\documentclass{amsart}
\usepackage{latexsym,amsfonts,amsmath,amssymb}

%%%%%%%%%%%%%%%%%% fonts/sets %%%%%%%%%%%%%%%%%%%%%
\newcommand{\Reals}{{\mathbb{R}}}
\newcommand{\Disk}{{\mathbb{D}}}
\newcommand{\Ints}{{\mathbb{Z}}}
\newcommand{\Cmplx}{{\mathbb{C}}}

\newcount\probnum
\probnum=1
\newcommand{\Q}{\medskip \noindent \hbox to 0mm{\hss \number\probnum.\ {\global\advance\probnum by 1}}}


%%%%%%%%%%%%%%%%%%%%%% operators %%%%%%%%%%%%%%%%%%%%%%
\let\Re=\undefined\DeclareMathOperator{\Re}{Re}
\let\Im=\undefined\DeclareMathOperator{\Im}{Im}
\DeclareMathOperator{\sech}{sech}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\sign}{sign}


\let\llldots=\ldots
\def\ldots{\llldots{}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% end of  definitions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{document}
\begin{center}
247A Homework.
\end{center}

\medskip

\noindent
The two sources for notes are
\verb!http://www.math.ubc.ca/~ilaba/wolff/!\\
and \verb!http://www.its.caltech.edu/~schlag/notes_033002.pdf!

\medskip

\Q Let us define
$$
J_0(x) = \tfrac{1}{2\pi} \int_0^{2\pi}  \cos\bigl(x\sin(\theta)\bigr) \,d\theta
$$
Show that
$$
  f(x) \mapsto F(\xi) = 2\pi \int J_0(2\pi \xi x) f(x) x \,dx
$$
defines a unitary map from $L^2\bigl([0,\infty),r\,dr\bigr)$ to itself.
Describe the relation to the Fourier transform of radial functions in
two dimensions.

%%%%%%%%%%%%%%%%%%%%

\Q Let $d\rho$ be a probability measure on $\Reals$ with
$\int x \,d\mu(x)=0$ and $\int x^4 \,d\rho(x) < \infty$.
Prove the central limit theorem for the sum of independent
random variables with this distribution.

Specifically, if $X_1,X_2,\ldots$ are $d\rho$-distributed, show
that for any Schwartz function $f$,
$$
\mathbb{E} \bigl\{ f\bigl(\tfrac{X_1+\cdots+X_n}{n^{1/2}}\bigr) \bigr\}
\longrightarrow \tfrac{1}{\sqrt{2\pi\sigma^2}} \int \exp\{ -\tfrac{x^2}{2\sigma^2} \} f(x) \,dx
$$
as $n\to\infty$.  Hint: first show convergence for $f(x)=e^{- 2\pi i x\xi}$ uniformly for
$\xi$ in a compact set.

%%%%%%%%%%%%%%%%%%

\Q Show that every continuous (group) homomorphism from $\mathbb{T}$ into
$\Cmplx^*$ (the non-zero complex numbers under multiplication) takes the form $x\mapsto e^{2\pi i n x}$ with
$n$ an integer.  What is the analogous statement for continuous homomorphisms $\Reals\to\Cmplx^*$.

%%%%%%%%%%%%%%

\Q Let us define a sequence functions on $\Reals$ by
$$
\psi_n(x) = \Bigl[ \frac{d\ }{dx} - 2\pi x \Bigr]^n e^{-\pi x^2}
$$
where $n=0,1,\ldots$.  Show that $\psi_n(x)$ form an orthogonal sequence of eigenfunctions
for the Fourier transform on $L^2(\Reals)$.

In fact they are a basis, but this is much harder to prove.  One approach to this latter problem
is to realize that they are the eigenfunctions of the harmonic oscillator:
$$
u(x) \mapsto \Bigl[ \frac{d\ }{dx} - 2\pi x \Bigr]\Bigl[ - \frac{d\ }{dx} - 2\pi x \Bigr] u(x) =
-\frac{d^2u}{dx^2} + (4\pi^2 x^2 - 2\pi) u(x).
$$

%%%%%%%%%%%%%%%%

\Q Let $G$ be a finite cyclic group and $H$ a subgroup.  For $\chi\in\hat G$ we
write
$$
\hat f (\chi) = \sum_g f(g) \bar \chi(g).
$$
We say $\chi\in \hat{G}^H$ if $\chi$ is constant on the cosets of $H$.

Prove the following analogue of the classical Poisson Summation formula:
$$
\frac{1}{|G|} \sum_{\chi\in\hat{G}^H} \hat f(\chi) = \frac{1}{|H|} \sum_{h \in H} f(h).
$$
(The classical version has $G=\Reals$ and $H=\Ints$, which leads to $\hat{G}^H=\{e^{2\pi i n x} : n \in \Ints\}$.)

%%%%%%%%%%%%%%%%%%%%%%%

\Q Suppose $f \in L^2(\Reals)$ is supported on $[-\tfrac12,\tfrac12]$ then we know that
$f$ can be recovered from the values of $\hat f(n)$ for $n\in\Ints$ (the characters form
an orthonormal basis).  Prove the Shannon Sampling Theorem:
$$
\hat f(\xi) = \sum_n \hat f(n) \frac{\sin[\pi(n-\xi)]}{\pi(n-\xi)}
$$
(which includes proving convergence of this infinite sum).

