Tube incidence implications of Fourier restriction - Fourier Restriction Theory

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)?

f = ˇ (\hat{f} φ_{B_{λ}}) = f * ˇ φ_{B_{λ}},

where $φ_{B_{λ}} (ξ) = φ_{B_{1}} (λ^{- 1} ξ)$ .

ˇ φ_{B_{λ}} (x) = \int_{ℝ^{n}} e^{2 π i x \cdot ξ} φ_{B_{1}} (λ^{- 1} ξ) d ξ = λ^{n} \int_{ℝ^{n}} e^{2 π i (λ x) \cdot ξ} φ_{B_{1}} (ξ) d ξ = λ^{n} ˇ φ_{B_{1}} (λ x) .

$* | ˇ φ_{B_{1}} |$ is approximately averaging over a $1$ -ball. $* | ˇ φ_{B_{λ}} |$ is approximately averaging over $λ^{- 1}$ -balls.

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

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

\int_{B_{R}} | f (x) |^{p^{'}} d x ≲ \int_{B_{R}} | f (x) φ_{B_{R}} (x) |^{p^{'}} d x

$φ_{B_{R}} (x) \in S (ℝ^{n})$ , $| φ_{B_{R}} | \sim 1$ on $B_{R}$ , $supp \hat{φ_{B_{R}}} \subseteq B_{R^{- 1}} (0)$ .

Last lecture $\to$

\int_{I_{R^{- 1}}} (\int_{\begin{array}{c} | ξ^{'} | < 1 \\ ξ^{'} \in ℝ^{n - 1} \end{array}} | \hat{f} * \hat{φ_{B_{R}}} (ξ^{'}, | ξ^{'} |^{2} + ξ_{n}) |^{q^{'}} d ξ^{'}) \overset{(*)}{≲} {(R^{- 1})}^{1 - \frac{p^{'}}{q^{'}}} {(\int_{ξ \in ℝ^{n}} | \hat{f} (ξ) |^{q^{'}} d ξ)}^{\frac{p^{'}}{q^{'}}} .

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^{'}}} \underset{\sim 1}{\underset{⏟}{{(\int | \hat{φ_{B_{R}}} |^{\frac{q}{q}})}^{\frac{1}{q}}}} & (Holder) \\ ≲ | I_{R^{- 1}} |^{1 - \frac{p^{'}}{q^{'}}} {(\int_{ℝ^{n}} \int_{ℝ^{n}} | \hat{f} |^{q^{'}} (ξ - η) | \hat{φ_{B_{R}}} | (η) d η d ξ)}^{\frac{p^{'}}{q^{'}}} \\ \sim | I_{R^{- 1}} |^{1 - \frac{p^{'}}{q^{'}}} {(\int_{ℝ^{n} | \hat{f} |^{q^{'}} (ξ) d ξ})}^{\frac{p^{'}}{q^{'}}} \end{array}

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 $g = \sum_{Q} g_{Q}$ .

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

Note

\int_{| t | < 1} g (t, t^{2} + ξ_{2}) d t \sim g_{Q} .

Therefore,

\int_{| t | < 1} g (t, t^{2} + ξ_{2}) d t \sim C

if $| ξ_{2} | \leq c R^{- 1}$ . ( $\leq C$ 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.

LHS of (*) ≲ | I_{R^{- 1}} | \max_{ξ_{2} \in I_{R^{- 1}}} {(\int_{| ξ^{'} | < 1} {(| \hat{f} | * | \hat{φ_{B_{R}}} | (ξ^{'}, | ξ^{'} |^{2} + ξ_{n}))}^{q^{'}} d ξ^{'})}^{\frac{p^{'}}{q^{'}}}

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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