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Restriction and extension estimates are dual

Statement

Assume Countable Choice and let $1 < p < \infty$ with conjugate exponent $p^{'}$ . For a compact hypersurface $S$ with surface measure $σ$ and the operators $R_{0}, E$ of Fourier restriction and adjoint extension operators the following are equivalent: (a) there is $C < \infty$ with $∥ \hat{f} ∥_{L^{2} (σ)} \leq C ∥ f ∥_{L^{p}}$ for all $f \in S (R^{n})$ , so that $R_{0}$ has a unique bounded extension $R : L^{p} (R^{n}) \to L^{2} (σ)$ ; (b) there is $C < \infty$ with $∥ E g ∥_{L^{p^{'}} (R^{n})} \leq C ∥ g ∥_{L^{2} (σ)}$ for all $g \in L^{2} (σ)$ . The least constants agree, and $E$ is the adjoint of $R$ under the $L^{p}$ – $L^{p'}$ and $L^{2} (σ)$ pairings: $\int E g \overline{f} d x = \int_{S} g \overline{R f} d σ$ .

Facts & Assumptions

Given: Countable Choice, $1 < p < \infty$ with conjugate $p^{'}$ , a compact hypersurface $S$ with surface measure $σ$ (finite), and the operators $R_{0} : S (R^{n}) \to L^{2} (σ)$ , $f \mapsto \hat{f} ∣_{S}$ , and $E : L^{1} (σ) \to C_{b} (R^{n})$ , $g \mapsto (g σ)^{\lor}$ , of Fourier restriction and adjoint extension operators.

[F1]

The operator $E$ is defined on $L^{1} (σ) \supseteq L^{2} (σ)$ by an everywhere-defined bounded uniformly continuous function, the surface measure is finite, and $R_{0}$ is pointwise defined on Schwartz data; $L^{p} (σ; C)$ and $L^{p} (R^{n}; C)$ are the complex Lebesgue classes. (Fourier restriction and adjoint extension operators)

[F2]

Pairing identity: for every finite complex Borel measure $μ$ and all $F, G \in S (R^{n})$ , $\int (G μ)^{\lor} \overline{F} d x = \int G \overline{\hat{F}} d μ$ . (Fourier pairing for a finite measure and Schwartz data)

[F4]

$C_{c}^{\infty} (R^{n}) \subset S$ is dense in $L^{p} (R^{n})$ for $1 \leq p < \infty$ ; smooth ball cutoffs exist. ( $C_{c}^{\infty} (R^{n})$ is dense in $L^{p} (R^{n})$ for $1 \leq p < \infty$ , Explicit compactly supported smooth cutoffs)

[F5]

Complex $L^{p}$ spaces are complete under Countable Choice; a bounded map from a dense subspace to a Banach space extends uniquely with the same norm. Cauchy–Schwarz and complex Hölder hold, including endpoints. (Complex Lp completeness and almost-everywhere subsequences, A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm, The complex $L^{2}$ pairing is well-defined and satisfies Cauchy–Schwarz, Complex Holder, Minkowski, and the quotient norm)

[F6]

Absolutely integrable product kernels admit Fubini, and nonnegative integrals obey monotone convergence. (Fubini's theorem for L^1 functions on a sigma-finite product, Monotone convergence for the integral)

Proof

technique · direct; prove the pairing by Fubini and obtain extension integrability by compactly supported norm tests

1.1F1F2F6

For $g \in L^{2} (σ) \subset L^{1} (σ)$ and $F \in S$ , the kernel $e^{2 π i x \cdot ω} g (ω) \overline{F (x)}$ has absolute integral $∥ g ∥_{1} ∥ F ∥_{1} < \infty$ . Fubini gives $\int E g \overline{F} d x = \int_{S} g \overline{\hat{F}} d σ$ . This is an absolutely convergent integral pairing, without asserting $E g \in L^{2} (R^{n})$ .

2.1F1F4F5F6step 1.1algebra

Assume (a), fix $g$ , and set $h = E g$ , $M = C ∥ g ∥_{2}$ . Step 1.1 and Cauchy–Schwarz give $∣ \int h \overline{F} ∣ \leq M ∥ F ∥_{p}$ on Schwartz functions. For a bounded ball $B$ , put $v = 1_{B} h ∣ h ∣^{p^{'} - 2}$ , assigning zero where $h = 0$ . Since $h$ is bounded, $v \in L^{p}$ and $A = \int_{B} ∣ h ∣^{p^{'}} < \infty$ . Approximate $v$ in $L^{p}$ by $C_{c}^{\infty}$ functions and multiply by a fixed smooth cutoff equal to one on $B$ , supported in a larger bounded ball. These approximants still converge to $v$ in $L^{p}$ , and their integrals against $h$ converge, because $h$ is bounded and their supports have uniformly finite measure. Passing to the limit gives $A \leq M A^{1 / p}$ . Thus $A^{1 / p^{'}} \leq M$ (also when $A = 0$ ). Letting $B$ increase to $R^{n}$ and using monotone convergence proves $h \in L^{p^{'}}$ and $∥ h ∥_{p^{'}} \leq M$ , establishing (b).

2.2F1F5step 1.1algebra

Assume (b). For Schwartz $F$ , step 1.1 and Hölder give $∣ \int_{S} g \overline{R_{0} F} d σ ∣ \leq C ∥ g ∥_{2} ∥ F ∥_{p}$ for every $g \in L^{2} (σ)$ . If $R_{0} F \neq 0$ , take $g = R_{0} F / ∥ R_{0} F ∥_{2}$ ; otherwise the desired estimate is immediate. Hence $∥ R_{0} F ∥_{2} \leq C ∥ F ∥_{p}$ , proving (a). No density assertion on surface functions is needed.

3.1F4F5step 1.1step 2.1step 2.2∎

By [F4] and [F5], (a) gives the unique extension $R : L^{p} \to L^{2} (σ)$ . For each $g$ , the pairing in step 1.1 extends by continuity in $f$ from Schwartz data to every $f \in L^{p}$ , since $E g \in L^{p^{'}}$ by step 2.1. This identifies $E$ as the adjoint under the displayed Banach dual pairings. Steps 2.1 and 2.2 preserve each admissible constant, so the least constants, and the operator norms, agree. Countable Choice is the hypothesis of the density and completeness suppliers.

Depends on

Used by

Stein-Tomas for compact hypersurfaces with nonzero curvature Corollary
Knapp rules out extension below the Tomas exponent Counterexample
Pointwise restriction is not defined on Lp equivalence classes Counterexample
Knapp necessary condition for spherical L2 restriction Theorem
Stein-Tomas spherical restriction theorem Theorem

Dependency tree · two levels

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Sources

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