\noindent
{\it Remark:} The audible spectrum extends only to about 20kHz.  Consequently, as heard
by a human, one may regard music as a function whose Fourier transform is supported on
a finite interval.  The above theorem says that to faithfully reproduce music, one
need only sample the signal forty thousand times per second.  This is what happens in
CD recording.

%%%%%%%%%%%%%%%%%%%%

\Q Given $\omega\in\Reals^d$, show that the following are equivalent:

\noindent (a) For $m\in\Ints^d$, $m\cdot \omega=0$ implies $m=0$.

\noindent (b) The curve $t\mapsto t\omega +\Ints^d$ is dense in $\Reals^d/\Ints^d$.

\noindent (c) For any continuous function $f$ on $\Reals^d/\Ints^d$,
$$
\lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T  f(t\omega +\Ints^d) \, dt
    = \int_0^1\!\cdots\!\int_0^1 f(x+\Ints^d) \,dx.
$$
[Hint: prove (a)$\Leftrightarrow$(c) and then (c)$\Rightarrow$(b)$\Rightarrow$(a).]

%%%%%%%%%%%%%%%%%%%%%%%%%

\Q Let $d\mu$ be a finite complex measure on $\Reals$.

\noindent(a) Show that
$$
 \lim_{L\to\infty} \frac{1}{2L} \int_{-L}^L \bigl| \hat\mu(\xi) \bigr|^2 d\xi
    = \sum_{x\in\Reals} \bigl| \mu(\{x\}) \bigr|^2
$$
(finiteness of the measure implies that only countably many terms in the sum are non-zero).

\noindent(b) Suppose that $d\mu$ is purely atomic, that is, $d\mu$ is a (countable) linear combination of
delta measures.  Show that $\hat \mu$ is almost periodic.

A function on $f$ on $\Reals$ is said to be almost periodic if for any $\epsilon>0$, there exists
$L>0$ so that any interval of length $L$ contains an $\epsilon$-almost period:
$$
\forall a\in\Reals\quad\exists p\in[a,a+L]\quad\text{such that}\quad\sup_x \bigl|f(x)-f(x+p)\bigr| < \epsilon.
$$

\noindent {\it Hint:} For part (b) begin by considering the case
$\hat\mu(\xi)=e^{i\xi} + e^{2\pi i\xi}$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Q The dyadic cubes in $\Reals^d$ are the sets of the form
$$
Q_{n,k} = [k_1 2^n,(k_1+1) 2^n) \times \cdots \times [k_d 2^n,(k_d+1) 2^n)
$$
were $n$ ranges over $\Ints$ and $k\in\Ints^d$.

\noindent(a) Given a collection of dyadic cubes whose diameters are bounded,
show that one may find a sub-collection which covers
the same region of $\Reals^d$ but with all cubes disjoint.

\noindent(b) Define the (uncentered) dyadic maximal function by
$$
[M_Df] (x) = \sup_{Q\ni x} \frac{1}{|Q|} \int_Q f(y) \, dy
$$
where the supremum is over all dyadic cubes that contain $x$.  Show that this operator
is of weak type (1,1).

\noindent(c) Deduce boundedness of the Hardy-Littlewood maximal function from the above.

\noindent
{\it Remarks:} Part (a) provides a replacement for the Vitali Covering Lemma.
I propose you address (c) `geometrically'; draw some pictures in the planar ($d=2$)
case.

%%%%%%%%%%%%%%%%%%%%%5

\Q (a) Evaluate
$$
D_N(x)=\sum_{n=-N}^N e^{2\pi inx}
$$
and show that it is not an approximate identity on $\mathbb{T}$.

\noindent(b) Show that $\frac1{2N+1} |D_N(x)|^2$ is an approximate identity and derive its relation to
the Fejer kernel.

\noindent(c) Calculate
$$
\sum_{n\in\Ints} r^{|n|} e^{2\pi inx}
$$
for $0<r<1$ and show that for $r\to1$ it gives rise to an approximate identity.

\noindent(d) Suppose $\phi_n$ is an approximate identity and $d\mu$, a finite complex
measure on $\mathbb{T}$.  Show that $\phi_n * d\mu$ converges weak-$*$ to $d\mu$.

Note: $d\mu_n$ converges weak-$*$ to $d\mu$ iff for every bounded continuous function, $f$,
$\int f \, d\mu_n \to \int f \, d\mu$.

%%%%%%%%%%%%%%%%%%%%

\Q (a) Given $f\in L^p(\Reals)$, $1\leq p < \infty$, show that $t\mapsto f(x+t)$ defines a continuous map
of $\Reals$ into $L^p(\Reals,dx)$.

\noindent(b) Show that it is not equi-continuous as $f$ varies
over the set of $f$ with $\|f\|_{L^p}\leq 1$. (That is, $\epsilon$
cannot be chosen from $\delta$ independently of $f$.)

\noindent(c) Show that part (a) is false for $L^\infty$ and $M(\Reals)$.

