querent

Two short problems.

(a) Show that there is a closed subset $𝐸$ of $[0,1]$ with positive Lebesgue measure and with empty interior.

(b) Show that if $𝑓:[0,1]\to ℝ$ is absolutely continuous and $𝐴\subseteq [0,1]$ is Lebesgue measurable with measure 0 then $𝑓(𝐴)$ is measurable with measure 0.

Let ${\ell }^2(\mathbb{Z})$ denote the space of square summable bi-infinite sequences. Let $𝐿$ denote the operator of multiplication by $𝑛$, and let $𝐻=\{{\{{𝑎}_{𝑛}\}}_{𝑛=-\infty }^{\infty }\in {\ell }^2(\mathbb{Z}):{\{𝑛{𝑎}_{𝑛}\}}_{𝑛=-\infty }^{\infty }\in {\ell }^2(\mathbb{Z})\}\subseteq {\ell }^2(\mathbb{Z})$ with ${\Vert {\{{𝑎}_{𝑛}\}}_{𝑛=1}^{\infty }\Vert }_{𝐻}^2={\sum }_{𝑛\in \mathbb{Z}}(1+{𝑛}^2){|{𝑎}_{𝑛}|}^2$. Let $𝑅\in \mathrm{ℒ︀}({\ell }^2(\mathbb{Z}),{\ell }^2(\mathbb{Z}))$.

(a) Show that there is a discrete set $𝐷\subseteq ℂ$ such that $𝐿+𝑅-𝜆𝐼:𝐻\to {\ell }^2$ is invertible for $𝜆\notin 𝐷$.

(b) With $\top =ℝ/(2𝜋\mathbb{Z})$, show that if $𝑉\in 𝐶(\top )$ then the set of $𝜆\in ℂ$ for which there exists $𝑓\in {𝐶}^1(\top )$, not identically 0, for which ${𝑓}^{\prime }+𝑉𝑓=𝜆𝑓$, is discrete.

Suppose that $𝑋,𝑌$ are Banach spaces, and let ${\mathrm{𝒯︀}}_{𝑠}$ denote the norm topology on $𝑋$, ${\mathrm{𝒰︀}}_{𝑠}$ the norm topology on $𝑌$. Let ${\mathrm{𝒯︀}}_{𝑤}$ denote the weak topology on $𝑋$, and ${\mathrm{𝒰︀}}_{𝑤}$ denote the weak topology on $𝑌$.

(a) Show that $(𝑋,{\mathrm{𝒯︀}}_{𝑤})$ has the following property, sometimes called $\left(𝑇3\frac{1}{2}\right)$ or completely regular: if $𝑥\in 𝑋$ then $\{𝑥\}$ is ${\mathrm{𝒯︀}}_{𝑤}$-closed, and given any $𝑥\in 𝑋$ and $𝐶\subseteq 𝑋$ ${\mathrm{𝒯︀}}_{𝑤}$-closed with $𝑥\notin 𝐶$, there is a continuous function $𝑓:𝑋\to [0,1]$ with $𝑓(𝑥)=1$ and $𝑓$ identically 0 on $𝐶$.

(b) Show that a linear map $𝑇:𝑋\to 𝑌$ is continuous as a map from $(𝑋,{\mathrm{𝒯︀}}_{𝑠})$ to $(𝑌,{\mathrm{𝒰︀}}_{𝑠})$ if and only if it is continuous as a map from $(𝑋,{\mathrm{𝒯︀}}_{𝑤})$ to $(𝑌,{\mathrm{𝒰︀}}_{𝑤})$.

In this problem, let ${\Vert \cdot \Vert }_{𝑝}$ be the ${𝐿}^{𝑝({ℝ}^{𝑛})}$ norm, $1\le 𝑝\le \infty$, and let ${𝐶}_0^{\infty }(ℝ)$ be the set of compactly supported ${𝐶}^{\infty }$ functions.

(a) Show, including the explicit constant, that for $𝜑\in {𝐶}_0^{\infty (ℝ)}$, ${\Vert 𝜑\Vert }_{\infty }\le \frac{1}{2}{\int }_{-\infty }^{\infty }|{𝜑}^{\prime }(𝑡)|d𝑡$.

(b) Suppose that there is $𝐶>0$ such that for all functions $𝜑\in {𝐶}_0^{\infty ({ℝ}^{𝑛})}$ there is an inequality of the form ${\Vert 𝜑\Vert }_{𝑞}\le 𝐶{\Vert \nabla 𝜑\Vert }_{𝑝}$. Show that one would necessarily then have the relationship ${𝑞}^{-1}={𝑝}^{-1}-{𝑛}^{-1}$. (Hint: consider the functions ${𝜑}_{𝑡}$ defined by ${𝜑}_{𝑡}(𝑥)=𝜑(𝑡𝑥)$.)

(c) When $𝑛=2$, part (b) suggests that one might have an inequality of the form ${\Vert 𝜑\Vert }_{\infty }\le 𝐶{\Vert \nabla 𝜑\Vert }_2$. Show that there is no $𝐶>0$ such that this inequality holds for all $𝜑\in {𝐶}_0^{\infty ({ℝ}^{𝑛})}$.

Suppose that $𝑋,𝑌$ are Hilbert spaces. An operator $𝐴\in \mathrm{ℒ︀}(𝑋,𝑌)$ is Fredholm if $𝐴$ has closed range, and $Ker𝐴$ as well as $𝑌/Ran𝐴$ are finite dimensional.

