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Cap wave packets concentrate on the dual tube

Statement

Assume Countable Choice and let $n \geq 2$ . There is $c_{n} > 0$ such that for every $δ \in (0, 1]$ , every $v \in S^{n - 1}$ , every $η \in R^{n}$ and every rotation $R$ with $R e_{n} = v$ the following holds. With $C_{δ} (v) = {ω \in S^{n - 1} : 1 - ω \cdot v \leq δ^{2}}$ , the tube $T = η + {ξ : ∣ ξ \cdot v ∣ \leq c_{n} δ^{- 2}, ∣ ξ - (ξ \cdot v) v ∣ \leq c_{n} δ^{- 1}}$ and the data $g_{η} (ω) = e^{- 2 π i η \cdot ω} 1_{C_{δ} (v)} (ω)$ , one has $∣ E g_{η} (x) ∣ \geq \frac{1}{2} σ (C_{δ} (v))$ for every $x \in T$ . In particular the extension of cap data of angular radius $δ$ is essentially coherent on a dual tube of dimensions $δ^{- 1} \times \dots \times δ^{- 1} \times δ^{- 2}$ .

Facts & Assumptions

Given: Countable Choice, $n \geq 2$ , $δ \in (0, 1]$ , $v \in S^{n - 1}$ , $η \in R^{n}$ , and $ξ \in R^{n}$ with $∣ ξ \cdot v ∣ \leq c_{n} δ^{- 2}$ and $∣ ξ - (ξ \cdot v) v ∣ \leq c_{n} δ^{- 1}$ for a constant $c_{n} \in (0, 1 / 100]$ to be fixed below; write $x = η + ξ$ .

[F1]

Extension: for $g \in L^{1} (σ)$ one has $E g (x) = \int_{S^{n - 1}} e^{2 π i x \cdot ω} g (ω) d σ (ω)$ , with $L^{1} (σ)$ computed componentwise for complex functions, and for a real $φ$ one has $∣ E g (x) ∣ \geq Re (e^{- i φ} E g (x))$ . (Fourier restriction and adjoint extension operators, Complex Lp classes and Euclidean test-function conventions, The Lebesgue integral is linear on $L^{1} (μ)$ )

[F2]

Cap geometry: $ω \cdot v \geq 1 - δ^{2}$ on $C_{δ} (v)$ , the diameter satisfies $diam C_{δ} (v) \leq 2 \sqrt{2} δ$ , and $σ (C_{δ} (v)) > 0$ . (Spherical cap and dual slab scales)

[F3]

Unit-circle estimates: $\sin 0 = 0$ and $∣ e^{i θ} - 1 ∣ = 2 ∣ \sin (θ / 2) ∣ \leq ∣ θ ∣$ for every real $θ$ , because $∣ \sin u ∣ \leq ∣ u ∣$ and $e^{i θ} = \cos θ + i \sin θ$ ; consequently $1 - \cos θ = \frac{1}{2} ∣ e^{i θ} - 1 ∣^{2} \leq ∣ e^{i θ} - 1 ∣ \leq ∣ θ ∣$ , so $Re e^{i θ} = \cos θ \geq 1 - ∣ θ ∣$ . In particular $Re e^{i θ} \geq \frac{1}{2}$ whenever $∣ θ ∣ \leq \frac{1}{2}$ . ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , Sine and cosine are $1$ -Lipschitz on $R$ , The zero sets of sine and cosine and the least positive common period 2 pi)

Proof

technique · direct; evaluate the extension of the modulated cap data, expand the phase in the cap parameter, and use the elementary lower bound for the real part of a unit-modulus exponential

1.1F1given

The extension of the data. With $g_{η} = e^{- 2 π i η \cdot ω} 1_{C_{δ} (v)}$ and $x = η + ξ$ , [F1] gives $E g_{η} (x) = \int_{C_{δ} (v)} e^{2 π i x \cdot ω} e^{- 2 π i η \cdot ω} d σ (ω) = \int_{C_{δ} (v)} e^{2 π i ξ \cdot ω} d σ (ω)$ , since $x - η = ξ$ .

1.2F2F3givenalgebra

The phase on the cap. Let $ω \in C_{δ} (v)$ and write $ω = v + (ω - v)$ . Then $ξ \cdot ω = ξ \cdot v + ξ \cdot (ω - v)$ , and by [F2] and the tube inequalities $∣ ξ \cdot (ω - v) ∣ \leq ∣ (ξ \cdot v) (v \cdot (ω - v)) ∣ + ∣ (ξ - (ξ \cdot v) v) \cdot (ω - v) ∣ \leq c_{n} δ^{- 2} δ^{2} + 2 \sqrt{2} c_{n} δ^{- 1} δ \leq (1 + 2 \sqrt{2}) c_{n} .$ Choose $c_{n} : = 1 / 100$ , so $∣ ξ \cdot (ω - v) ∣ \leq (1 + 2 \sqrt{2}) / 100 < 1 / (4 π)$ and $∣ 2 π ξ \cdot (ω - v) ∣ \leq \frac{1}{2}$ . By the last clause of [F3], $Re e^{2 π i ξ \cdot (ω - v)} \geq \frac{1}{2}$ for every $ω \in C_{δ} (v)$ .

2.1F1step 1.1step 1.2

The lower bound. Multiplying step 1.1 by the unimodular factor $e^{- 2 π i ξ \cdot v}$ and applying [F1], $∣ E g_{η} (x) ∣ \geq Re (e^{- 2 π i ξ \cdot v} E g_{η} (x)) = Re \int_{C_{δ} (v)} e^{2 π i ξ \cdot (ω - v)} d σ (ω) = \int_{C_{δ} (v)} Re e^{2 π i ξ \cdot (ω - v)} d σ (ω) \geq \frac{1}{2} σ (C_{δ} (v)),$ the middle equality by componentwise integration of complex-valued functions [F1] and the last inequality by step 1.2 integrated against the positive measure $σ$ ; this holds for every $x = η + ξ \in T$ . The tube is a product of a tangential ball of radius $c_{n} δ^{- 1}$ and a normal interval of length $2 c_{n} δ^{- 2}$ . In any orthonormal tangential frame it contains the box with each tangential half-width $c_{n} δ^{- 1} / \sqrt{n - 1}$ and normal half-width $c_{n} δ^{- 2}$ ; this has the stated scales.

3.1step 1.2step 2.1∎

Conclusion. Steps 1.1–2.1 prove that for the explicit constant $c_{n} = 1 / 100$ the extension of the cap data $g_{η}$ is bounded below by $\frac{1}{2} σ (C_{δ} (v))$ on the whole dual tube $T$ , uniformly in $δ, v, η$ .

Depends on

Used by

Knapp rules out extension below the Tomas exponent Counterexample
Knapp cap and dual tube volume calculation Example
Knapp necessary condition for spherical L2 restriction Theorem

Dependency tree · two levels

64 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Merz, Some notes on restriction theory (standard reference, not scraped)