Introduction to Fourier Restriction

4 Introduction to Fourier Restriction

Theorem (Hausdorff-Young inequality). Assuming that:

$1 \leq p \leq 2$
$\frac{1}{p} + \frac{1}{q} = 1$

Then

Proof. The inequality is true for $p = 1$ , $q = \infty$ and for $p = 2$ , $q = 2$ the inequality is true since we have equality (Plancherel).

For values in between we can interpolate. □

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

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

Scaling: Plug in $f_{λ} (x) = f (λ x)$ which is $L^{\infty}$ -normalised ( $∥ f_{λ} ∥_{\infty} = ∥ f ∥_{\infty}$ ). Then $\hat{f_{λ}} (ξ) = λ^{- n} \hat{f} (λ^{- 1} ξ)$ which is $L^{1}$ -normalised ( $∥ \hat{f_{λ}} ∥_{1} = ∥ \hat{f} ∥_{1}$ ).

∫ ∫
(LHS of (∗))q = |^fλ(ξ)|qdξ = λ−nq+n |^f(ξ)|qdξ.
ℝn

∫ ∫
p p − n p
(RHS of (∗)) = ℝn |fλ(x)| dx = λ |f(x)| dx.

So we need for all

λ > 0

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 $∥ \hat{f} ∥_{q} \leq C_{p, q} ∥ f ∥_{p}$ ?

Which functions $f$ satisfy $\frac{∥ \hat{f} ∥_{q}}{∥ f ∥_{p}} = C_{p, q}$ ?

2014:

∥^f∥q ≤ Cp,q∥f ∥p − distp(f,maximisers (Gaussian)).

Fourier restriction asks which

(p, q)

permit estimates

∥ \hat{f} ∥_{L^{q} (R)} ≲ ∥ f ∥_{L^{p} (ℝ^{n})}

(

R

is the restricted frequency set,

R \subseteq ℝ^{n}

Example. $R = B_{1} (0) \subseteq ℝ^{n}$ , unit ball $\to$ : Basically, sitll governed by Hausdorff-Young inequality.

Example. $R = B_{1} (0) \cap {x_{n} = 0}$ (measure $0$ subset of $ℝ^{n}$ ).

$R_{R} (p \to q)$ means

n
∥^f∥Lq(ℛ ) ≲ ∥f∥Lp(ℝn) ∀f : ℝ → ℂ 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

where

L^{q} (S^{n - 1})

uses the usual surface measure

d σ

S^{n - 1}

Notation.

^^ˇx↦→− fx(x) f^(x) f(x) f(x)
◟◝◜◞ ◟◝◜◞ ◟◝◜◞ ◟◝◜◞
∈𝒮(ℝn) ∈𝒮(ℝn) ∈𝒮(ℝn) ∈ 𝒮(ℝn)

We may call

ˇ

the “inverse Fourier transform”.

Let $φ \in C_{c}^{\infty} (ℝ^{n})$ , $ℝ$ -valued, $\geq \frac{1}{2}$ on $B_{1} (0)$ , with support in $B_{2} (0)$ .

May also assume $ˇ φ$ is $ℝ$ -valued, $ˇ φ ≳ 1$ on $B_{c} (0)$ . $ˇ φ$ bounded, $| ˇ φ (x) | ≲_{m} {(| x |^{2} + 1)}^{- m} \forall m$ . $| ˇ φ (x) |$ behaves like $χ_{B_{1} (0)}$ in $L^{p}$ .

Consider dilates $f_{R} (x) = ˇ φ (R^{- 1} x) e^{2 π i x \cdot v_{r}}$ , $R ≫ 1$ .

Frequency side $| \hat{f_{R}} (ξ) | \sim R^{n} χ_{B_{R^{- 1}} (0)} (x)$ ( $L^{1}$ -norm).

∫
|^fR(ξ)|qdσ(ξ) ∼ RnqR −(n− 1) .
Sn−1 →σ◟(B-−◝1◜(n)∩S◞n−1)
R

B_{R^{- 1}} (n) \cap S^{n - 1}

cap of radius

R^{- 1}

. Spacial side

| f_{R} (x) | \sim χ_{B_{R} (0)} (x)

(in

L^{\infty}

-norm).

height $1$ , $\int_{ℝ^{n}} | f_{R} (x) |^{p} d x \sim R^{n}$ .

$R^{n - \frac{n - 1}{q}} ≲ R^{\frac{n}{p}}$ , $n - \frac{n - 1}{q} \leq \frac{n}{p}$ .

Consider: $g_{R} (x) = e^{i x \cdot w_{r}} ˇ φ (R^{- 1} x_{1}, \dots, R^{- 1} x_{n - 1}, R^{- 2} x_{n})$ .

Frequency: $| \hat{g_{r}} (ξ) | \approx | c y l i n d e r |^{- 1} χ_{c y l i n d e r} (ξ)$ . $| c y l i n d e r |^{- 1} |$ $\to$ ${(R - 1 \cdot R^{- 1} R^{- 2})}^{- 1} = R^{n + 1}$ .

∫
|^gR(ξ)|qdσ(ξ) ∼ R (n+1)qσ (cap of radius R−1) = R(n+1)q−(n−1).
Sn−1

Spatial side: $| g_{R^{'}} (x) | \approx χ_{c y l i n d e r} (x)$ ( $L^{\infty}$ -norm).

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

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

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

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

Consider the statement

(recall that we called this

R_{S^{n - 1}} (p \to q)

Fix $φ \in C_{c}^{\infty} (ℝ^{n})$ , $φ \sim χ_{B_{1} (0)}$ . For computing $L^{p}$ norms, $| \hat{φ} | \sim χ_{B_{c} (0)}$ .

Wave packet function with localised spatial and frequency behaviour.

Last time: $f_{R} (x) = e^{i x \cdot v_{R}} ˇ φ (R^{- 1} x_{1}, \dots, R^{- 1} x_{n - 1}, R^{- 2} x_{n})$ .

Frequency: $| \hat{f_{R}} (ξ) |$

$\int_{S^{n - 1}} | \hat{f_{R}} |^{q} d σ$

|-------------------|
n + 1− n-−-1 ≤ n+-1-|
---------q-------q---

Spatial: $| f_{R} (x) |$

$\int_{ℝ^{1}} | f_{R} |^{p}$

Note: sphere near $n$ looks like $(ξ^{1}, 1 - \frac{1}{2} | ξ^{'} |^{2})$ , $S^{n - 1} \cap supp \hat{f_{R}} \sim cap of radius R^{- 1}$ .

Naive attempt: $g_{R} (x) = R^{n} ˇ φ (R x)$

Frequency: $| \hat{g_{R}} (ξ) |$

$\int_{S^{n - 1}} | \hat{g_{R}} (ξ) |^{q} d ξ \sim 1$

Spatial: $| g_{R} (x) |$

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

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

(trivial).

$| \hat{g_{R}} | \sim 1$ on $S^{n - 1}$ made $∥ \hat{g_{R}} ∥_{L^{q} (S^{n - 1})}$ easy to compute.

Could we improve things?

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