What is Fourier Restriction Theory? - Fourier Restriction Theory

Main object:

f : ℝ^{d} \to ℂ

f (x) = \sum_{ξ \in ℝ} b_{ξ} e^{2 π i x \cdot ξ}

b_{ξ} \in ℂ

x \in ℝ^{d}

is a spatial variable, and

ξ \in ℝ^{d}

is the frequency variable.

The frequencies (or Fourier transform) of

f

is restricted to a set

R

(where

R

we will always be finite – so no need to worry about convergence issues).

Example.

(i) Schrödinger equation:
$u (x, t) = \sum_{n = 1}^{N} b_{n} e (n x + n^{2} t) .$

Easy: $(2 π i \partial_{+} - Δ) u = 0$ , with initial data $u (x, 0) = \sum_{n - 1}^{N} b_{n} e (n x)$ .

$(x, t) = (x_{1}, x_{2})$ . Then since $e (n x + n^{2} t) i e ((n, n^{2}) \cdot (x, t))$ , we might consider $R = {(n, n^{2}) : n = 1, \dots, N}$ .
(ii) Dirichlet polynomials:
$D (t) = \sum_{n = N}^{2 N} b_{n} e^{i (\log n) t} .$

with $b_{n} \equiv 1$ , partial sums of Riemann Zeta function. We might consider $R = {\frac{1}{2 π} \log n}_{n = N}^{2 N}$ .

Both avoid linear structure.

${\log n}$ is a concave set (getting closer and closer together).

${(n, n^{2})}$ lie on a parabola.

Guiding principle: if properties of an object avoid (linear) structure, then we expect some random or average behaviour.

The above examples avoid linear structure using some notion fo curvature. See Bourgain

Λ (p)

paper:

\to

extra behaviour.

Square root cancellation: If we add

\pm 1

randomly

N

times, then we expect a quantity with size

N^{\frac{1}{2}}

Theorem 1.1 (Khinchin’s inequality). Assuming that:

${𝜀_{n}}_{n = 1}^{N}$ be IID random variables with $ℙ (𝜀_{n} = 1) = ℙ (𝜀_{n} = - 1) = \frac{1}{2}$
$1 < p < \infty$
$x_{1}, \dots, x_{N} \in ℂ$

Then

{(𝔼 {| \sum_{n = 1}^{N} 𝜀_{n} x_{n} |}^{p})}^{\frac{1}{p}} \sim_{p} {(\sum_{n = 1}^{N} | x_{n} |^{2})}^{\frac{1}{2}} = ∥ x_{n} ∥_{2} .

Proof. Without loss of generality, $x_{1}, \dots, x_{n} \in ℝ$ . Without loss of generality, $∥ x ∥_{2} = 1$ .

$p = 2$ : want to show $𝔼 ({| \sum_{n} 𝜀_{n} x_{n} |}^{2}) \sim 1$ .

𝔼 (\sum_{n} 𝜀_{n} x_{n} \bar{\sum_{m} 𝜀_{m} x_{m}}) = \sum_{n, m} 𝔼 (𝜀_{n} 𝜀_{m} x_{n} \bar{x_{m}}) = \sum_{n} | x_{n} |^{2} + \sum_{n} \sum_{m \neq n} x_{n} \bar{x_{m}} \underset{= 0}{\underset{⏟}{𝔼 𝜀_{n}}} 𝔼 𝜀_{m} .

What about general exponents $p$ ?

𝔼 ({| \sum_{n} 𝜀_{n} x_{n} |}^{p}) = \int_{0}^{\infty} ℙ ({| \sum_{n} 𝜀_{n} x_{n} |}^{p} > α) d α .

The equality here is the Layer cake formula, which is true for any $p \in (0, \infty)$ .

Let $λ > 0$ . Study the random variable $e^{λ \sum_{n} 𝜀_{n} x_{n}} \in (0, \infty)$ .

𝔼 (e^{λ \sum_{n} 𝜀_{n} x_{n}}) = 𝔼 (\prod_{n} e^{λ 𝜀_{n} x_{n}}) = \prod_{n} 𝔼 e^{λ 𝜀_{n} x_{n}} = \prod_{n} (\frac{1}{2} e^{λ x_{n}} + \frac{1}{2} e^{- λ x_{n}}) .

Fact: $\frac{1}{2} e^{z} + \frac{1}{2} e^{- z} \leq e^{\frac{z^{2}}{2}}$ (to check, use the Taylor series). So we can get

\begin{array}{l} α ℙ (e^{λ \sum_{n} 𝜀_{n} x_{n}} > α) & \leq 𝔼 (e^{λ \sum_{n} 𝜀_{n} x_{n}}) & (Chebyshev’s inequality) \\ \leq \prod_{n} e^{λ^{2} | x_{n} |^{2} ∕ 2} \\ = e^{λ^{2} ∕ 2} \end{array}

By symmetry,

α ℙ (e^{| λ \sum_{n} 𝜀_{n} x_{n} |} > α) ≲ e^{λ^{2} ∕ 2} .

Choose $α = e^{λ^{2}}$ :

ℙ (| \sum_{n} 𝜀_{n} x_{n} | > λ) = ℙ ({| \sum_{n} 𝜀_{n} x_{n} |}^{p} > λ^{p}) ≲ e^{- λ^{2} ∕ 2} .

Use in Layer cake:

𝔼 ({| \sum_{n} 𝜀_{n} x_{n} |}^{p}) ≲ \int_{0}^{\infty} e^{- α^{2 ∕ p} ∕ 2} d α \sim_{p} 1 .

Lower bound: use Hölder’s inequality. $X = \sum_{n} 𝜀_{n} x_{n}$ .

\underset{= 1}{\underset{⏟}{𝔼 (X \bar{X})}} \leq \underset{≲_{p} 1}{\underset{⏟}{{(𝔼 (| X |^{p}))}^{1 ∕ p}}} {(𝔼 | X |^{q})}^{1 ∕ q} .

$\frac{1}{p} + \frac{1}{q} = 1$ . □

Both

f, g

are

1

-periodic. So study them on

𝕋 = [0, 1]

f (0) = N

| f (x) | \sim N

for

\in [0, c \frac{1}{N}]

g (0) i N

| g (x) | \sim N

for

x \in [0, c \frac{1}{N^{2}}]

For the first one,

N^{p - 1}

(organised behaviour) dominates as soon as

p > 2

, and for the second one,

N^{p ∕ 2}

dominates for

2 \leq p \leq 4

(“square root cancellation behaviour lasts for longer”).