Title: Daniel's Flashcards / Year 4 / Term 2 / Fourier Restriction Theory

html: 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 ) = 1 2
- 1 < p < ∞
- x 1 , … , x N ∈ ℂ
Then
proof: Proof. Without loss of generality, x 1 , … , x n ∈ ℝ. Without loss of generality, ∥ x ∥ 2 = 1.
p = 2: want to show 𝔼 ( | ∑ ⁡ n 𝜀 n x n | 2 ) ∼ 1.
What about general exponents p?
The equality here is the Layer cake formula, which is true for any p ∈ ( 0 , ∞ ).
Let λ > 0. Study the random variable e λ ∑ ⁡ n 𝜀 n x n ∈ ( 0 , ∞ ).
Fact: 1 2 e z + 1 2 e − z ≤ e z 2 2 (to check, use the Taylor series). So we can get
α ℙ ( e λ ∑ n 𝜀 n x n > α ) ≤ 𝔼 ( e λ ∑ n 𝜀 n x n ) ( Chebyshev’s inequality ) ≤ ∏ n e λ 2 | x n | 2 ∕ 2 = e λ 2 ∕ 2
By symmetry,
Choose α = e λ 2:
Use in Layer cake:
Lower bound: use Hölder’s inequality. X = ∑ ⁡ n 𝜀 n x n.
1 p + 1 q = 1. □
nothing: True

html: Assuming:
Theorem 2.1. - p > 2 - b n ∈ ℂ Then:
proof: Proof. Consider: ∑ ⁡ n = 1 N b n e ( n x ), b n ∈ ℂ.
∫ [ 0 , 1 ] | ∑ n b n e ( n x ) | p d x ≤ ∥ ∑ n b n e ( n x ) ∥ ∞ p − 2 ∫ [ 0 , 1 ] | ∑ n b n e ( n x ) | 2 d x ≤ ( N 1 2 ∥ b n ∥ 2 ) p − 2 ∥ b n ∥ 2 2 CS = N p 2 − 1 ∥ b n ∥ 2 p □

html: Assuming:
Theorem 2.2. - 2 > p Then:
proof: Proof. The inequality is true for p = 1, q = ∞ and for p = 2, q = 2 the inequality is true since we have equality (Plancherel).
For values in between we can interpolate. □

html: Theorem (Hausdorff-Young inequality). Assuming that:
- 1 ≤ p ≤ 2
- 1 p + 1 q = 1
Then
proof: Proof. The inequality is true for p = 1, q = ∞ and for p = 2, q = 2 the inequality is true since we have equality (Plancherel).
For values in between we can interpolate. □
nothing: True

html: Conjecture (Restriction Conjecture). R S n − 1 ( p → q ) if and only if n + 1 − n − 1 q ≤ n + 1 p and p < 2 n n + 1.
proof: None
nothing: True

html: Lemma (Locally constant property). Assuming that:
- f ∈ S ( ℝ n )
- supp ⁡ f ^ ⊆ B 1 ( 0 )
Then for any unit ball B 1 ⊆ ℝ n and any x ∈ B ′,
proof: Proof of locally constant property. Let f ∈ S ( ℝ n ), supp ⁡ f ^ ⊆ B 1 ( 0 ). Let φ ∈ C c ∞ ( ℝ n ) such that φ ≡ 1 on B 1 ( 0 ), supp ⁡ φ ⊆ B 2 ( 0 ).
By Fourier inversion:
Let x 0 , x ∈ B 1 (unit ball in ℝ n).
Math input error Math input error
nothing: True