%%%%%%%%%%%%%%%%%%%%%%%%

\Q (From Wolff \S4.)  Find a sequence of Schwartz functions $\phi_n$ such that (a) $\|\phi_n\|_{L^p}$
and $\|\hat\phi_n\|_{L^{p'}}$ are constant.  The supports of $\hat \phi_n$ are disjoint and
those of $\phi_n$ are almost disjoint.  Use $\sum_{n=1}^N \phi_n$ to show that if $\|\hat f\|_{L^{p'}}
\lesssim \|f\|_{L^{p}}$ then $p\leq 2$.

By almost disjoint we mean $\|\sum_{n=1}^N \phi_n\|_{L^p}^p \leq
\frac{100}{99} \sum_{n=1}^N \|\phi_n\|_{L^p}^p$. Notice that if
the supports were actually disjoint, then $100/99$ could be
replaced by $1$.

\noindent {\it Hint:} Take a single $C^\infty_c$ function and
modify it by translation and multiplication by characters.

\Q Prove the Rising Sun Lemma: Given a non-negative $f\in L^1(\Reals)$, define
$$
[M_R f] (x) = \sup_{t>0} \frac1t\! \int_0^t f(x+s) \, ds.
$$
If $S=\{ x : M_R f > \lambda \}$ then $|S| = \lambda^{-1} \int_S f(x) \, dx$.  [Hint: $S$ is open.]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Q Prove the following theorem of Milicer-Gruzewska: Let $d\mu$ be a complex measure on
$\mathbb{T}$ with the property that $\hat\mu(n)\to 0$ as $n\to\infty$ ($\mu$ is called a Rajchman
measure). If $f\in L^1(d|\mu|)$ and $d\nu=f\,d\mu$ then $\hat \nu (n) \to 0$.  [Hint: mimic the
proof of the Riemann--Lebesgue Lemma from Schlag's notes.]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Q Let $R(k)$ be the smallest number such that in any colouring of the edges of the complete
graph on $R(k)$ vertices by two colours, one can find a monochromatic complete graph on $k$
vertices. These are known as Ramsey numbers; it is not difficult to show that $R(k)\leq 2^{2k}$.
The problem here is to prove that $2^{k/2} \leq R(k)$, which is due to Erd\H{o}s.

\noindent (a) Determine the expected number of monochromatic complete graphs on $k$ vertices
contained within a random colouring of the complete graph on $n$ vertices.

\noindent (b) Show that this is less that one when $n=2^{k/2}$ and so complete the problem.

%%%%%%%%%%%%%%%%%%%%%%%%%%

\Q Let $f\in C^\alpha$ with $\alpha<1$, and let $K_n$ denote the Fej\'er kernel.

\noindent (a) Show that
$$
\| f*K_n - f \|_{C^0}  \lesssim n^{-\alpha} \|f\|_{C^\alpha} .
$$

\noindent(b)[Optional] Show that $\| f*K_n - f \|_{C^\alpha} \to 0$ may fail.  However, it is
true if one restricts to those $f$ with
\begin{equation}\label{E1}
  \sup_{|x-y|<\delta} |f(x)-f(y)|  = o(\delta^\alpha).
\end{equation}

\noindent(c)[Optional] Show that the set of $f\in C^\alpha$ that obey \eqref{E1} is exactly the
closure of $C^\infty$ in $C^\alpha$.

%%%%%%%%%%%%%%%%%%%%%

\Q Let $f$ be a continuous function on $\mathbb{T}$. Suppose that for each $n>0$ there is a
trigonometric polynomial $p_n$ of degree $n$ (or less) such that
$$
  \| f - p_n \|_{C^0} \lesssim n^{-\alpha}
$$
where $\alpha<1$.  Show that $f$ is $\alpha$ H\"older continuous. Hint: write
$$
f=p_1 + \sum_{k=1}^\infty  \bigl( p_{2^k} - p_{2^{k-1}} \bigr).
$$

%%%%%%%%%%%%%%%%%%%%%%%%

\Q Let $\Omega$ be a simply-connected open domain bounded by a Jordan curve.  By a theorem of
Carath\'eodory, any conformal map $f$ of $\Disk$ onto $\Omega$ can be extended to a homeomorphism
of $\bar{\Disk}$ onto $\bar\Omega$.

We say that a curve $\gamma:S^1\to\Cmplx$ is rectifiable if there exists a constant $L$ so that
for any $0\leq\theta_0<\theta_1<\cdots<\theta_n<2\pi$,
$$
\sum_{k=0}^n |\gamma(e^{i\theta_k})-\gamma(e^{i\theta_{k+1}})| \leq L
$$
where $\theta_{n+1}=\theta_0$.

Prove the following theorem of F. and M. Riesz: $f' \in H^1$ if and only if $\partial \Omega$ is rectifiable.
[Hint: the function $z\mapsto\sum|f(ze^{i\theta_k})-f(ze^{i\theta_{k+1}})|$ is continuous and sub-harmonic
on $\Disk$.]

%%%%%%%%%%%%%%%%%%%%%%%%

\Q Prove the following result of Privalov: For $0<\alpha<1$,
$f\in C^\alpha$ implies $\tilde f \in C^\alpha$.

