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\newpage
\section{Equivalent Versions of Fourier Restriction}

Searching for $(p, q)$ for which
\[
  \|\ft f\|_{L^q(S^{n - 1})} \le C_{p, q, n} \|f\|_{L^p(\Rbb^n)}
.\]

\begin{enumerate}[(1)]
  \item 
    \begin{center}
      \includegraphics[width=0.6\linewidth]{images/12f0fbf31200403e.png}
    \end{center}
    $\mathcal{R} = [0, 1]^{n - 1} \times \{0\}$
    \begin{center}
      \includegraphics[width=0.6\linewidth]{images/052a11d42f3047ad.png}
    \end{center}
  \item
    \begin{center}
      \includegraphics[width=0.6\linewidth]{images/89573b4194e544c2.png}
    \end{center}
    $\to \int_{B_{R^2}} \left| \sum_\theta
    \chi_{T_\theta} \right|^{\frac{p}{2}}$.
    ``continuum incidence geometry problem''.
    \begin{center}
      \includegraphics[width=0.6\linewidth]{images/d1d9d745ceb6461e.png}
    \end{center}
    integral equals $=\sum_{B_R \subseteq B_{R^2}}
    \int_{B_R} \left| \sum_\theta \chi_{T_\theta}
    \right|^{\frac{p}{2}}$.

    ``fractal geometry''.
    \begin{center}
      \includegraphics[width=0.6\linewidth]{images/bb2d83f51d7046bd.png}
    \end{center}
\end{enumerate}

Reminder: $R_{S^{n - 1}}(p \to q)$ means
\[
  \|\ft f\|_{L^q(S^{n - 1})}
  \le C_{p, q, n} \|f\|_{L^q(\Rbb^n)}
  \qquad \forall f \in \mathcal{S}(\Rbb^n)
.\]

\begin{fcconjstar}[Restriction Conjecture]
  \begin{iffc}
    \lhs
    $R_{S^{n - 1}}(p \to q)$
    \rhs
    $n + 1 - \frac{n - 1}{q} \le \frac{n + 1}{p}$
    and $p < \frac{2n}{n + 1}$.
  \end{iffc}
\end{fcconjstar}

\noproof

First proved for $S^1 \subseteq \Rbb^2$ by
Fefferman (1970) and Zygmund (1974).

Special things happen in $\Rbb^2$, classical
harmonic analysis techniques apply.

Same conjecture for $R_{P^{n - 1}}(p \to q)$,
where
\[
  P^{n - 1}
  = \{(\xi, |\xi|^2) \in \Rbb^n : |\xi| < 1\}
.\]
$\|\ft f\|_{L^q(P^{n - 1})}^q = \int_{|\xi| < 1}
|\ft f(\xi, |\xi|^2)|^q \dd \xi$.
\begin{center}
  \includegraphics[width=0.3\linewidth]{images/6851dbae95ed4fba.png}
\end{center}

For $n \ge 3$, open and active!

Restriction theory can be used to deduce continuum incidence
geometry estimates.

Surprisingly, we can go the other way too (very
recent progress, whereas the above direction has
been well-known since at least the $90$s).

\subsubsection*{Equivalent formulations of
$R_{P^{n - 1}}(p \to q)$}

Dual version is called ``Fourier extension'':
\[
  \|\ft f\|_{L^q(P^{n - 1})}
  = \sup_{\substack{g \in L^{q'}(P^{n - 1}) \\
  \|g\|_{L^{q'}(P^{n - 1})} = 1}}
  \left| \int_{\substack{|\xi| < 1 \\ \xi \in
  \Rbb^{n - 1}}} \ft f(\xi, |\xi|^2) g(\xi) \dd
  \xi \right|
.\]
($\frac{1}{q} + \frac{1}{q'} = 1$).
The integral equals:
\begin{align*}
  \int_{|\xi| < 1} \int_{\Rbb^n} e^{-2 \pi i x
  \cdot (\xi, |\xi|^2)} f(x) \dd x g(\xi) \dd \xi
  &= \int_{\Rbb^n} f(x) \ub{\int_{|\xi| < 1}
  e^{-2 \pi i x \cdot (\xi, |\xi|^2)} g(\xi) \dd
  \xi}_{Eg(x)} \dd x \\
  \|Eg\|_{L^{p'}(\Rbb^n)}
  &\le C_{p, q, n} \|g\|_{L^{q'(P^{n - 1})}}
  \qquad \forall \mathcal{S}(P^{n - 1})
\end{align*}
Call the last inequality $R^*_{P^{n - 1}}(q' \to
p')$.