(a) Show that $𝐴\in \mathrm{ℒ︀}(𝑋,𝑌)$ is Fredholm if and only if there are finite dimensional vector spaces $𝑉,𝑊$ and a finite rank operator $𝑃\in \mathrm{ℒ︀}(𝑊\oplus 𝑋,𝑉\oplus 𝑌)$ such that $\widetilde{𝐴}+𝑃$ is invertible, where $\widetilde{𝐴}\in \mathrm{ℒ︀}(𝑊\oplus 𝑋,𝑉\oplus 𝑌)$ is defined by $\widetilde{𝐴}(𝑤,𝑥)=(0,𝐴𝑥)$, $𝑥\in 𝑋$, $𝑤\in 𝑊$.

(b) Suppose $𝐴\in \mathrm{ℒ︀}(𝑋,𝑌)$ is Fredholm. Show that there exists $𝛿>0$ such that if $𝑅\in \mathrm{ℒ︀}(𝑋,𝑌)$ with ${\Vert 𝑅\Vert }_{\mathrm{ℒ︀}(𝑋,𝑌)}<𝛿$ then $𝐴+𝑅$ is Fredholm.

Let $\mathrm{\Omega }$ be a compact polygonal domain in ${ℝ}^2$, i.e. $\mathrm{\Omega }$ is an open set with compact closure such that each has a neighborhood ${𝑂}_{𝑥}$ such that $\mathrm{\Omega }\cap {𝑂}_{𝑥}$ is given by the intersection of one or two half-planes with ${𝑂}_{𝑥}$.

Show that there is $𝐶>0$ such that the Fourier transform of the characteristic function ${𝜒}_{\mathrm{\Omega }}$ of $\mathrm{\Omega }$ satisfies $|(\mathrm{ℱ︀}{𝜒}_{\mathrm{\Omega }})(𝜉)|\le {𝐶(1+|𝜉|)}^{-1}$, $𝜉\in {ℝ}^2$. (Hint: reduce to the case of the Fourier transform of a product of a cutoff function with one or two characteristic functions of half-planes.)

Are there non-trivial open cones $\mathrm{\Sigma }$ in ${ℝ}^2-\{0\}$ such that a faster decay estimate holds as $\mathrm{\Sigma }\ni 𝜉\to \infty$?

Suppose that $𝐾$ is a bounded measurable function on $\{(𝑥,𝑦):0\le 𝑦\le 𝑥\le 1\}\subseteq {[0,1]}^2$. Let $𝑇$ be the Volterra operator

$𝑓\in {𝐿}^{𝑝([0,1])}$, $1<𝑝<\infty$. Show that for $𝑓\in {𝐿}^{𝑝([0,1])}$, $1<𝑝<\infty$, this expression indeed defines a function $𝑇𝑓\in {𝐿}^{𝑝([0,1])}$, and $𝑇:{𝐿}^{𝑝([0,1])}\to {𝐿}^{𝑝([0,1])}$ is bounded in this range of $𝑝$. Show moreover that the spectrum of $𝑇$ is $\{0\}$.

Two short problems.

(a) Show that ${𝐶}^{\infty ([0,1])}$ is dense in ${𝐿}^{𝑝([0,1])}$ for $1\le 𝑝<\infty$.

(b) Let $𝐻$ be a separable Hilbert space with an orthonormal basis ${\{{𝑥}_{𝑛}\}}_{𝑛=1}^{\infty }$. Let $\{{𝑦}_{𝑛}\}$ be a sequence in $𝐻$ and prove that the following two statements are equivalent.

1. ${lim}_{𝑛\to \infty }(𝑥,{𝑦}_{𝑛})=0$ for all $𝑥\in 𝐻$.

2. ${\{\Vert {𝑦}_{𝑛}\Vert \}}_{𝑛=1}^{\infty }$ is bounded in $ℝ$, and ${lim}_{𝑛\to \infty }({𝑥}_{𝑚},{𝑦}_{𝑛})=0$ for each $𝑚\ge 1$.

Let $0<𝛼<1$. Show that there exists a continuous real-valued function $𝑓$ on $[0,1]$ such that for all $𝑥\in [0,1]$ and for all $𝛿>0$ and $𝐶>0$ there is $𝑦\in [0,1]\cap (𝑥-𝛿,𝑥+𝛿)$ such that the estimate $|𝑓(𝑦)-𝑓(𝑥)|\le 𝐶{|𝑦-𝑥|}^{𝛼}$ does not hold.

Suppose $𝑋,𝑌$ are Hilbert spaces.

(a) Show that if $𝐴\in \mathrm{ℒ︀}(𝑋,𝑌)$ has closed range and finite dimensional kernel then there is a constant $𝐶$ and a finite rank orthogonal projection $𝐹\in \mathrm{ℒ︀}(𝑋)$ such that ${\Vert 𝑥\Vert }_{𝑋}\le 𝐶({\Vert 𝐴𝑥\Vert }_{𝑌}+{\Vert 𝐹𝑥\Vert }_{𝑋})$.

(b) Show that if in addition $𝑌/Ran𝐴$ is finite dimensional, then there is a constant $\widetilde{𝐶}$ and a finite rank orthogonal projection $\widetilde{𝐹}\in \mathrm{ℒ︀}(𝑌)$ such that ${\Vert 𝑦\Vert }_{𝑌}\le \widetilde{𝐶}\left({\Vert {𝐴}^{\ast }𝑦\Vert }_{𝑋}+{\Vert \widetilde{𝐹}𝑦\Vert }_{𝑌}\right)$.