%%%%%%%%%%%%%%%%%%%%%%%%%

\Q (a) Suppose $T$ is a rotation invariant operator on $L^2(\Reals/\Ints)$, that is,
$R_y T = T R_y$ for any rotation $[R_y f](x)=f(x-y)$.
Show that $e^{2\pi inx}$, $n\in\Ints$, are eigenfunctions of $T$.

\noindent(b) Let $T$ be a bounded operator on $L^2(\Reals^n)$ such that there is a function $K$ obeying
$|K(x,y)|\lesssim |x-y|^{-n}$ so that whenever $f$ and $g$ have disjoint supports,
$$
  \langle g , Tf \rangle = \int\!\int \bar g(x) K(x,y) f(y) \,dy\,dx.
$$
Show that if $T$ is translation invariant, then $K(x,y)=F(x-y)$, which means that $T$ is
a convolution operator. [Hint: Treat (a) and (b) independently.]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Q (a) Let $I\subseteq\Reals$ be an interval and let $z\in\Cmplx^+=\{z:\Im z >0\}$.
Show that the harmonic measure of $I\subseteq\partial\Cmplx^+$ with respect to $z$
is equal to the angle subtended by $I$ at $z$ divided by $\pi$.  Deduce that the
the harmonic measure of $I$ is constant on arcs of circles.

\noindent(b) Calculate the conjugate function of $\chi_{[0,a]}(\theta) \in L^2(S^1;\tfrac{d\theta}{2\pi})$.

%%%%%%%%%%%%%%%%%%%%%%%

\Q (a) Suppose $f:\Disk\to\Cmplx^{+}=\{z:\Im z >0\}$ is analytic.
Show that there exists a finite positive measure $d\mu$
and a real constant $a$ so that
$$
f(z) = a + i \int_0^{2\pi} \frac{e^{i\theta} + z}{e^{i\theta} - z} \,d\mu(\theta).
$$
This result is due to Herglotz. [Hint: First look at $\Im(f)$.]

\noindent(b) Deduce that any holomorphic mapping of $\Cmplx^{+}$ into itself admits the representation
$$
f(z) = a + bz + \int_{\Reals} \frac{1+tz}{t-z} \,d\rho(t).
$$
where $a\in\Reals$, $b\geq 0$, and $d\rho$ is a positive measure.

%%%%%%%%%%%%%%%%%%%%%%%

\Q Prove the following theorem of Kolmogorov:  suppose $0<p<1$ then
$$
f(z) = \int_0^{2\pi} \frac{d\mu(\theta)}{e^{i\theta} - z}
  \quad\text{implies}\quad
\sup_{0<r<1} \int |f(re^{i\theta})|^p \,d\theta < \infty
$$
for any finite complex measure $d\mu$.

%%%%%%%%%%%%%%%%%%%%%%

\Q Let $\ell^p_\mu$ denote the weighted $\ell^p$ space
$$
\| c \|^p = \sum (|n|+1)^{-2} |c_n|^p.
$$
Let $\phi_n$, $n\in\Ints$, be an orthonormal basis for $L^2(\Reals/\Ints)$ which obeys $\|\phi_n\|_{L^\infty} \lesssim 1$
and define $T:L^2\to\ell^2_\mu$ by
$$
 [Tf](n) = (|n|+1) \bigl\langle\phi_n(x) , f(x) \bigr\rangle.
$$

\noindent (a) Show that $T$ extends to a bounded map of $L^p$ into $\ell^p_\mu$ for all $1<p\leq 2$.
This result is due to Hardy and Littlewood.
[\textit{Hint:} Prove a weak-type bound and use Marcinkiewicz.]


\noindent (b) Given a sequence $c_j$ indexed by $j\in\Ints$, define the rearrangement $c_j^*$ as follows:
For $j\geq0$, $c_j^*$ is the $(j+1)$th largest element of the set $\{|c_0|,|c_1|,\ldots\}$ while for $j<0$,
it is the $|j|$th largest element of $\{|c_{-1}|,|c_{-2}|,\ldots\}$.  Derive the following inequality of
Payley:
$$
   \sum (1+|j|)^{p-2} |c_j^*|^p \lesssim \|f\|_{L^p}^p
$$
where $c_j=\hat f(j)$.

\noindent (c) By splitting the sum dyadically, show that this implies the usual Hausdorff--Young
inequality for $1<p\leq 2$.

%%%%%%%%%%%%%%%%%%%%%

\Q Suppose $f\in L^1(\Reals/\Ints)$ and let $Mf$ denote its (uncentred) dyadic
maximal function.

\noindent (a) Show that for $\lambda>\int |f|$,
$$
\frac{1}{\lambda} \int_{|f|>\lambda} |f(x)| \,dx \lesssim |\{x: Mf > \lambda\}|.
$$
[Hint: Do a Calder\'on--Zygmund style decomposition.]

\noindent (b) Deduce that if $Mf\in L^1$, then $|f| \log[1+|f|] \in L^1$.  This result
is due to Stein.

\noindent (c) Use the fact that $M:L^\infty\to L^\infty$ and $L^1\to L^1_{w}$ to show
$$
|\{x: Mf > \lambda\}| \lesssim \frac{1}{\lambda} \int_{|f|>c\lambda} |f(x)| \,dx.
$$
for some small constant $c$.

