathbb{N}$ we have

Show that for almost every $𝑥\in 𝑋$ (with respect to the measure $𝜇$) there exist only finitely many $𝑛\in \mathbb{N}$ such that $|{𝑓}_{𝑛(𝑥)}|>2019\sqrt{log𝑛}$.

Hint: use the Borel-Cantelli lemma.

Show that for each $𝑘\in ℝ-\{0\}$ the limit

exists, and find all $𝑚\in ℝ$ such that ${lim}_{𝑘\to 0}{|𝑘|}^{𝑚}𝐼(𝑘)=0$.

A form of the Hahn-Banach theorem states that if $𝑋$ is a normed vector space over either $𝔽=ℝ$ (real version) or $𝔽=ℂ$ (complex version), $𝑌$ a subspace, $\ell :𝑌\to 𝔽$ a $𝔽$-linear map with $|\ell (𝑦)|\le \Vert 𝑦\Vert$ for all $𝑦\in 𝑌$ then there is a $𝔽$-linear map $𝐿:𝑋\to 𝔽$ such that $𝐿{|}_{𝑌}=\ell$ and $|𝐿(𝑥)|\le \Vert 𝑥\Vert$ for all $𝑥\in 𝑋$.

Given the real version of the Hahn-Banach theorem, prove the complex version.

Suppose $𝑋,𝑌$ are reflexive Banach spaces, ${𝐴}_{𝑛}\in \mathrm{ℒ︀}(𝑋,𝑌)$ for $𝑛\in \mathbb{N}$. Suppose that for all $𝑥\in 𝑋$ and for all $\ell \in {𝑌}^{\ast }$, ${lim}_{𝑛\to \infty }\ell ({𝐴}_{𝑛}𝑥)$ exists. Show that there exists $𝐴\in \mathrm{ℒ︀}(𝑋,𝑌)$ such that ${𝐴}_{𝑛}\to 𝐴$ in the weak operator topology. Give an example of ${𝐴}_{𝑛}$ satisfying these assumptions such that ${𝐴}_{𝑛}$ does not converge to $𝐴$ in the strong operator topology.

Recall that $𝐴$ is a Hilbert-Schmidt operator (on a separable Hilbert space $𝐻$) if ${\sum }_{𝑛\in \mathbb{N}}{\Vert 𝐴{𝑒}_{𝑛}\Vert }^2<\infty$ for an orthonormal basis $\{{𝑒}_{𝑛}:𝑛\in \mathbb{N}\}$.

(a) Prove that every Hilbert-Schmidt operator is compact, but there exist compact operators which are not Hilbert-Schmidt.

(b) Let $𝐻={𝐿}^2((0,1))$ and suppose that $(𝐴𝑓)(𝑥)={\int }_0^{𝑥}𝑓(𝑦)d𝑦$. Prove that $𝐴$ is Hilbert-Schmidt.

Let ${𝐻}^1({ℝ}^2)$ be the subspace of ${𝐿}^2({ℝ}^2)$ consisting of $𝑓\in {𝐿}^2({ℝ}^2)$ with the distributional derivatives of $𝑓$ satisfying , $𝑗=1,2$. Define ${𝑋}_{𝑗}$ as the completion of $\mathrm{𝒮︀}({ℝ}^2)$ (Schwartz functions) with respect to the norm

where is the $𝑗$th partial, $𝑗=1,2$. Show that ${𝑋}_{𝑗}$ is (naturally identified with) a subspace of ${𝐿}^2({ℝ}^2)$ (i.e. the inclusion from $\mathrm{𝒮︀}({ℝ}^2)$ into ${𝐿}^2({ℝ}^2)$ extends to an injective continuous linear map from ${𝑋}_{𝑗}$ to ${𝐿}^2({ℝ}^2)$), and ${𝑋}_1\cap {𝑋}_2={𝐻}^1({ℝ}^2)$.

For $𝑠\ge 0$ define ${𝐻}^{𝑠(ℝ)}$ to be the subspace of ${𝐿}^2(ℝ)$ consisting of $𝑓\in {𝐿}^2(ℝ)$ with ${\int }_{ℝ}{(1+{|𝜉|}^2)}^{𝑠}{\left|\widehat{𝑓}(𝜉)\right|}^{2}d𝜉<\infty$, and let ${\Vert 𝑓\Vert }_{{𝐻}^{𝑠(ℝ)}}≔{\Vert {(1+{|𝜉|}^2)}^{𝑠/2}\widehat{𝑓}(𝜉)\Vert }_{{𝐿}^2(ℝ)}$, where $\widehat{𝑓}$ denotes the Fourier transform of $𝑓$.

(a) Prove that for every $𝑠>1/2$, there exists a constant $𝐶>0$ (depending only on $𝑠$) such that for every $𝑓\in {𝐻}^{𝑠(ℝ)}$,

(b) Show if $𝑢,𝑣\in {𝐻}^1(ℝ)$ then their product $𝑢𝑣\in {𝐻}^1(ℝ)$ and there is ${𝐶}^{\prime }>0$ such that

Hint: show and use that ${\Vert 𝑢\Vert }_{{𝐻}^1(ℝ)}$ is a constant multiple of ${({\Vert 𝑢\Vert }_{{𝐿}^2}^2+{\Vert {𝑢}^{\prime }\Vert }_{{𝐿}^2}^2)}^{1/2}$.