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\newpage
\section{Tube incidence implications of Fourier
restriction}

Last time: LOCALLY CONSTANT PROPERTY (think of it
more as ``heuristic'').

If $\supp \ft f \subseteq B_1$ then
\begin{enumerate}[(1)]
  \item Imagine $|f| \sim \text{const}$ on unit
    balls.
  \item $f = \invft{(\ft{\varphi_{B_1}})} = f *
    \invft{\varphi_{B_1}}$.
\end{enumerate}
What if $\supp \ft f \subseteq B_\lambda(0)$ (so $|f|
\sim \text{const}$ on $\lambda^{-1}$-balls)?
\[
  f = \invft{(\ft f \varphi_{B_\lambda})}
  = f * \invft{\varphi_{B_\lambda}}
,\]
where $\varphi_{B_\lambda}(\xi) =
\varphi_{B_1}(\lambda^{-1} \xi)$.
\[
  \invft{\varphi_{B_\lambda}}(x)
  = \int_{\Rbb^n} e^{2\pi i x \cdot \xi}
  \varphi_{B_1}(\lambda^{-1} \xi) \dd \xi
  = \lambda^n \int_{\Rbb^n} e^{2\pi i(\lambda x)
  \cdot \xi} \varphi_{B_1}(\xi) \dd \xi
  = \lambda^n \invft{\varphi_{B_1}}(\lambda x)
.\]
$* |\invft{\varphi_{B_1}}|$ is approximately
averaging over a $1$-ball. $*
|\invft{\varphi_{B_\lambda}}|$ is approximately
averaging over $\lambda^{-1}$-balls.

What about $\supp \ft f \subseteq
B_\lambda(\ol{v})$?

Same thing happens, because $e^{ix \cdot
{\text{something}}} f$ will have Fourier support in
$B_\lambda(0)$, and taking absolute values means
we don't notice the $e^{ix \cdot
\text{something}}$ (modulation).

Returning to $R_{\mathcal{P}^{n - 1}}^*(q' \to p')
\implies R_{\mathcal{P}^{n - 1}}^{*,
\text{loc}}(q' \to p')$.
\[
  \int_{B_R} |f(x)|^{p'} \dd x
  \lesssim \int_{B_R} |f(x) \varphi_{B_R}(x)|^{p'}
  \dd x
\]
$\varphi_{B_R}(x) \in \mathcal{S}(\Rbb^n)$,
$|\varphi_{B_R}| \sim 1$ on $B_R$, $\supp
\ft{\varphi_{B_R}} \subseteq B_{R^{-1}}(0)$.

Last lecture $\to$
\[
  \int_{I_{R^{-1}}} \left( \int_{\substack{|\xi'|
  < 1 \\ \xi' \in \Rbb^{n - 1}}} |\ft f *
  \ft{\varphi_{B_R}} (\xi', |\xi'|^2 + \xi_n)|^{q'}
  \dd \xi'\right)
  \stackrel{(*)}{\lesssim}
  (R^{-1})^{1 - \frac{p'}{q'}} \left( \int_{\xi
  \in \Rbb^n} |\ft f(\xi)|^{q'} \dd \xi
  \right)^{\frac{p'}{q'}}
.\]

Can choose $\ft{\varphi_{B_R}}$ such that
\begin{itemize}
  \item $|\ft{\varphi_{B_R}}| \sim R^n
    \chi_{B_{R^{-1}}(0)}$.
  \item $\|\ft{\varphi_{B_R}}\|_1 \sim 1$.
\end{itemize}