\textbf{Local, dual version:} allows us to work with functions,
F.T.

For any $R \ge 1$, any $B_R \subseteq \Rbb^n$, we
have
\[
  \left( \int_{\Rbb^n} |f(x)|^{p'}
  \right)^{\frac{1}{p'}}
  \lesssim R^{\frac{1}{q}} \left( \int |\ft
  f(\xi)|^{q'} \dd \xi \right)^{\frac{1}{q'}}
\]
for all $f \in \mathcal{S}(\Rbb^n)$ with $\supp
\ft f \subseteq N_{R^{-1}}(P^{n - 1})$.

Call this $R^{*, \text{loc}}_{P^{n - 1}}(q' \to
p')$.

$R^*_{P^{n - 1}}(q' \to p') \implies R_{p^{n -
1}}^{*, \text{loc}}(q' \to p')$. First thing
bounds $Eg$, while secound thing bounds $f$ when
we have $\supp \ft f \subseteq N_{R^{-1}}(P^{n
- 1})$.

Let $f \in \mathcal{S}(\Rbb^n)$, $\supp \ft f
\subseteq N_{R^{-1}}(P^{n - 1})$. Try to express
$f$ in trems of ext. op.

\begin{align*}
  f(x)
  &\stackrel{\text{Fourier inversion}}{=}
  \int_{\Rbb^n} e^{2\pi i x \cdot \xi} \ft f(\xi)
  \dd \xi \\
  &= \int_{\Rbb^{n - 1}} \int_\Rbb e^{2 \pi i x
  \cdot (\xi', \xi_n)} \ft f(\xi', \xi_n) \dd
  \xi_n \dd \xi' \\
  &= \int_{\Rbb} e^{2 \pi x_n \xi_n} \int_{\Rbb^{n
- 1}} e^{2 \pi i x \cdot (\xi', |\xi'|^2)}
  \ub{\ft f(\xi', |\xi'|^2 + \xi_n)}_{\eqdef
  g_{\xi_n}(\xi')} \dd \xi' \dd \xi_n
  &&\xi_n \mapsto \xi_n + |\xi'|^2 \\
  &= \int_{[-R^{-1}, R^{-1}]} e^{2 \pi x_n \xi_n}
  \ub{\int_{|\xi'| < 1} e^{2 \pi i x \cdot (\xi', |\xi'|^2)}
  \ub{\ft f(\xi', |\xi'|^2 + \xi_n)}_{\eqdef
  g_{\xi_n}(\xi')} \dd \xi'}_{E g_{\xi_n}(x)} \dd
  \xi_n \\
  \int_{\Rbb^n} |f(x)|^{p'} \dd x
  &= \int_{\Rbb^n} \left| \int_{I_{R^-1}} e^{2 \pi
  i x_n \xi_n} E g_{\xi_n}(x) \dd \xi_n
  \right|^{p'} \dd x
  &&\text{$R^*$ bounds $\int |Eg|^{p'}$} \\
  &\stackrel{\text{H\"olders's}}{\le}
  \int_{\Rbb^n} |I_{R^-1}|^{p' - 1}
  \int_{I_{R^{-1}}} |E g_{\xi_n(x)}|^{p'} \dd
  \xi_n \dd x \\
  &\lesssim \ub{(R^{-1})^{p' - 1} \int_{I_{R^{-1}}}
  \left( \int_{|\xi'| < 1} |g_{\xi_n}(\xi')|^{q'}
  \dd \xi' \right)^{\frac{p'}{q'}} \dd \xi_n}_{(*)}
  &&\text{using $R^*_{P^{n - 1}}(q' \to p')$}
\end{align*}
Goal is to bound this last expression by
\[
  R^{-\frac{p'}{q}} \ub{\int_{I_{R^{-1}}} \int_{|\xi'|
  < 1} |g_{\xi_n}(\xi')|^{q'} \dd \xi' \dd
  \xi_n}_{\|\ft f\|_{L^{q'}(N_{R^{-1}}(P^{n -
  1}))}}
\]
Lucy case: $\frac{p'}{q'} < 1$, i.e. $1 <
\frac{q'}{p'}$. Then
\[
  (*) \le R^{-p' + 1} \int_{I_{R^{-1}}}
\]

\[
  \int_A h \stackrel{\text{H\"older's}}{\le}
  |A|^{1 - \frac{1}{s}} \left( \int_A h^s
  \right)^{\frac{1}{s}}
,\]
$s \ge 1$
(actually $\approx$ since $h$ approximately
constant on $A$)