querent

Let $\top =ℝ/(2𝜋\mathbb{Z})$ be the unit circle, and consider the integral $𝐼(𝑟)={\int }_{\top }{𝑒}^{𝑖𝑟cos𝜃}𝜑(𝜃)d𝜃$, $𝜑\in {𝐶}^{\infty (\top )}$, where $d𝜃$ is the Lebesgue measure on $\top$. Show that there exists $𝐶>0$ such that $|𝐼(𝑟)|\le 𝐶{𝑟}^{-1/2}$, $𝑟\ge 1$.

Hint: Show that if $𝜑$ is supported away from $[0],[𝜋]\in ℝ/(2𝜋\mathbb{Z})$, then $𝐼(𝑟)$ is rapidly decreasing as $𝑟\to \infty$; then assume $𝜑$ is supported near $[0]$ or $[𝜋]$, and change variables to obtain an integral of the form $\int {𝑒}^{\pm 𝑖𝑟{𝑠}^2}\widetilde{𝜑}(𝑠)d𝑠$ (times a prefactor). (Note: $𝐼(𝑟)$ is essentially the Fourier transform, evaluated at $(𝑟,0)$, of a delta distribution on the unit circle in ${ℝ}^2$.)

Two short problems.

(a) Suppose $(𝑋,\mathrm{ℬ︀},𝜇)$ is a measure space. If $𝜇(𝑋)<\infty$, are there any inclusions among the spaces ${𝐿}^1(𝑋,𝜇)$, ${𝐿}^2(𝑋,𝜇)$, ${𝐿}^{\infty (𝑋,𝜇)}$? (List any inclusions you can, and provide a proof for these.) If $𝜇(𝑋)=\infty$, but $𝜇$ is $𝜎$-finite, can the reverse of these inclusions hold? (Give an example or provide a proof to the contrary.)

(b) Suppose ${𝐸}_{𝑘}$, $𝑘=1,2,\dots$, are measurable subsets of ${ℝ}^{𝑛}$ with ${\sum }_{𝑘}𝜇({𝐸}_{𝑘})<\infty$, where $𝜇$ is the Lebesgue measure. Show that almost all $𝑥$ lie in ${𝐸}_{𝑘}$ for finitely many $𝑘$, i.e. that

(Make sure that you show the measurability of any set whose measure you use.)

(a) Suppose that $𝑌$ is a normed complex vector space with norm $\Vert \cdot \Vert$, and $𝜆:𝑌\to ℂ$ is linear but is not continuous. Show that $𝑁={𝜆}^{-1}(\{0\})$ is dense in $𝑌$.

(b) Show that ${\ell }^2$ has an orthonormal basis $\{{𝑥}_{𝑛}\}$, where each ${𝑥}_{𝑛}$ actually lies in ${\ell }^1$, and has the property that if we list the components of each ${𝑥}_{𝑛}$ as ${𝑥}_{𝑛}^{(𝑘)}$, then ${\sum }_{𝑘=1}^{\infty }{𝑥}_{𝑛}^{(𝑘)}=0$.

Prove that a weakly convergent sequence ${𝑥}_{𝑛}\in {\ell }^1$ also converges strongly.

Suppose that $1<𝑝<\infty$, $𝑓,{𝑓}_{𝑛}\in {𝐿}^{𝑝([0,1])}$, $𝑛\in \mathbb{N}$, ${\Vert {𝑓}_{𝑛}\Vert }_{{𝐿}^{𝑝}}\le 1$ for all $𝑛$, and ${𝑓}_{𝑛}\to 𝑓$ almost everywhere. Show that ${𝑓}_{𝑛}\to 𝑓$ weakly and ${\Vert 𝑓\Vert }_{{𝐿}^{𝑝}}\le 1$.

Recall that for $𝑠\ge 0$ the Sobolev space ${𝐻}^{𝑠({ℝ}^{𝑛})}$ consists of $𝑢\in {𝐿}^2({ℝ}^{𝑛})$ with

where $\mathrm{ℱ︀}$ is the Fourier transform. Let $𝑅:𝑆({ℝ}^{𝑛})\to 𝑆({ℝ}^{𝑛-1})$ denote the restriction map $(𝑅𝑢)({𝑥}^{\prime })=𝑢({𝑥}^{\prime },0)$, ${𝑥}^{\prime }\in {ℝ}^{𝑛-1}$. Show that for $𝑠>1/2$, $𝑅$ has a unique continuous extension to a map $𝑅:{𝐻}^{𝑠({ℝ}^{𝑛})}\to {𝐻}^{𝑠-1/2}({ℝ}^{𝑛-1})$.

(a) Give an example of Hilbert spaces $𝑋,𝑌$ and an operator $𝐴\in \mathrm{ℒ︀}(𝑋,𝑌)$ such that $Ran𝐴$ is not closed.