\noindent (d) Deduce that if $|f| \log[1+|f|] \in L^1$ then $Mf\in L^1$.

%%%%%%%%%%%%%%%%%%%%%%%%%

\Q For $1\leq p <\infty$, let $L^p_{w}(\Reals)$ denote the set of measurable functions on $\Reals$ for which
$$
\|f\|^*_p = \sup_{\lambda>0}  \big\{ \lambda^p |\{x : |f|>\lambda \}| \big\}^{1/p}
$$
is finite. The $*$ is to warn that this isn't a norm; however,

\noindent (a) For $1<p<\infty$, the following defines a norm on $L^p_{w}(\Reals)$:
$$
\|f\|_{p,w} = \sup_{E} \frac{1}{|E|^{(p-1)/p}} \int_E |f(x)|.
$$
Moreover, $\|f\|^*_p \lesssim \|f\|_{p,w} \lesssim \|f\|^*_p$. [\textit{Remark:}
with this norm, $L^p_{w}(\Reals)$ is actually a Banach space.]

\noindent (b) Show that there is no norm on $L^1_w(\Reals)$ comparable to $\|f\|^*_1$
by considering the following family of functions
$$
  \sum_{k=0}^N \frac1{|x-k|}
$$
as $N\to\infty$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Q (a) Let $c_n$ denote the surface area of $S^{n-1}\subseteq \Reals^n$. Show that for $n\geq 3$,
$$
G(x) = \frac{1}{(n-2)c_n|x|^{n-2}}
$$
is the Green function for the Laplace equation in $\Reals^n$: if $f\in\mathcal{S}$, then $-\Delta(G*f)=f$.

\noindent (b) For any $f,g\in \mathcal{S}$,
$$
 \left| \int f(x) g(x) \,dx \right|^2 \leq \|\nabla f\|_{L^2}^2 \int\!\!\int g(x) G(x-y) g(y) \,dx\,dy.
$$

\noindent (c) Deduce the following Sobolev inequality:
$$
  \forall\,f\in\mathcal{S}, \qquad \| f \|_{L^q} \lesssim \|\nabla f\|_{L^2} \qquad \text{where $q=2n/(n-2)$}
$$
by choosing $g$ appropriately.

\noindent (d) Show that on $\Reals$, one does not have
$$
\| f \|_{L^\infty}^2 \lesssim \|f'\|_{L^2}^2
$$
however it is true that
$$
\| f \|_{L^\infty}^2 \lesssim \|f'\|_{L^2}^2 + \|f\|_{L^2}^2.
$$
[\textit{Remark:} In this regard, $\Reals^2$ is like $\Reals$; there is no estimate without adding $\|f\|_{L^2}$.
However, one has only
$$
\| f \|_{L^q}^2 \lesssim \|\nabla f\|_{L^2}^2 + \|f\|_{L^2}^2
$$
for all $2\leq q<\infty$.]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Q Suppose $a:\Reals^2\to\Reals$ obeys
$$
\frac{\partial^{n+m}}{\partial x^n \partial\xi^m} a(x,\xi) \in L^\infty
$$
for all $n,m\geq 0$.  We then define an operator on $L^2(\Reals)$ by
$$
[Tf](x) = \int a(x,\xi) e^{2\pi i x \xi} \hat{f}(\xi)\, d\xi.
$$
(This is the pseudo-differential operator with symbol, $a$, which belongs to the
exotic symbol class $S^0_{0,0}$.)  Show that it is bounded.  [\textit{Hint:}
let $\psi_j$ be a partition of unity adapted to the partition of
$\Reals$ by $[j,j+1)$, then apply the Cotlar-Stein Lemma using the operators
with symbols $a_{i,j}(x,\xi) = \psi(x-i) a(x,\xi) \psi(\xi-j)$.]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Q (a) Prove that
$$
\left| \int_{\Reals^n} f(x) g(x) \,dx \right| \leq \int_{\Reals^n} f^*(x) g^*(x) \,dx
$$

\noindent (b) Suppose $f\mapsto f*K$ is a bounded operator on $L^2(\Reals^n)$ and
$K(x)\lesssim |x|^{-n}$. Show that there exists $C$ so that
$$
\int_{\epsilon<|x|<N} K(x) \, dx \leq C
$$
for all $0<\epsilon<N<\infty$.

%%%%%%%%%%%%%%%%%%%%

\Q Given a measurable function $t:\Reals\to(0,\infty)$, let us define
$$
  [T_tf](x) = \frac{1}{\sqrt{2\pi t(x)}} \int \exp\{ - \tfrac{(x-y)^2}{2 t(x)} \} f(y) \,dy.
$$
(a) Determine the adjoint of the operator $T_t$; write it as an integral operator.

\noindent (b) Consider $T_t^{} T_t^\dagger$ and show that for $f\geq0$,
$$
  [T_t^{} T_t^\dagger f](x) \lesssim [T_{2t} f](x) + [T_{2t}^\dagger f](x).
$$

\noindent (c) Deduce that maximal operator
$$
[Mf](x) = \sup_{t>0} \frac{1}{\sqrt{2\pi t}} \int \exp\{ - \tfrac{(x-y)^2}{2 t} \} f(y) \,dy.
$$
is bounded on $L^2(\Reals)$. [\textit{Remark:} There is nothing special about
the Gaussian, it was just chosen for concreteness.]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Q Given $n\in\Ints^3$, let us write $|n|$ for the $\ell^1$ norm: $|n|=|n_1|+|n_2|+|n_3|$.
Consider the following operator on $\ell^2(\Ints^3)$:
$$
  [Hu](n) = \sum_{|n-m|=1} u(m).
$$
Schur's test (or part (b)) shows that this is a bounded operator.

