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$R_{\mathcal{P}^{n - 1}}^*(q' \to p') \implies
R_{\mathcal{P}^{n - 1}}^{*, \text{loc}}(q' \to
p')$ continued.

Spatial: $x, f(x), Eg(x)$
\[
  \boxed{\int_{\Rbb^n} |f(x)|^{p'} \dd x}
\]
\[
  \int_{\Rbb^n} |Eg(x)|^{p'} \dd x
\]
\begin{center}
  \includegraphics[width=0.3\linewidth]{images/991d92625ae644d1.png}
\end{center}

Frequency $\xi$:
\begin{center}
  \includegraphics[width=0.3\linewidth]{images/065de0e9d77b44da.png}
\end{center}
$g(\xi', |\xi'|^2) = g(\xi')$, $\supp \ft f
\subseteq \mathcal{N}_{R^{-1}}(\mathcal{P}^{n -
1})$,
\[
  \int_{\mathcal{N}_{R^{-1}}(\mathcal{P}^{n - 1})}
  |\ft f(\xi)|^{q'} \dd \xi
\]
\[
  \int_{|\xi'| < 1} |g(\xi')|^{q'} \dd \xi'
.\]

\[
  \int_{B_R} |f(x)|^{p'} \dd x
  \le \cdots
  \stackrel{(*)}{\le} R^{-(p' - 1)}
  \int_{I_{R^{-1}}} \left( \int_{|\xi'| < 1}
  |g_{\xi_n}(\xi')|^{q'} \dd \xi'
  \right)^{\frac{p'}{q'}} \dd \xi_n
.\]

Aiming for
\[
  (*) \le R^{-\frac{p'}{q}} \left(
  \int_{I_{R^{-1}}} \int_{|\xi'| < 1}
  |g_{\xi_n}(\xi')|^{q'} \dd \xi'
  \right)^{\frac{p'}{q'}}
.\]

Lucky case: $\frac{p'}{q'} \le 1$.
\[
  (*)
  \stackrel{\text{H\"older's inequality in $\xi_n$}}{\le}
  R^{-(p' - 1)} |I_{R^{-1}}|^{1 - \frac{p'}{q'}}
  \left( \int_{I_{R^{-1}}} \int_{|\xi'| < 1}
  |g_{\xi_n}(\xi')|^{q'} \dd \xi' \dd \xi_n
  \right)^{\frac{p'}{q'}}
\]
$R^{-(p' - 1)} (R^{-1})^{1 - \frac{p'}{q'}} =
R^{-\frac{p'}{q}}$ ($1 = \frac{1}{q} + \frac{1}{q'}$)

Unlucky case: $\frac{p'}{q'} > 1$. H\"older's
inequality goes in the wrong direction:
\[
  \int_A h
  \le |A|^{1 - \frac{1}{s}} \left( \int_A h^s
  \right)^{\frac{1}{s}}
\]
for all $s \ge 1$, $h \ge 0$.

This is equality if $h$ is constant on $A$.

PAUSE THIS.

\subsubsection*{Useful Harmonic Analysis tool}

The locally constant property.

Convolution: Let $f, g \in \mathcal{S}(\Rbb^n)$.
Define $f * g \in \mathcal{S}(\Rbb^n)$ by
\[
  f * g(x)
  = \int_{\Rbb^n} f(x - y) g(y) \dd y
  = \int_{\Rbb^n} f(Y) g(x - y) \dd y
.\]
See Young's convolution:
\[
  \|f * g\|_{L^r}
  \le \|f\|_p \|g\|_q
\]
when $1 + \frac{1}{r} = \frac{1}{p} +
\frac{1}{q}$.

\begin{example*}
  $g = \chi_B$ ($B$ is the unit ball in $\Rbb^n$),
 
 \[
   f * \chi_B(x)
   = \int_{y \in B} f(x - y) \dd y
 \]
 RHS is ``average value of $f$ on $B(x)$''.

 ``$f * \chi_B$ is approximately constant on balls
 of radius $\sim 1$''.
\end{example*}

\textbf{Support property:} $\supp f = A$, $\supp g
= B$, $\supp f * g \subseteq A + B = \{a + b : a
\in A, b \in B\}$.

\textbf{Convolution and Fourier Transform:}
$\ft{fg} = \ft f * \ft g$.

\textbf{Locally constant property:} Let $f \in
\mathcal{S}(\Rbb^n)$, $\supp \ft f \subseteq
B_1(0)$. Then for any unit ball $B' \subseteq
\Rbb^n$ and any $x \in B'$,
\[
  \|f\|_{L^\infty(B')}
  \lesssim_m \int_{\Rbb^n} |f|(x - y) \frac{1}{(1 +
  |y|^2)^m} \dd y
.\]

$\frac{1}{(1 + |y|^2)^m}$ is $\sim 1$ on $|y|
\lesssim 1$.
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/11afd5390c034637.png}
\end{center}
Digesting $f \ge 0$.