(b) Show that if $𝑋,𝑌$ are Hilbert spaces, $𝐴\in \mathrm{ℒ︀}(𝑋,𝑌)$, and $Ran𝐴$ is closed then $Ran{𝐴}^{\ast }$ is closed (where ${𝐴}^{\ast }\in \mathrm{ℒ︀}(𝑌,𝑋)$ is the Hilbert space adjoint).

Hint: Reduce to the case when $𝐴$ is invertible.

(c) Show that if there exists $𝐶>0$ such that for all $𝑦\in 𝑌$, ${\Vert 𝑦\Vert }_{𝑌}\le 𝐶{\Vert {𝐴}^{\ast }𝑦\Vert }_{𝑋}$ then $𝐴$ is surjective. Is the conclusion still true if only ${𝐴}^{\ast }$ injective is assumed? (Give a proof or provide a counterexample.)

Let ${\top }^{𝑛}={(ℝ/(2𝜋\mathbb{Z}))}^{𝑛}$ be the $𝑛$-torus, and let ${\mathrm{𝒟︀}}^{\prime }({\top }^{𝑛})$ denote the set of distributions on the torus.

(a) For $𝜑\in {𝐶}^{\infty ({\top }^{𝑛+𝑚})}$ let ${𝜑}_{𝑥}(𝑦)=𝜑(𝑦,𝑥)$, $(𝑦,𝑥)\in {\top }^{𝑛}\times {\top }^{𝑚}$, so ${𝜑}_{𝑥}\in {𝐶}^{\infty ({\top }^{𝑛})}$. Show that if $𝑢\in {\mathrm{𝒟︀}}^{\prime }({\top }^{𝑛})$ and $𝜑\in {𝐶}^{\infty ({\top }^{𝑛+𝑚})}$ then the function $𝑓:{\top }^{𝑚}\to ℂ$ defined by $𝑓(𝑥)=𝑢({𝜑}_{𝑥})$ is ${𝐶}^{\infty }$.

(b) For $𝜑\in 𝐶({\top }^{𝑚};{𝐶}^{\infty ({\top }^{𝑛})})$ (i.e. $𝑥\mapsto {𝜑}_{𝑥}$ is continuous as a map ${\top }^{𝑚}\to {𝐶}^{\infty ({\top }^{𝑛})}$) show that ${\int }_{{\top }^{𝑚}}𝑢({𝜑}_{𝑥})d𝑥=𝑢({\int }_{{\top }^{𝑚}}{𝜑}_{𝑥}d𝑥)$.

Let $𝑆({ℝ}^{𝑛})$, resp. ${𝑆}^{\prime }({ℝ}^{𝑛})$, denote the set of Schwartz functions, resp. tempered distributions, on ${ℝ}^{𝑛}$.

(a) Suppose that $𝑓\in {𝑆}^{\prime }(ℝ)$, ${𝜓}_0\in 𝑆(ℝ)$ with ${\int }_{ℝ}{𝜓}_0(𝑥)d𝑥\ne 0$, and $𝑎\in ℝ$. Show that there is a unique $𝑢\in {𝑆}^{\prime }(ℝ)$ such that ${𝑢}^{\prime }=𝑓$ and $𝑢({𝜓}_0)=𝑎$.

(b) For $𝜀>0$, $𝑘\in {\mathbb{N}}^{+}$, define ${𝑢}_{\pm ,𝜀}:𝑆(ℝ)\to ℂ$ by ${𝑢}_{\pm ,𝜀}(𝜑)={\int }_{ℝ}{(𝑥\pm 𝑖𝜀)}^{-𝑘}𝜑(𝑥)d𝑥$, $𝜑\in 𝑆(ℝ)$. Show that for all $𝜀>0$, ${𝑢}_{\pm ,𝜀}\in {𝑆}^{\prime }(ℝ)$, and that there exist ${𝑢}_{\pm }\in {𝑆}^{\prime }(ℝ)$ such that for all $𝜑\in 𝑆(ℝ)$, ${𝑢}_{\pm ,𝜀}(𝜑)\to {𝑢}_{\pm }(𝜑)$ as $𝜀\to 0$, and compute ${𝑢}_{+}-{𝑢}_{-}$.

Suppose that $𝑤$ is a measurable function on ${ℝ}^{𝑛}$ which is finite and strictly positive almost everywhere. Suppose that $𝐾$ is a measurable function on ${ℝ}^{2𝑛}$ such that

for almost every $𝑥$, and for almost every $𝑦$, respectively. Prove that the integral operator $𝑇𝑓(𝑥)={\int }_{{ℝ}^{𝑛}}𝐾(𝑥,𝑦)𝑓(𝑦)d𝑦$ is bounded on ${𝐿}^2({ℝ}^{𝑛})$ with $\Vert 𝑇\Vert \le 𝐴$.