Tube incidence implications of Fourier restriction

6 Tube incidence implications of Fourier restriction

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

If $supp \hat{f} \subseteq B_{1}$ then

(1) Imagine $| f | \sim const$ on unit balls.
(2) $f = ˇ (\hat{φ_{B_{1}}}) = f * ˇ φ_{B_{1}}$ .

What if $supp \hat{f} \subseteq B_{λ} (0)$ (so $| f | \sim const$ on $λ^{- 1}$ -balls)?

where

φ_{B_{λ}} (ξ) = φ_{B_{1}} (λ^{- 1} ξ)

∫ ∫
φˇ (x ) = e2πix⋅ξφ (λ −1ξ)dξ = λn e2πi(λx)⋅ξφ (ξ)dξ = λnφˇ (λx).
Bλ ℝn B1 ℝn B1 B1

* | ˇ φ_{B_{1}} |

is approximately averaging over a

1

-ball.

* | ˇ φ_{B_{λ}} |

is approximately averaging over

λ^{- 1}

-balls.

What about $supp \hat{f} \subseteq B_{λ} (v)$ ?

Same thing happens, because $e^{i x \cdot something} f$ will have Fourier support in $B_{λ} (0)$ , and taking absolute values means we don’t notice the $e^{i x \cdot something}$ (modulation).

Returning to $R_{P^{n - 1}}^{*} (q^{'} \to p^{'}) ⟹ R_{P^{n - 1}}^{*, loc} (q^{'} \to p^{'})$ .

∫ ∫
|f(x)|p′dx ≲ |f (x)φBR (x )|p′dx
BR BR

φ_{B_{R}} (x) \in S (ℝ^{n})

| φ_{B_{R}} | \sim 1

B_{R}

supp \hat{φ_{B_{R}}} \subseteq B_{R^{- 1}} (0)

Last lecture $\to$

( ) ′
∫ ∫ ′ ′2 q′ ′ (∗) −1 1− p′(∫ q′ ) pq′
( |ξ′|<1 |^f ∗ ^φBR (ξ ,|ξ |+ ξn)| dξ) ≲ (R ) q′ n |^f(ξ)| dξ .
IR−1 ξ′∈ℝn−1 ξ∈ℝ

Can choose $\hat{φ_{B_{R}}}$ such that

$| \hat{φ_{B_{R}}} | \sim R^{n} χ_{B_{R^{- 1}} (0)}$ .
$∥ \hat{φ_{B_{R}}} ∥_{1} \sim 1$ .

Case 1: $\frac{p^{'}}{q^{'}} \leq 1$ . LHS of $(*)$ (using Hölder) is

\begin{array}{l} \leq | I_{R^{- 1}} |^{1 - \frac{p^{'}}{q^{'}}} {(\int_{I_{R^{- 1}}} \int_{| ξ^{'} | < 1} {(| \hat{f} | * | \hat{φ_{B_{R}}} |^{\frac{1}{q^{'}} + \frac{1}{q}} (ξ^{'}, | ξ^{'} |^{2} + ξ_{n}))}^{q^{'}} d ξ^{'} d ξ_{n})}^{\frac{p^{'}}{q^{'}}} \\ \sim | I_{R^{- 1}} |^{1 - \frac{p^{'}}{q^{'}}} {(\int_{ℝ^{n}} {(| \hat{f} | * | \hat{φ_{B_{R}}} | (ξ))}^{q^{'}} d d ξ)}^{\frac{p^{'}}{q^{'}}} \\ \leq {(\int | \hat{f} |^{q^{'}} (ξ - η) | \hat{φ_{B_{R}}} | (η) d η)}^{\frac{1}{q^{'}}} munder \end{array}

∼1 (Holder) ≲|IR−1|1−p′ q′ (∫ ℝn ∫ ℝn|f^|q′ (ξ − η)|φBR^|(η)dηdξ) p′ q′ ∼|IR−1|1−p′ q′ (∫ ℝn|f^|q′(ξ)dξ) p′ q′

Case 2: $\frac{p^{'}}{q^{'}} > 1$ . 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 $g = \sum_{Q} g_{Q}$ .

( ) p′-
∫ ∫ 2 q′
g(t,t + ξ2)dt dξ2.
IR−1 |t|<1

Note

Therefore,

| ξ_{2} | \leq c R^{- 1}

. (

\leq C

for all

ξ_{2} \in I_{R^{- 1}}

$| (P^{'} + (0, ξ_{2})) \cap Q | \sim R^{- 1}$ for $| ξ_{2} | \leq c R^{- 1}$ .

$\sim | I_{R^{- 1}} |^{1 - \frac{p^{'}}{q} + \frac{p}{q^{'}}} C^{\frac{p^{'}}{q^{'}}} = | I_{R^{- 1}} |^{1 - \frac{p^{'}}{q^{'}}} {(| I_{R^{- 1}} | C)}^{\frac{p^{'}}{q^{'}}} {(\int_{R^{- 1}} \int_{| t | < 1} g (t, t^{2} + ξ_{2}) d t d ξ_{2})}^{\frac{p^{'}}{q^{'}}}$

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

Make the intuition rigorous.

p′
(∫ ′ ) q′-
LHS of (∗) ≲ |IR−1| max (|^f|∗|^φBR|(ξ′,|ξ′|2 + ξn))qdξ′
ξ2∈IR−1 |ξ′|<1

Consider the integral:

\begin{array}{l} \int_{| ξ^{'} | < 1} {(| \hat{f} | * | \hat{φ_{B_{R}}} | (ξ^{'}, | ξ^{'} |^{2} + ξ_{n}))}^{q^{'}} d ξ^{'} & = \int_{| ξ^{'} | < 1} {(\int_{ℝ^{n}} | \hat{f} | (η) | \hat{φ_{B_{R}}} | ((ξ^{'}, | ξ^{'} |^{2} + ξ_{n}) - η) d η)}^{q^{'}} d ξ^{'} \\ \leq \int_{| ξ^{'} | < 1} (\int_{ℝ^{n}} | \hat{f} |^{q^{'}} (η) | \hat{φ_{B_{R}}} | ((ξ^{'}, | ξ^{'} |^{2} + ξ_{n}) - η) d η) d ξ^{'} & Same pointwise Holder \\ \sim R^{n} {(R^{- 1})}^{n - 1} \int_{ℝ^{n}} | \hat{f} |^{q^{'}} (η) d η \\ \sim R^{1} \int_{ℝ^{n}} | \hat{f} |^{q^{'}} \\ ≲ {(R^{- 1})}^{1} \cdot R^{\frac{p^{'}}{q^{'}}} {(\int | \hat{f} |^{q^{'}})}^{\frac{p^{'}}{q^{'}}} \end{array}

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