4 Introduction to Fourier Restriction

Are there any other $(p,q)$ for which

$$\parallel \widehat{f}{\parallel }_q\lesssim \parallel f{\parallel }_p?\text{(∗)}$$

We saw that $1\le p\le 2$ was necessary (translations / modulations, Khinchin’s inequality).

Scaling: Plug in $f_{\lambda }(x)=f(\lambda x)$ which is $L^{\infty }$-normalised ($\parallel f_{\lambda }{\parallel }_{\infty }=\parallel f{\parallel }_{\infty }$). Then $\widehat{f_{\lambda }}(\xi )={\lambda }^{-n}\widehat{f}({\lambda }^{-1}\xi )$ which is $L^1$-normalised ($\parallel \widehat{f_{\lambda }}{\parallel }_1=\parallel \widehat{f}{\parallel }_1$).

$${(\text{LHS of (}\ast \text{)})}^q={\int }_{{ℝ}^n}|\widehat{f_{\lambda }}(\xi ){|}^{q}d\xi ={\lambda }^{-nq+n}\int |\widehat{f}(\xi ){|}^{q}d\xi .$$

$${(\text{RHS of (}\ast \text{)})}^p={\int }_{{ℝ}^n}|f_{\lambda }(x){|}^{p}dx={\lambda }^{-n}\int |f(x){|}^{p}dx.$$

So we need for all $\lambda >0$:

$${\lambda }^{-n+\frac{n}{q}}\parallel \widehat{f}{\parallel }_q\lesssim {\lambda }^{-\frac{n}{p}}\parallel f{\parallel }_p.$$

So we need $-n+\frac{n}{q}=-\frac{n}{p}$, i.e. $\frac{1}{p}+\frac{1}{q}=1$.

Classical questions

What is $C_{p,q}$ the smallest constant such that $\parallel \widehat{f}{\parallel }_q\le C_{p,q}\parallel f{\parallel }_p$?

Which functions $f$ satisfy $\frac{\parallel \widehat{f}{\parallel }_q}{\parallel f{\parallel }_p}=C_{p,q}$?

2014:

$$\parallel \widehat{f}{\parallel }_q\le C_{p,q}\parallel f{\parallel }_p-{\mathrm{dist}}_p(f,\text{maximisers (Gaussian)}).$$

Fourier restriction asks which $(p,q)$ permit estimates $\parallel \widehat{f}{\parallel }_{L^q(R)}\lesssim \parallel f{\parallel }_{L^p({ℝ}^n)}$ ($R$ is the restricted frequency set, $R\subseteq {ℝ}^n$).

$R_R(p\to q)$ means

$$\parallel \widehat{f}{\parallel }_{L^q(R)}\lesssim \parallel f{\parallel }_{L^p({ℝ}^n)}\forall f:{ℝ}^n\to ℂ\text{ Schwartz}$$

Always: $R_R(1\to \infty )$ true for all $R$.

In the second example, only this trivial statement is true (i.e. $R_R(p\to q)$ is false for all other values for $p,q$).

Let $R=S^{n-1}$ be the unit sphere in ${ℝ}^n$. Consider

$$\parallel \widehat{f}{\parallel }_{L^q(S^{n-1})}\lesssim \parallel f{\parallel }_{L^p({ℝ}^n)}$$

where $L^q(S^{n-1})$ uses the usual surface measure $d\sigma$ on $S^{n-1}$.

Let $\phi \in C_c^{\infty }({ℝ}^n)$, $ℝ$-valued, $\ge \frac{1}{2}$ on $B_1(0)$, with support in $B_2(0)$.

May also assume $ˇ\phi$ is $ℝ$-valued, $ˇ\phi \gtrsim 1$ on $B_c(0)$. $ˇ\phi$ bounded, $|ˇ\phi (x)|{\lesssim }_m{(|x{|}^2+1)}^{-m}\forall m$. $|ˇ\phi (x)|$ behaves like ${\chi }_{B_1(0)}$ in $L^p$.

Consider dilates $f_R(x)=ˇ\phi (R^{-1}x)e^{2\pi ix\cdot v_r}$, $R\gg 1$.