For any unit interval $I' \subseteq \Rbb^n$, $x
\in I'$,
\[
  \|f\|_{L^\infty(I')}
  \le \int_{\left( -\half, \half \right)} f(x - y)
  \dd y
.\]

Suppose this:
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/d750730b20024d73.png}
\end{center}
LHS has to be constant.

\[
  \|f\|_{L^\infty(I')}
  \lesssim \int_{\left( -\half, \half \right)} f(x
  - y) \dd y
.\]
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/76a43d4734324961.png}
\end{center}

\[
  \|f\|_{L^\infty(I')}
  \stackrel{(*)}{\lesssim} \int_{\left( -\half,
  \half \right)} f(x - y) \dd y
\]

\begin{fclemmastar}[Locally constant property]
  Assuming:
  - $f \in \mathcal{S}(\Rbb^n)$
  - $\supp \ft f \subseteq B_1(0)$
  Then:
  for any unit ball $B^1 \subseteq \Rbb^n$ and any
  $x \in B'$,
  \[
    \|f\|_{L^\infty(B')}
    \lesssim_m \int_{\Rbb^n} |f|(x - y)
    \frac{1}{(1 + |y|^2)^m} \dd y
  .\]
\end{fclemmastar}

The proof of this fact is more important than the
statement -- we will be using the strategy in
future.

\begin{proof}[Proof of locally constant property]
  Let $f \in \mathcal{S}(\Rbb^n)$, $\supp \ft f
  \subseteq B_1(0)$. Let $\varphi \in
  \ccinf(\Rbb^n)$ such that $\varphi \equiv 1$ on
  $B_1(0)$, $\supp \varphi \subseteq B_2(0)$.

  By Fourier inversion:
  \[
    f = \invft{(\ft f)}
    = \invft{(\ft f \varphi)}
    = \invft{(\ft f)} * \invft \varphi
    = f * \invft \varphi
  .\]
  Let $x_0, x \in B^1$ (unit ball in $\Rbb^n$).
  \begin{align*}
    |f(x_0)|
    &= \left| \int f(x_0 - y) \invft{\varphi}(y)
    \dd y \right| \\
    &\le \int |f|(x_0 - y)|\invft \varphi(y)| \dd
    y \\
    &= \int |f|(x - y)|\invft \varphi|(y - (y -
    x_0)) \dd y \\
    &\lesssim_m \int |f|(x - y) \frac{1}{(1 + |y -
    \ub{(x - x_0)}_{|x - x_0| < 1}|^2)^m} \dd y \\
    &\lesssim_m \int |f|(x - y) \frac{1}{(1 +
    |y|^2)^m}
    \qedhere
  \end{align*}
\end{proof}

\subsubsection*{Returning to $R_{\mathcal{P}^{n -
1}}^*(q' \to p') \implies R_{\mathcal{P}^{n -
1}}^{*, \text{loc}}(q' \to p')$}

\[
  \int_{B_R} |f(x)|^{p'} \dd x
  \lesssim \int_{B_R} |f \varphi_{B_R}|^{p'} \dd x
\]
where $\varphi_{B_R} \in \mathcal{S}(\Rbb^n)$
satisfies $\varphi_{B_R} \sim 1$ on $B_R$, $\supp
\ft{\varphi_{B_R}} \subseteq B_{R^{-1}}(0)$.

\[
  \ft{f\varphi_{B_R}}
  = \ub{\ft f}_{\text{$R^{-1}$ neighbourhood of
  $\mathcal{P}^{n - 1}$}} *
  \ub{\ft{\varphi_{B_R}}}_{\text{$R^{-1}$ ball}}(\xi)
\]
supported in $2R^{-1}$-neighbourhood of
$\mathcal{P}^{n - 1}$.

Repeat steps of proof:
\begin{align*}
  \int_{B_R} |f(x)|^{p'} \dd x
  &\lesssim R^{-(p' - 1)} \int_{I_{R^{-1}}} \left(
  \int_{|\xi'| < 1} |h_{\xi_n}(\xi')|^{q'} \dd
  \xi' \right)^{\frac{p'}{q'}} \dd \xi_n
\end{align*}
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/16471f52c1fc49fa.png}
\end{center}

\[
  h(\xi_n) = \boxed{\ft f *
  \ft{\varphi_{B_R}}(\xi', |\xi'|^2 + \xi_n)}
\]