\noindent (a) Given $n\in\Ints^d$, let us write $\delta_n$ for the function $k\mapsto\delta_{k,n}$.
Show that $\langle \delta_m | H^N \delta_n \rangle$ is equal to the number of paths of length $N$
from $n$ to $m$ in the $\Ints^3$ lattice.


\noindent (b) As $H$ is translation invariant, we know that we can write it as a Fourier
multiplier. Find the Fourier multiplier.

\noindent (c) Determine the leading term in the $t\to\infty$ asymptotics of
$$
\langle \delta_0 | e^{tH} \delta_0 \rangle.
$$

\noindent (d) [Optional]  Use the Borel--Cantelli Lemma to deduce that in three dimensions, a random
walker starting at the origin will return to the origin only finitely many times (with probability one).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Q Let $\Omega$ denote a hyperplane in $\Reals^d$ and let $d\sigma$ denote the induced Lebesgue
measure.  For $s\geq 0$, $H^s$ denotes the Sobolev space of functions $f\in L^2$ for which
$$
\bigl\|f\bigr\|_{H^s}^2 = \int |\hat f|^2 (1+|\xi|^2)^s \,d\xi
$$
is finite.

Show that for $\epsilon>1/2$, $f\mapsto f|_\Omega$ defines a continuous map from $H^s(\Reals^d)$
to $H^{s-\epsilon}(\Omega)$.  Also show that for $\epsilon\leq 1/2$, it does not.

%%%%%%%%%%%%%%%%%%%%%%%

\Q Let $\Omega$ denote the cone $|\xi_0|^2 = |\xi_1|^2+\cdots+|\xi_d|^2$ in $\Reals^{d+1}$ and let
$d\sigma$ denote the induced surface measure.

\noindent (a) If $f$ is a smooth function supported in a compact subset of
$\Reals^{d+1}\setminus\{0\}$, show that Fourier transform of $f\,d\sigma$ has a natural
interpretation as a solution of the wave equation:
$$
\frac{d^2u}{dt^2} = \sum_j \frac{d^2u}{dx_j^2}.
$$

\noindent (b) Calculate the leading term asymptotics of $\widehat{fd\sigma}$ as $|\xi|\to\infty$ in a fixed
direction. For simplicity, just treat the case $d=2$ with $f$ supported in the region $\{\xi_0>0\}$.
Warning: the cone does not have non-vanishing Gaussian curvature!

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Q (a) Given $\psi_0$ with $\hat\psi_0 \in C^\infty_c(\Reals)$, write the solution of the free
Schr\"odinger equation
$$
i\frac{\partial\psi}{\partial t} = -\frac{\partial^2\psi}{\partial^2 x}, \ \ %
    \qquad \psi(x,t=0) = \psi_0(x)
$$
as an integral involving $\hat\psi_0$.

\noindent (b) Study the asymptotics in the regime $t\to\infty$ with $x=vt$ and $v\in\Reals$ fixed.
Specifically, prove that
$$
\left| \psi(x,t) - \tfrac{1}{\sqrt{4\pi t}} e^{-i\pi/4} e^{i x^2/4t} \hat\psi_0(\tfrac{x}{4\pi t}) \right|
\lesssim  (t^2+x^2)^{-3/4}
$$
for $t$ sufficiently large.

\noindent (c) Let us call the map of $\psi_0$ into the leading asymptotic behaviour $V(t)$.  That is,
the LHS of the equation above is $|\psi(x,t)-V(t)\psi_0|$.  Check that this determines a unitary map and
that $\psi(t)-V(t)\psi_0$ converges to zero in $L^2$.

\noindent (d) Use the fact that for $t$ fixed, $\psi_0\mapsto\psi(t)$ is also a unitary map to deduce that the above
asymptotic holds in $L^2$ sense for any initial data $\psi_0\in L^2$.

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\Q Prove the van der Corput Lemma: (a) Suppose $\phi$ is real-valued and smooth in $(a,b)$ and that
for some $k\geq1$, $\phi^{(k)}(x)\geq 1$ on $[a,b]$.  Show that
$$
\left| \int_a^b e^{i\lambda\phi(x)} \,dx \right| \leq 3^k \lambda^{-1/k}
$$
for $k\geq2$ and also for $k=1$ if we assume that $\phi'$ is monotone.  [\textit{Hints}:
Proceed by induction.  For $k=1$, integrate by parts wisely.  For the step from $k$ to $k+1$,
treat any interval with $|\phi^{(k)}(x)|\leq \delta(\lambda)$ separately from those where
it is bigger than $\delta(\lambda)$.]