Frequency side $|\widehat{f_R}(\xi )|\sim R^n{\chi }_{B_{R^{-1}}(0)}(x)$ ($L^1$-norm).

$${\int }_{S^{n-1}}|\widehat{f_R}(\xi ){|}^{q}d\sigma (\xi )\sim \underset{\to \sigma (B_{R^{-1}}(n)\cap S^{n-1})}{\underbrace{R^{nq}{R}^{-(n-1)}}}.$$

$B_{R^{-1}}(n)\cap S^{n-1}$ cap of radius $R^{-1}$. Spacial side $|f_R(x)|\sim {\chi }_{B_R(0)}(x)$ (in $L^{\infty }$-norm).

height $1$, ${\int }_{{ℝ}^n}|f_R(x){|}^{p}dx\sim R^n$.

$R^{n-\frac{n-1}{q}}\lesssim R^{\frac{n}{p}}$, $n-\frac{n-1}{q}\le \frac{n}{p}$.

Consider: $g_R(x)=e^{ix\cdot w_r}ˇ\phi (R^{-1}{x}_1,\dots ,R^{-1}{x}_{n-1},R^{-2}{x}_n)$.

Frequency: $|\widehat{g_r}(\xi )|\approx |cylinder{|}^{-1}{\chi }_{cylinder}(\xi )$. $|cylinder{|}^{-1}|$ $\to$ ${(R-1\cdot R^{-1}{R}^{-2})}^{-1}=R^{n+1}$.

$${\int }_{S^{n-1}}|\widehat{g_R}(\xi ){|}^{q}d\sigma (\xi )\sim R^{(n+1)q}\sigma (\text{cap of radius }{R}^{-1})=R^{(n+1)q-(n-1)}.$$

Spatial side: $|g_{R^{\prime }}(x)|\approx {\chi }_{cylinder}(x)$ ($L^{\infty }$-norm).

$\int |g_R(x){|}^p=R^{n-1+2}=R^{n+1}$.

$\to$ $R^{n+1-\frac{n-1}{q}}\lesssim R^{\frac{n+1}{q}}$, $⟹n+1-\frac{n-1}{q}\le \frac{n+1}{p}$. Implies B.

On Monday, we will build examples $h$ such that $\widehat{h}$ sees all of $S^{n-1}$.

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

Consider the statement

$$\parallel \widehat{f}{\parallel }_{L^q(S^{n-1})}\lesssim \parallel f{\parallel }_{L^p({ℝ}^n)}.$$

(recall that we called this $R_{S^{n-1}}(p\to q)$).

Fix $\phi \in C_c^{\infty }({ℝ}^n)$, $\phi \sim {\chi }_{B_1(0)}$. For computing $L^p$ norms, $|\widehat{\phi }|\sim {\chi }_{B_c(0)}$.

Wave packet function with localised spatial and frequency behaviour.

Last time: $f_R(x)=e^{ix\cdot v_R}ˇ\phi (R^{-1}{x}_1,\dots ,R^{-1}{x}_{n-1},R^{-2}{x}_n)$.

Frequency: $|\widehat{f_R}(\xi )|$

${\int }_{S^{n-1}}|\widehat{f_R}{|}^{q}d\sigma$

$$\boxed{n+1-\frac{n-1}{q}\le \frac{n+1}{q}}$$

Spatial: $|f_R(x)|$

${\int }_{{ℝ}^1}|f_R{|}^p$

Note: sphere near $n$ looks like $({\xi }^1,1-\frac{1}{2}|{\xi }^{\prime }{|}^2)$, $S^{n-1}\cap \mathrm{supp}\widehat{f_R}\sim \text{cap of radius }{R}^{-1}$.

Naive attempt: $g_R(x)=R^nˇ\phi (Rx)$

Frequency: $|\widehat{g_R}(\xi )|$

${\int }_{S^{n-1}}|\widehat{g_R}(\xi ){|}^{q}d\xi \sim 1$

Spatial: $|g_R(x)|$

$\int |g_R{|}^p\sim R^{-n}\cdot R^{np}$.

Deduce: $1\lesssim R^{-\frac{n}{p}+n}$, so

$$\boxed{p\ge 1}$$

(trivial).

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

Could we improve things?