\textbf{Case 1:} $\frac{p'}{q'} \le 1$. LHS of
$(*)$ (using H\"older) is
\begin{align*}
  &\le |I_{R^{-1}}|^{1 - \frac{p'}{q'}}
  \left( \int_{I_{R^{-1}}} \int_{|\xi'| < 1} (|\ft
  f| * |\ft{\varphi_{B_R}}|^{\frac{1}{q'} +
  \frac{1}{q}}(\xi', |\xi'|^2 + \xi_n))^{q'} \dd
  \xi' \dd \xi_n \right)^{\frac{p'}{q'}} \\
  &\sim |I_{R^{-1}}|^{1 - \frac{p'}{q'}}
  \left( \int_{\Rbb^n} (|\ft f| *
  |\ft{\varphi_{B_R}}|(\xi))^{q'} \dd \dd\xi
  \right)^{\frac{p'}{q'}} \\
  &\le \left( \int |\ft f|^{q'}(\xi - \eta)
  |\ft{\varphi_{B_R}}|(\eta) \dd \eta
  \right)^{\frac{1}{q'}} \ub{\left( \int
  |\ft{\varphi_{B_R}}|^{\frac{q}{q}}
  \right)^{\frac{1}{q}}}_{\sim 1}
  &&\text{(Holder)} \\
  &\lesssim |I_{R^{-1}}|^{1 - \frac{p'}{q'}}
  \left( \int_{\Rbb^n} \int_{\Rbb^n} |\ft f|^{q'}
  (\xi - \eta) |\ft{\varphi_{B_R}}|(\eta) \dd \eta
  \dd \xi \right)^{\frac{p'}{q'}} \\
  &\sim |I_{R^{-1}}|^{1 - \frac{p'}{q'}} \left(
  \int_{\Rbb^n |\ft f|^{q'}(\xi) \dd \xi}
  \right)^{\frac{p'}{q'}}
\end{align*}

\textbf{Case 2:} $\frac{p'}{q'} > 1$. Use
$\mathcal{P}^1 \subseteq \Rbb^2$ for intuition.
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/27d5e85a481544c7.png}
\end{center}
Imagine a function $g$ which is approximately
constant on each $R^{-1}$ cube $Q$. Think of $g$
as $g = \sum_Q g_Q$.
\[
  \int_{I_{R^{-1}}} \left( \int_{|t| < 1}
  g(t, t^2 + \xi_2) \dd t \right)^{\frac{p'}{q'}}
  \dd \xi_2
.\]
Note
\[
  \int_{|t| < 1} g(t, t^2 + \xi_2) \dd t \sim g_Q
.\]
Therefore,
\[
  \int_{|t| < 1} g(t, t^2 + \xi_2) \dd t \sim C
\]
if $|\xi_2| \le cR^{-1}$.
($\le C$ for all $\xi_2 \in I_{R^{-1}}$).

$|(P' + (0, \xi_2)) \cap Q| \sim R^{-1}$ for
$|\xi_2| \le cR^{-1}$.

$\sim |I_{R^{-1}}|^{1 - \frac{p'}{q} +
\frac{p}{q'}} C^{\frac{p'}{q'}} = |I_{R^{-1}}|^{1
- \frac{p'}{q'}} (|I_{R^{-1}}| C)^{\frac{p'}{q'}}
\left( \int_{R^{-1}} \int_{|t| < 1} g(t, t^2 +
\xi_2) \dd t \dd \xi_2 \right)^{\frac{p'}{q'}}$

Important: locally constant property means we
didn't need $\frac{p'}{q'} < 1$, like before.

Make the intuition rigorous.
\[
  \text{LHS of $(*)$}
  \lesssim |I_{R^{-1}}| \max_{\xi_2 \in
  I_{R^{-1}}} \left( \int_{|\xi'| < 1} (|\ft f| *
  |\ft{\varphi_{B_R}}|(\xi', |\xi'|^2 +
  \xi_n))^{q'} \dd
  \xi'\right)^{\frac{p'}{q'}}
\]
Consider the integral:
\begin{align*}
  \int_{|\xi'| < 1} (|\ft f| *
  |\ft{\varphi_{B_R}}|(\xi', |\xi'|^2 +
  \xi_n))^{q'} \dd
  \xi'
  &= \int_{|\xi'| < 1} \left( \int_{\Rbb^n} |\ft
  f|(\eta) |\ft{\varphi_{B_R}}|((\xi', |\xi'|^2 +
  \xi_n) - \eta) \dd \eta \right)^{q'} \dd \xi' \\
  &\le \int_{|\xi'| < 1} \left( \int_{\Rbb^n} |\ft
  f|^{q'} (\eta) |\ft{\varphi_{B_R}}|((\xi',
  |\xi'|^2 + \xi_n) - \eta) \dd \eta \right) \dd
  \xi'
  &&\text{Same pointwise Holder}
  \\
  &\sim R^n (R^{-1})^{n - 1} \int_{\Rbb^n} |\ft
  f|^{q'}(\eta) \dd \eta \\
  &\sim R^1 \int_{\Rbb^n} |\ft f|^{q'} \\
  &\lesssim (R^{-1})^1 \cdot R^{\frac{p'}{q'}}
  \left( \int |\ft f|^{q'} \right)^{\frac{p'}{q'}}
\end{align*}