6 Tube incidence implications of Fourier restriction

Last time: LOCALLY CONSTANT PROPERTY (think of it more as “heuristic”).

If $\mathrm{supp}\widehat{f}\subseteq B_1$ then

• (1) Imagine $|f|\sim \text{const}$ on unit balls.
• (2) $f=ˇ(\widehat{{\phi }_{B_1}})=f\ast ˇ{\phi }_{B_1}$.

What if $\mathrm{supp}\widehat{f}\subseteq B_{\lambda }(0)$ (so $|f|\sim \text{const}$ on ${\lambda }^{-1}$-balls)?

$$f=ˇ(\widehat{f}{\phi }_{B_{\lambda }})=f\ast ˇ{\phi }_{B_{\lambda }},$$

where ${\phi }_{B_{\lambda }}(\xi )={\phi }_{B_1}({\lambda }^{-1}\xi )$.

$$ˇ{\phi }_{B_{\lambda }}(x)={\int }_{{ℝ}^n}{e}^{2\pi ix\cdot \xi }{\phi }_{B_1}({\lambda }^{-1}\xi )d\xi ={\lambda }^n{\int }_{{ℝ}^n}{e}^{2\pi i(\lambda x)\cdot \xi }{\phi }_{B_1}(\xi )d\xi ={\lambda }^nˇ{\phi }_{B_1}(\lambda x).$$

$\ast |ˇ{\phi }_{B_1}|$ is approximately averaging over a $1$-ball. $\ast |ˇ{\phi }_{B_{\lambda }}|$ is approximately averaging over ${\lambda }^{-1}$-balls.

What about $\mathrm{supp}\widehat{f}\subseteq B_{\lambda }(\overline{v})$?

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 $R_{P^{n-1}}^{\ast }(q^{\prime }\to p^{\prime })⟹R_{P^{n-1}}^{\ast ,\text{loc}}(q^{\prime }\to p^{\prime })$.

$${\int }_{B_R}|f(x){|}^{p^{\prime }}dx\lesssim {\int }_{B_R}|f(x){\phi }_{B_R}(x){|}^{p^{\prime }}dx$$

${\phi }_{B_R}(x)\in S({ℝ}^n)$, $|{\phi }_{B_R}|\sim 1$ on $B_R$, $\mathrm{supp}\widehat{{\phi }_{B_R}}\subseteq B_{R^{-1}}(0)$.

Last lecture $\to$

$${\int }_{I_{R^{-1}}}\left({\int }_{\begin{array}{c}|{\xi }^{\prime }|<1\\ {\xi }^{\prime }\in {ℝ}^{n-1}\end{array}}|\widehat{f}\ast \widehat{{\phi }_{B_R}}({\xi }^{\prime },|{\xi }^{\prime }{|}^2+{\xi }_n){|}^{q^{\prime }}d{\xi }^{\prime }\right)\stackrel{(\ast )}{\lesssim }{(R^{-1})}^{1-\frac{p^{\prime }}{q^{\prime }}}{\left({\int }_{\xi \in {ℝ}^n}|\widehat{f}(\xi ){|}^{q^{\prime }}d\xi \right)}^{\frac{p^{\prime }}{q^{\prime }}}.$$

Can choose $\widehat{{\phi }_{B_R}}$ such that

• $|\widehat{{\phi }_{B_R}}|\sim R^n{\chi }_{B_{R^{-1}}(0)}$.

• $\parallel \widehat{{\phi }_{B_R}}{\parallel }_1\sim 1$.

Case 1: $\frac{p^{\prime }}{q^{\prime }}\le 1$. LHS of $(\ast )$ (using Hölder) is

$$\begin{array}{llllll}\hfill & \le |I_{R^{-1}}{|}^{1-\frac{p^{\prime }}{q^{\prime }}}{\left({\int }_{I_{R^{-1}}}{\int }_{|{\xi }^{\prime }|<1}{(|\widehat{f}|\ast |\widehat{{\phi }_{B_R}}{|}^{\frac{1}{q^{\prime }}+\frac{1}{q}}({\xi }^{\prime },|{\xi }^{\prime }{|}^2+{\xi }_n))}^{q^{\prime }}d{\xi }^{\prime }d{\xi }_n\right)}^{\frac{p^{\prime }}{q^{\prime }}}& \hfill & & \hfill & \\ \hfill & \sim |I_{R^{-1}}{|}^{1-\frac{p^{\prime }}{q^{\prime }}}{\left({\int }_{{ℝ}^n}{(|\widehat{f}|\ast |\widehat{{\phi }_{B_R}}|(\xi ))}^{q^{\prime }}dd\xi \right)}^{\frac{p^{\prime }}{q^{\prime }}}& \hfill & & \hfill & \\ \hfill & \le {\left(\int |\widehat{f}{|}^{q^{\prime }}(\xi -\eta )|\widehat{{\phi }_{B_R}}|(\eta )d\eta \right)}^{\frac{1}{q^{\prime }}}\underset{\sim 1}{\underbrace{{\left(\int |\widehat{{\phi }_{B_R}}{|}^{\frac{q}{q}}\right)}^{\frac{1}{q}}}}& \hfill & \text{(Holder)}& \hfill & & \hfill & \\ \hfill & \lesssim |I_{R^{-1}}{|}^{1-\frac{p^{\prime }}{q^{\prime }}}{\left({\int }_{{ℝ}^n}{\int }_{{ℝ}^n}|\widehat{f}{|}^{q^{\prime }}(\xi -\eta )|\widehat{{\phi }_{B_R}}|(\eta )d\eta d\xi \right)}^{\frac{p^{\prime }}{q^{\prime }}}& \hfill & & \hfill & \\ \hfill & \sim |I_{R^{-1}}{|}^{1-\frac{p^{\prime }}{q^{\prime }}}{\left({\int }_{{ℝ}^n|\widehat{f}{|}^{q^{\prime }}(\xi )d\xi }\right)}^{\frac{p^{\prime }}{q^{\prime }}}& \hfill & & \hfill & \end{array}$$

