%! TEX root = FR.tex
% vim: tw=50
% 03/02/2025 10AM

$\mathcal{R} = S^{n - 1} \subseteq R^n$.

Consider the statement
\[
  \|\ft f\|_{L^q(S^{n - 1})} \lesssim
  \|f\|_{L^p(\Rbb^n)}
.\]
(recall that we called this $R_{S^{n - 1}}(p \to
q)$).

Fix $\varphi \in C_c^\infty(\Rbb^n)$, $\varphi
\sim \chi_{B_1(0)}$.
For computing $L^p$ norms, $|\ft\varphi| \sim
\chi_{B_c(0)}$.

Wave packet function with localised spatial and
frequency behaviour.

Last time: $f_R(x) = e^{ix \cdot v_R}
\invft \varphi(R^{-1} x_1, \ldots, R^{-1} x_{n -
1}, R^{-2} x_n)$.

Frequency: $|\ft{f_R}(\xi)|$
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/397b1056c74c45bc.png}
\end{center}
$\int_{S^{n - 1}} |\ft{f_R}|^q \dd \sigma$
\[
  \boxed{n + 1 - \frac{n - 1}{q} \le \frac{n +
  1}{q}}
\]

Spatial: $|f_R(x)|$
\begin{center}
  \includegraphics[width=0.3\linewidth]{images/43990f389a444627.png}
\end{center}
$\int_{\Rbb^1} |f_R|^p$

Note: sphere near $\mathbf{n}$ looks like $(\xi^1,
1 - \half|\xi'|^2)$, $S^{n - 1} \cap \supp
\ft{f_R} \sim \text{cap of radius $R^{-1}$}$.
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/d8903ad031f44c7d.png}
\end{center}

\textbf{Naive attempt:} $g_R(x) = R^n \invft
\varphi(Rx)$

Frequency: $|\ft{g_R}(\xi)|$
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/8ca5c7e7fd984948.png}
\end{center}
$\int_{S^{n - 1}} |\ft{g_R}(\xi)|^q \dd \xi \sim 1$

Spatial: $|g_R(x)|$
\begin{center}
  \includegraphics[width=0.3\linewidth]{images/14d1b9a237b34bf4.png}
\end{center}
$\int |g_R|^p \sim R^{-n} \cdot R^{np}$.

Deduce: $1 \lesssim R^{-\frac{n}{p} + n}$, so
\[
  \boxed{p \ge 1}
\]
(trivial).

$|\ft{g_R}| \sim 1$ on $S^{n - 1}$ made
$\|\ft{g_R}\|_{L^q(S^{n - 1})}$ easy to compute.

Could we improve things?

Could think about $R \sim 1$: then
$\|\ft{g_R}\|_{L^q(S^{n - 1})} \sim 1$, $\|g_R\|_p
\sim 1$. This is more efficient, but we can't take
a limit. So not so useful.

Build a function $H(x)$ which satisfies $|\ft
H(\xi)| \sim 1$ on $S^{n - 1}$.

Let $\{\theta\}$ be a maximal collection of $\sim
R^{-1}$-spaced points on $S^{n - 1}$
($\#\{\theta\} \sim R^{n - 1}$).
\begin{center}
  \includegraphics[width=0.4\linewidth]{images/0d483d4581ec4568.png}
\end{center}
For each $\theta$, let $A_\theta^{-1} : \Rbb^n \to
\Rbb^n$ be an affine map which sends
\[
  B_1(0) \to R^{-1} \times \cdots \times R^{-1}
  \times R^{-2} \text{ ellipsoid centered at
  $\theta$, tangent to $S^{n - 1}$ at $\theta$}
.\]
Define $\varphi_\theta = \varphi \circ A_\theta$,
$H(x) = \sum_\theta \invft{\varphi_\theta}(x)$.

$\ft H(\xi) = \sum_\theta \varphi_\theta(\xi)
\sim 1$ on $S^{n - 1}$ (actually on
$R^{-2}$-neighbourhood of $S^{n - 1}$).

$\int_{S^{n - 1}} |\ft H(\xi)|^q \dd \sigma \sim
1$.

$H(x) = \sum_\theta \invft{\varphi_\theta}(x)$.

$|\varphi_\theta(x)|$ Frequency:
\begin{center}
  \includegraphics[width=0.3\linewidth]{images/15fd103e8226456c.png}
\end{center}


$|\invft{\varphi_\theta}(x)|$ Spatial:
\begin{center}
  \includegraphics[width=0.4\linewidth]{images/159800844fa043be.png}
\end{center}

$\ess \supp H \supseteq \bigcup_\theta \ess \supp
\invft{\varphi_\theta}$.
\begin{center}
  \includegraphics[width=0.4\linewidth]{images/d26ca6b026fc4f52.png}
\end{center}
``bush of tubes''

$\sim R^{n - 1}$ many $R \times \cdots \times R
\times R^{2}$ tubes in $R^{-1}$-separated
directions
\[
  \int_{\Rbb^n} |H(x)|^p \dd x
  = \int_{\Rbb^n} \left| \sum_\theta
  \ub{\invft{\varphi_\theta}(x)}_{\text{$\Cbb$-valued
  function}} \right|^p \dd x
\]

\begin{center}
  \includegraphics[width=0.4\linewidth]{images/ec10754f46754878.png}
\end{center}

Compute
\begin{align*}
  \int_{\Rbb^n} \left| \sum_\theta
  \chi_{T_\theta}(x)^2 \right|^{\frac{p}{2}}
  &= \int_{\Rbb^n} \left| \sum_\theta
  \chi_{T_\theta}(x) \right|^{\frac{p}{2}} \\
  &= \ub{\int_{B_R} \left| \sum \chi_{T_\theta}
  \right|^{\frac{p}{2}}}_{(R^{n -
  1})^{\frac{p}{2}} \cdot R^n}
  + \ub{\int_{R_{R^2} \setminus B_R} \left| \sum
  \chi_{T_\theta} \right|^{\frac{p}{2}}}_{(*)}
\end{align*}
Consider overlap of $T_\theta$ on $\lambda S^{n -
1}$ ($\lambda \in (R, R^{2})$).

Average overlap on $\lambda S^{n - 1}$:
\[
  \average_{\lambda S^{n - 1}} \sum_\theta
  \chi_{T_\theta} \sim \lambda^{-(n - 1)}
  \sum_\theta \int_{\lambda S^{n - 1}}
  \chi_{T_\theta}
  \sim \lambda^{-(n - 1)} \sum_\theta R^{n - 1}
  = \lambda^{-(n - 1)} R^{2(n - 1)}
.\]
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/0f5b7a257bb94284.png}
\end{center}
Not too hard to check that the number of active
$T_\theta$ on $\lambda S^{n - 1}$ is $\sim
\lambda^{-(n - 1)} R^{2(n - 1)}$.

Now calculate:
\[
  (*) \sim \sum_{R < \lambda < R^2} \int_{\lambda
  < |x| < 2\lambda} \ub{\left| \sum_\theta
  \chi_{T_\theta}
  \right|^{\frac{p}{2}}}_{[\lambda^{-(n - 1)}
  R^{2(n - 1)}]^{\frac{p}{2}}
  \lambda^n} \dd x
.\]

\[
  1 \lesssim R^{-(n + 1) p} [R^{2n} + R^{(n - 1)
  \frac{p}{2}} R^n]
.\]
Two cases: either $R^{2n}$ dominates or the other term
dominates. So $\boxed{p \le \frac{2n}{n + 1}}$.