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

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

Let $\{𝜃\}$ be a maximal collection of $\sim R^{-1}$-spaced points on $S^{n-1}$ ($\#\{𝜃\}\sim R^{n-1}$).

For each $𝜃$, let $A_{𝜃}^{-1}:{ℝ}^n\to {ℝ}^n$ be an affine map which sends

$$B_1(0)\to R^{-1}\times \cdots \times R^{-1}\times R^{-2}\text{ ellipsoid centered at }𝜃\text{, tangent to }{S}^{n-1}\text{ at }𝜃.$$

Define ${\phi }_{𝜃}=\phi \circ A_{𝜃}$, $H(x)={\sum }_{𝜃}ˇ{\phi }_{𝜃}(x)$.

$\widehat{H}(\xi )={\sum }_{𝜃}{\phi }_{𝜃}(\xi )\sim 1$ on $S^{n-1}$ (actually on $R^{-2}$-neighbourhood of $S^{n-1}$).

${\int }_{S^{n-1}}|\widehat{H}(\xi ){|}^{q}d\sigma \sim 1$.

$H(x)={\sum }_{𝜃}ˇ{\phi }_{𝜃}(x)$.

$|{\phi }_{𝜃}(x)|$ Frequency:

$|ˇ{\phi }_{𝜃}(x)|$ Spatial:

$\mathrm{ess}\mathrm{supp}H\supseteq {\bigcup }_{𝜃}\mathrm{ess}\mathrm{supp}ˇ{\phi }_{𝜃}$.

“bush of tubes”

$\sim R^{n-1}$ many $R\times \cdots \times R\times R^2$ tubes in $R^{-1}$-separated directions

$${\int }_{{ℝ}^n}|H(x){|}^{p}dx={\int }_{{ℝ}^n}{\left|\sum _{𝜃}\underset{ℂ\text{-valued function}}{\underbrace{ˇ{\phi }_{𝜃}(x)}}\right|}^{p}dx$$

Compute

$$\begin{array}{llll}\hfill {\int }_{{ℝ}^n}{\left|\sum _{𝜃}{\chi }_{T_{𝜃}}{(x)}^2\right|}^{\frac{p}{2}}& ={\int }_{{ℝ}^n}{\left|\sum _{𝜃}{\chi }_{T_{𝜃}}(x)\right|}^{\frac{p}{2}}& \hfill & \\ \hfill & =\underset{{(R^{n-1})}^{\frac{p}{2}}\cdot R^n}{\underbrace{{\int }_{B_R}{\left|\sum {\chi }_{T_{𝜃}}\right|}^{\frac{p}{2}}}}+\underset{(\ast )}{\underbrace{{\int }_{R_{R^2}\setminus B_R}{\left|\sum {\chi }_{T_{𝜃}}\right|}^{\frac{p}{2}}}}& \hfill & \end{array}$$

Consider overlap of $T_{𝜃}$ on $\lambda S^{n-1}$ ($\lambda \in (R,R^2)$).

Average overlap on $\lambda S^{n-1}$:

$$-{\int }_{\lambda S^{n-1}}\sum _{𝜃}{\chi }_{T_{𝜃}}\sim {\lambda }^{-(n-1)}\sum _{𝜃}{\int }_{\lambda S^{n-1}}{\chi }_{T_{𝜃}}\sim {\lambda }^{-(n-1)}\sum _{𝜃}{R}^{n-1}={\lambda }^{-(n-1)}{R}^{2(n-1)}.$$

Not too hard to check that the number of active $T_{𝜃}$ on $\lambda S^{n-1}$ is $\sim {\lambda }^{-(n-1)}{R}^{2(n-1)}$.

Now calculate:

$$(\ast )\sim \sum _{R<\lambda <R^2}{\int }_{\lambda <|x|<2\lambda }\underset{{[{\lambda }^{-(n-1)}{R}^{2(n-1)}]}^{\frac{p}{2}}{\lambda }^n}{\underbrace{{\left|\sum _{𝜃}{\chi }_{T_{𝜃}}\right|}^{\frac{p}{2}}}}dx.$$

$$1\lesssim R^{-(n+1)p}[R^{2n}+R^{(n-1)\frac{p}{2}}{R}^n].$$

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