Case 2: $\frac{p^{\prime }}{q^{\prime }}>1$. Use $P^1\subseteq {ℝ}^2$ for intuition.

Imagine a function $g$ which is approximately constant on each $R^{-1}$ cube $Q$. Think of $g$ as $g={\sum }_Q{g}_Q$.

$${\int }_{I_{R^{-1}}}{\left({\int }_{|t|<1}g(t,t^2+{\xi }_2)dt\right)}^{\frac{p^{\prime }}{q^{\prime }}}d{\xi }_2.$$

Note

$${\int }_{|t|<1}g(t,t^2+{\xi }_2)dt\sim g_Q.$$

Therefore,

$${\int }_{|t|<1}g(t,t^2+{\xi }_2)dt\sim C$$

if $|{\xi }_2|\le c{R}^{-1}$. ($\le C$ for all ${\xi }_2\in I_{R^{-1}}$).

$|(P^{\prime }+(0,{\xi }_2))\cap Q|\sim R^{-1}$ for $|{\xi }_2|\le c{R}^{-1}$.

$\sim |I_{R^{-1}}{|}^{1-\frac{p^{\prime }}{q}+\frac{p}{q^{\prime }}}{C}^{\frac{p^{\prime }}{q^{\prime }}}=|I_{R^{-1}}{|}^{1-\frac{p^{\prime }}{q^{\prime }}}{(|I_{R^{-1}}|C)}^{\frac{p^{\prime }}{q^{\prime }}}{\left({\int }_{R^{-1}}{\int }_{|t|<1}g(t,t^2+{\xi }_2)dtd{\xi }_2\right)}^{\frac{p^{\prime }}{q^{\prime }}}$

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

Make the intuition rigorous.

$$\text{LHS of }(\ast )\lesssim |I_{R^{-1}}|\underset{{\xi }_2\in I_{R^{-1}}}{\max }{\left({\int }_{|{\xi }^{\prime }|<1}{(|\widehat{f}|\ast |\widehat{{\phi }_{B_R}}|({\xi }^{\prime },|{\xi }^{\prime }{|}^2+{\xi }_n))}^{q^{\prime }}d{\xi }^{\prime }\right)}^{\frac{p^{\prime }}{q^{\prime }}}$$

Consider the integral:

$$\begin{array}{llllll}\hfill {\int }_{|{\xi }^{\prime }|<1}{(|\widehat{f}|\ast |\widehat{{\phi }_{B_R}}|({\xi }^{\prime },|{\xi }^{\prime }{|}^2+{\xi }_n))}^{q^{\prime }}d{\xi }^{\prime }& ={\int }_{|{\xi }^{\prime }|<1}{\left({\int }_{{ℝ}^n}|\widehat{f}|(\eta )|\widehat{{\phi }_{B_R}}|(({\xi }^{\prime },|{\xi }^{\prime }{|}^2+{\xi }_n)-\eta )d\eta \right)}^{q^{\prime }}d{\xi }^{\prime }& \hfill & & \hfill & \\ \hfill & \le {\int }_{|{\xi }^{\prime }|<1}\left({\int }_{{ℝ}^n}|\widehat{f}{|}^{q^{\prime }}(\eta )|\widehat{{\phi }_{B_R}}|(({\xi }^{\prime },|{\xi }^{\prime }{|}^2+{\xi }_n)-\eta )d\eta \right)d{\xi }^{\prime }& \hfill & \text{Same pointwise Holder}& \hfill & & \hfill & \\ \hfill & \sim R^n{(R^{-1})}^{n-1}{\int }_{{ℝ}^n}|\widehat{f}{|}^{q^{\prime }}(\eta )d\eta & \hfill & & \hfill & \\ \hfill & \sim R^1{\int }_{{ℝ}^n}|\widehat{f}{|}^{q^{\prime }}& \hfill & & \hfill & \\ \hfill & \lesssim {(R^{-1})}^1\cdot R^{\frac{p^{\prime }}{q^{\prime }}}{\left(\int |\widehat{f}{|}^{q^{\prime }}\right)}^{\frac{p^{\prime }}{q^{\prime }}}& \hfill & & \hfill & \end{array}$$