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\section{Introduction to Fourier Restriction}

\begin{fcthmstar}[Hausdorff-Young inequality]
  \label{thm:hy.ineq}
  Assuming:
  - $1 \le p \le 2$
  - $\frac{1}{p} + \frac{1}{q} = 1$
  Then:
  \[
    \|\ft f\|_q \le \|f\|_p
  .\]
\end{fcthmstar}

\begin{proof}
  The inequality is true for $p = 1$, $q = \infty$
  and for $p = 2$, $q = 2$ the inequality is true
  since we have equality (Plancherel).

  For values in between we can interpolate.
\end{proof}

Are there any other $(p, q)$ for which
\[
  \|\ft f\|_q \lesssim \|f\|_p
  ?
  \tag{$*$}
  \label{fbounded}
\]

We saw that $1 \le p \le 2$ was necessary
(translations / modulations,
\nameref{thm:khinchin}).

\textbf{Scaling:} Plug in $f_\lambda(x) = f(\lambda x)$
which is $L^\infty$-normalised
($\|f_\lambda\|_\infty = \|f\|_\infty$).
Then $\ft{f_\lambda}(\xi) = \lambda^{-n} \ft
f(\lambda^{-1} \xi)$ which is $L^1$-normalised
($\|\ft{f_\lambda}\|_1 = \|\ft f\|_1$).
\[
  (\text{LHS of \eqref{fbounded}})^q
  = \int_{\Rbb^n} |\ft{f_\lambda}(\xi)|^q \dd \xi
  = \lambda^{-nq + n} \int |\ft f(\xi)|^q \dd \xi
.\]
\[
  (\text{RHS of \eqref{fbounded}})^p
  = \int_{\Rbb^n} |f_\lambda(x)|^p \dd x
  = \lambda^{-n} \int |f(x)|^p \dd x
.\]
So we need for all $\lambda > 0$:
\[
  \lambda^{-n + \frac{n}{q}} \|\ft f\|_q
  \lesssim \lambda^{-\frac{n}{p}} \|f\|_p
.\]
So we need $-n + \frac{n}{q} = -\frac{n}{p}$, i.e.
$\frac{1}{p} + \frac{1}{q} = 1$.

\subsubsection*{Classical questions}

What is $C_{p, q}$ the smallest constant such that
$\|\ft f\|_q \le C_{p, q} \|f\|_p$?

Which functions $f$ satisfy $\frac{\|\ft
f\|_q}{\|f\|_p} = C_{p, q}$?

2014:
\[
  \|\ft f\|_q
  \le C_{p, q} \|f\|_p - \dist_p(f,
  \text{maximisers (Gaussian)})
.\]
Fourier restriction asks which $(p, q)$ permit
estimates $\|\ft f\|_{L^q(\mathcal{R})} \lesssim
\|f\|_{L^p(\Rbb^n)}$ ($\mathcal{R}$ is the
restricted frequency set, $\mathcal{R} \subseteq
\Rbb^n$).

\begin{example*}
  $\mathcal{R} = B_1(0) \subseteq \Rbb^n$, unit
  ball $\to$: Basically, sitll governed by
  \nameref{thm:hy.ineq}.
\end{example*}

\begin{example*}
  $\mathcal{R} = B_1(0) \cap \{x_n = 0\}$ (measure
  $0$ subset of $\Rbb^n$).
\end{example*}

$R_{\mathcal{R}}(p \to q)$ means
\[
  \|\ft f\|_{L^q(\mathcal{R})}
  \lesssim \|f\|_{L^p(\Rbb^n)}
  \qquad \forall f : \Rbb^n \to \Cbb
  \text{ Schwartz}
\]
Always: $R_{\mathcal{R}}(1 \to \infty)$ true
for all $\mathcal{R}$.

In the second example, only this trivial statement
is true (i.e. $R_{\mathcal{R}}(p \to q)$ is false
for all other values for $p, q$).