\noindent (b) Deduce that
$$
\left| \int_a^b e^{i\lambda\phi(x)} \psi(x) \,dx \right|
    \lesssim \lambda^{-1/k} \biggl[ |\psi(b)| + \int_a^b |\psi'(x)|\,dx \biggr].
$$

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\Q (a) Prove Debye's asymptotics for Bessel functions: given $\alpha\in(0,\infty)$,
$$
J_\nu\bigl(\nu\sech(\alpha)\bigr) = \frac{e^{\nu[\tanh(\alpha)-\alpha]}}{\sqrt{2\pi\nu\tanh(\alpha)}}
    \Bigl[ 1 + O(\nu^{-1}) \Bigr]
$$
as $\nu\to\infty$.

\noindent (b) Prove that
$$
\cos[z\sin(\theta)] = J_0(z) + 2 \sum_{k=1}^\infty J_{2k}(z)\cos(2k\theta)
$$
for all $\theta\in\Reals$.  Why does the series converge?

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\Q Let $E$ be a compact subset of $\Reals^n$ of non-zero $\alpha$-capacity.  In class we proved the existence of a
probability measure $d\nu$ so that
$$
\frac{1}{C_\alpha(E)} = \inf_{\supp(\mu)\subseteq E} I_\alpha(\mu) = I_\alpha(\nu).
$$
Recall that
$$
  V_\mu(x) = \int \frac{d\mu(y)}{|x-y|^\alpha}
$$
denotes the potential generated by $d\mu$.

\noindent (a) Show that $V_\nu(x)\geq  1/C_\alpha(E)$ for p.p. $x\in E$.  (Recall that `p.p.' means except for
a set of zero capacity.)

\noindent (b) Show that for any positive measure $\mu$, $V_\mu(x)$ is upper semi-continuous, that is,
for every $a\in\Reals$, the set $\{x:V_\mu(x)>a\}$ is open.  Equivalently,
$$
\liminf_{x_n\to x} V_\mu(x_n) \geq V_\mu(x).
$$

\noindent (c) From part (a), it follows that $C_\alpha(E) V_\nu(x) \geq  1$ for $\nu$-almost all $x$.  Explain.
From this and part (b), show that $C_\alpha(E) V_\nu(x) \leq  1$ for all $x\in\supp(\nu)$.

\noindent (d) Show that
$$
  C_\alpha(E) = \sup\{\|\mu\| : \supp(\mu)\subseteq E \text{ and } \forall x\in\supp(\mu),\  V_\mu(x) \leq  1 \}.
$$

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\Q Let $E\subseteq\Reals^n$, be a compact set of non-zero $\alpha$-capacity ($0<\alpha<n$).  Let us define
$$
D_n = \inf \ \tfrac{2}{n(n-1)} \sum_{1\leq i<j\leq n} |x_i-x_j|^{-\alpha}
$$
and
$$
M_n = \sup\ \inf_{x\in E} \ \tfrac{1}{n} \sum_{1\leq i\leq n} |x-x_i|^{-\alpha}.
$$
The infimum in the definition of $D_n$ and the supremum in the definition of $M_n$ are over
$\{x_1,\ldots,x_n\}\subset E$.

Show that $D_{n+1}\leq M_n$, that $M_n \leq 1/C_\alpha(E)$, and that $\liminf D_n \geq 1/C_\alpha(E)$.
Conclude that $C_\alpha(E)=\lim D_n = \lim M_n$.  [You may use the results of Question 3.]

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\Q Functions defined by
$$
f(s) = \sum_{n=1}^\infty \frac{a_n}{n^s},
$$
are known as Dirichlet series.  The most famous example is the Riemann zeta function, where
$a_n\equiv 1$.  By writing $s=\sigma+it$ we have $n^{-s}=e^{-\sigma\log(n)} e^{-it\log(n)}$
which shows the connection to Fourier integrals.

\noindent (a) Given $f(s)$ as above and $g(s)=\sum b_n n^{-s}$, show that $f(s)g(s)$ can also be
written as a Dirichlet series and find the formula for the coefficients. In this way, interpret
the coefficients of $\zeta(s)^2$.  (This operation is the multiplicative analogue of convolution.)

\noindent (b) Let $f$ and $g$ be Dirichlet series absolutely convergent for $\Re(s)>\sigma_0$.
Show that
$$
\lim_{T\to\infty} \frac{1}{2T}\int_{-T}^T f(\alpha+it) g(\beta-it) = \sum \frac{a_n b_n}{n^{\alpha+\beta}}
$$
for $\Re(\alpha)$ and $\Re(\beta)$ larger than $\sigma_0$.  This is the analogue of the Plancherel Theorem.

\noindent (c) Prove the following simple Abelian theorem:  Given $\alpha< 1$,
$$
    \lim_{n\to\infty} \log^\alpha(n)a_n = A
    \quad \Longrightarrow \quad
    \lim_{u\downarrow 0} \ u^{1-\alpha} f(1+u) = C_\alpha A
$$
and determine the value of $C_\alpha$.  What if $\alpha=1$?

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\Q (a) Let $d(n)=\#\{d>0 : d|n\}$.  Prove that if $n=\prod p_i^{a_i}$ then
$$
\frac{d(n)}{n^\delta} = \prod \biggl( \frac{a_i+1}{p_i^{\delta a_i}} \biggr)
    < \exp\Bigl\{\tfrac{2^{1/\delta}}{\delta \log(2)}\Bigr\}
$$
[Hint: be wasteful, $\frac{a+1}{p^{\delta a}} \leq 1 + \frac{a}{p^{\delta a}} \leq 1 + \frac{1}{\delta\log(2)}$.]