Let $\mathcal{R} = S^{n - 1}$ be the unit sphere
in $\Rbb^n$. Consider
\[
  \|\ft f\|_{L^q(S^{n - 1})}
  \lesssim \|f\|_{L^p(\Rbb^n)}
\]
where $L^q(S^{n - 1})$ uses the usual surface
measure $\dd \sigma$ on $S^{n - 1}$.

\begin{notation*}
  \glssymboldefn{invft}%
  \[
    \begin{tikzcd}
      \ub{f(x)}_{\in \mathcal{S}(\Rbb^n)}
      \ar[r, "\widehat{}"]
      & \ub{\ft f(x)}_{\in \mathcal{S}(\Rbb^n)}
      \ar[r, "\widehat{}"]
      \ar[rr, bend right=40, swap, "\widecheck{}"]
      & \ub{f(x)}_{\in \mathcal{S}(\Rbb^n)}
      \ar[r, "x \mapsto -x"]
      & \ub{f(x)}_{\in \mathcal{S}(\Rbb^n)}
    \end{tikzcd}
  \]
  We may call $\widecheck{}$ the ``inverse Fourier
  transform''.
\end{notation*}

Let $\varphi \in C_c^\infty(\Rbb^n)$,
$\Rbb$-valued, $\ge \half$ on $B_1(0)$, with
support in $B_2(0)$.

May also assume $\invft{\varphi}$ is
$\Rbb$-valued, $\invft{\varphi} \gtrsim 1$ on
$B_c(0)$. $\invft \varphi$ bounded, $|\invft
\varphi(x)| \lesssim_m (|x|^2 + 1)^{-m} \forall
m$. $|\invft\varphi(x)|$ behaves like
$\chi_{B_1(0)}$ in $L^p$.
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/8c177f87cf394742.png}
\end{center}

Consider dilates $f_R(x) = \invft \varphi(R^{-1}
x) e^{2 \pi i x \cdot v_r}$, $R \gg 1$.

Frequency side $|\ft{f_R}(\xi)| \sim R^n
\chi_{B_{R^{-1}}(0)}(x)$ ($L^1$-norm).
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/983a0a4ff91e41de.png}
\end{center}
\[
  \int_{S^{n - 1}} |\ft{f_R}(\xi)|^q \dd
  \sigma(\xi) \sim \ub{R^{nq} R^{-(n - 1)}}_{\to
  \sigma(B_{R^{-1}}(n) \cap S^{n - 1})}
.\]
$B_{R^{-1}}(n) \cap S^{n - 1}$ cap of radius
$R^{-1}$.
Spacial side $|f_R(x)| \sim \chi_{B_R(0)}(x)$ (in
$L^\infty$-norm).
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/2fd133ce333d4dc5.png}
\end{center}
height $1$, $\int_{\Rbb^n} |f_R(x)|^p \dd x \sim
R^n$.

$R^{n - \frac{n-1}{q}} \lesssim R^{\frac{n}{p}}$,
$n - \frac{n - 1}{q} \le \frac{n}{p}$.

Consider: $g_{\mathcal{R}}(x) = e^{ix \cdot w_r}
\invft \varphi (R^{-1} x_1, \ldots, R^{-1} x_{n
- 1}, R^{-2} x_n)$.

\newcommand\cyl{\mathrm{cylinder}}
Frequency: $|\ft{g_r}(\xi)| \approx |\cyl|^{-1}
\chi_\cyl(\xi)$. $|\cyl|^{-1}|$ $\to$ $(R{-1}
\cdot R^{-1} R^{-2})^{-1} = R^{n + 1}$.
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/8ffe2bc22bca4624.png}
\end{center}
\[
  \int_{S^{n - 1}} |\ft{g_R}(\xi)|^q \dd
  \sigma(\xi)
  \sim R^{(n + 1)q} \sigma(\text{cap of radius
  $R^{-1}$})
  = R^{(n + 1) q - (n - 1)}
.\]

Spatial side: $|g_{R'}(x)| \approx \chi_\cyl(x)$
($L^\infty$-norm).
\begin{center}
  \includegraphics[width=0.2\linewidth]{images/02e1229d9a804096.png}
\end{center}
$\int |g_R(x)|^p = R^{n - 1 + 2} = R^{n + 1}$.

$\to$ $R^{n + 1 - \frac{n - 1}{q}} \lesssim
R^{\frac{n + 1}{q}}$, $\implies n + 1 - \frac{n -
1}{q} \le \frac{n + 1}{p}$. Implies B.

On Monday, we will build examples $h$ such that
$\ft h$ sees \emph{all} of $S^{n - 1}$.