\noindent (b) Refine the above argument to show that
$$
\log[ d(n) ] \leq \frac{(1+\epsilon)\log(2)\log(n)}{\log\log(n)}
$$
for $n$ sufficiently large (depending on $\epsilon$).

\noindent (c) By the prime number theorem, $\vartheta(x)=\sum_{p\leq x} \log(p)$ obeys
$\vartheta(x)/x \to 1$.  Use this to show that
$$
\log[ d(n) ] \geq \frac{(1-\epsilon)\log(2)\log(n)}{\log\log(n)}.
$$
infinitely often.

\noindent (d) By counting lattice points under the hyperbola $xy=n$, show that
$$
d(1)+d(2)+d(3)+\cdots+d(n) = n\log(n) + O(n).
$$
While part (c) shows that $d(n)$ can be enormous, this result shows that it is typically much smaller.

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\Q Let $\omega=e^{2\pi i/3}$.

\noindent (a) Show that $\Ints[\omega]$ is a Euclidean domain using the norm $N(a+b\omega)=|a+b\omega|^2=a^2-ab+b^2$.

\noindent (b) Determine the units (there are six).

\noindent (c) Show that the following is a complete list of the primes in $\Ints[\omega]$ (without repetition):

(i) $1-\omega$ and its associates,

(ii) the rational primes of the form $3n+2$ and their associates, and

(iii) the (non-trivial) factors $a+b \omega$ of rational primes of the form $3n+1$.

\noindent (d) Deduce that the number of solutions $(n,m)\in\Ints^2$ to $N=n^2+3 m^2$ is bounded by
$C_\epsilon N^\epsilon$.  (You may use results from the previous problem.)

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\Q (a) Use the previous problem to prove the following result of Bourgain\footnote{`Fourier transform
restriction phenomena for certain lattice subsets and applications to non-linear evolution
equations', I. \textit{Geom. Funct. Analysis}, \textbf{3} (1993) 107--156.}:
$$
\left\| \sum_{|n|\leq N} a_n e^{2\pi i [nx + n^2 t]} \right\|_{L^6}^2
    \lesssim N^\epsilon \sum |a_n|^2
$$
where $L^6$ denotes $L^6(\Reals^2/\Ints^2; dx\,dt)$.

\noindent
(b) Gauss proved that if $a,b$ are integers and $q$ is an odd prime with $a,b\in[1,q-1]$, then
$$
  \biggl| \sum_{n=0}^{q-1} e^{ 2\pi i [an^2+bn]/q } \biggr| = \sqrt{q}.
$$
(Such sums are known as Gauss sums.)  From this we may deduce
$$
  \biggl| \sum_{n=0}^{N} e^{ 2\pi i [an^2+bn]/q } \biggr| \gtrsim \frac{N}{\sqrt{q}}
$$
when $N\geq q^2$, for example.  By studying small regions around $x=b/q$ and $t=a/q$ with $q\in[3,\sqrt{N}]$
show that $N^\epsilon$ cannot be replaced by an $N$-independent constant in part (a).

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\Q Prove the following two Strichartz estimates due to Bourgain\footnote{`Fourier transform
restriction phenomena for certain lattice subsets and applications to non-linear evolution
equations', I. \textit{Geom. Funct. Analysis}, \textbf{3} (1993) 107--156.}:
$$
\left\| \sum a_n e^{2\pi i (nx + n^2t)} \right\|_{L^4}^2 \lesssim \sum |a_n|^2
$$
where $L^4$ denotes $L^4(\Reals^2/\Ints^2; dx\,dt)$ and
$$
\left\| \sum_{n^2+m^2\leq N^2} a(n,m) e^{2\pi i [nx + my + (n^2+m^2)t]} \right\|_{L^4}^2
    \lesssim N^\epsilon \sum |a(n,m)|^2
$$
where $L^4$ denotes $L^4(\Reals^3/\Ints^3; dx\,dy\,dt)$.  [\textit{Hint:} Use the method from the proof of
Zygmund's restriction theorem.]

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\Q Let $d\mu$ and $d\mu_n$ be probability measures on $[0,\infty)$.

\noindent(a) Show that if $d\mu_n$ converges weak-$*$ to $d\mu$, then
$$
\limsup_{n\to\infty} \mu_n(K) \leq \mu(K)
$$
for any closed set $K$.  Also show that for any open set, $O$,
$$
\liminf_{n\to\infty} \mu_n(O) \geq \mu(O).
$$

\noindent(b) Give examples that show that the inequalities in part (a) can fail to be equalities.

\noindent(c) If we do not assume that $d\mu$ is a probability measure, half of (a) can fail.  Which
half and why?

\noindent(c) Show that if
$$
\lim_{n\to\infty} \int e^{-mx} \,d\mu_n(x) = \int e^{-mx} \,d\mu(x)
$$
for all $m\in\{0,1,2,\ldots\}$ then $d\mu_n$ converges weak-$*$ to $d\mu$.

\end{document}