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Knapp rules out extension below the Tomas exponent

Statement refuted

Assume Countable Choice. Let $n \geq 2$ and let $1 \leq q < 2 (n + 1) / (n - 1)$ . For every $C > 0$ there is $δ \in (0, 1]$ such that the spherical cap data $g = 1_{C_{δ}} \in L^{2} (σ)$ satisfy $∥ E g ∥_{L^{q} (R^{n})} > C ∥ g ∥_{L^{2} (σ)}$ . Hence no extension estimate $E : L^{2} (S^{n - 1}) \to L^{q} (R^{n})$ holds below the Stein-Tomas exponent, and consequently no restriction estimate $R : L^{p} (R^{n}) \to L^{2} (S^{n - 1})$ holds for $p > 2 (n + 1) / (n + 3)$ .

Facts & Assumptions

[F1]

Cap and tube scales: $c^{'} δ^{n - 1} \leq σ (C_{δ}) \leq C^{'} δ^{n - 1}$ and $λ_{n} (T_{δ}) = (2 c_{n})^{n} δ^{- (n + 1)}$ for $δ \in (0, 1]$ . (Spherical cap and dual slab scales)

[F2]

Concentration: $∣ E g_{δ} (x) ∣ \geq \frac{1}{2} σ (C_{δ})$ for every $x \in T_{δ}$ , so $∥ E g_{δ} ∥_{q}^{q} \geq (\frac{1}{2} σ (C_{δ}))^{q} λ_{n} (T_{δ})$ ; moreover $∥ g_{δ} ∥_{L^{2} (σ)} = σ (C_{δ})^{1 / 2}$ . (Cap wave packets concentrate on the dual tube, The nonnegative Lebesgue integral)

[F3]

The exponent relation $q < 2 (n + 1) / (n - 1)$ is equivalent to $a : = (n - 1) / 2 - (n + 1) / q < 0$ . Then $δ^{a} = \exp (a \log δ) \to \infty$ as $δ ↓ 0$ : for any $M > 0$ , $0 < δ < \exp (- M)$ implies $\log δ < - M$ by the inverse identity and strict monotonicity, so $a \log δ \to \infty$ , and the exponential diverges there. The Knapp theorem also rules out the restriction endpoint $p = \infty$ . (Conjugate exponents, including the endpoint conventions, Knapp necessary condition for spherical L2 restriction, Real powers for positive bases, with the zero-base positive-exponent convention, The natural logarithm as the inverse of the exponential function, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, The exponential tends to $+ \infty$ at $+ \infty$ and to $0$ at $- \infty$ )

[F4]

Duality: for $1 < p < \infty$ , the restriction estimate at exponent $p$ is equivalent to the extension estimate at $q = p^{'}$ with the same constant. (Restriction and extension estimates are dual)

Counterexample

Given: Countable Choice, $n \geq 2$ , $1 \leq q < 2 (n + 1) / (n - 1)$ , $C > 0$ , the caps $C_{δ} = {ω \in S^{n - 1} : 1 - ω \cdot e_{n} \leq δ^{2}}$ , the boxes $T_{δ} = {x : ∣ x_{n} ∣ \leq c_{n} δ^{- 2}, ∣ x_{j} ∣ \leq c_{n} δ^{- 1} (j < n)}$ , where $a_{n} > 0$ is a constant furnished by Cap wave packets concentrate on the dual tube and $c_{n} = a_{n} / \sqrt{n - 1}$ , the extension $E$ of Fourier restriction and adjoint extension operators, and $g_{δ} : = 1_{C_{δ}}$ .

Proof technique: direct; evaluate the extension on the cap family and observe that the quotient of norms diverges as $δ ↓ 0$ below the Tomas exponent.

1.1F1F2F3algebra

On the coordinate box $T_{δ}$ , $∣ (x_{1}, \dots, x_{n - 1}) ∣ \leq \sqrt{n - 1} c_{n} δ^{- 1} = a_{n} δ^{- 1}$ and $∣ x_{n} ∣ \leq c_{n} δ^{- 2} \leq a_{n} δ^{- 2}$ , so this box lies in the cylindrical tube supplied by [F2]. Therefore [F1] and [F2] give $∥ E g_{δ} ∥_{q} / ∥ g_{δ} ∥_{2} \geq \frac{1}{2} σ (C_{δ})^{1 / 2} λ_{n} (T_{δ})^{1 / q} \geq \frac{1}{2} (c^{'})^{1 / 2} (2 c_{n})^{n / q} δ^{(n - 1) / 2 - (n + 1) / q}$ . This is an inequality with an explicit positive constant, and its exponent $a = (n - 1) / 2 - (n + 1) / q$ is negative by [F3].

2.1F3step 1.1

Divergence. Since $a < 0$ , $δ^{a} \to + \infty$ as $δ ↓ 0$ ; hence for every $C > 0$ there is $δ \in (0, 1]$ with $∥ E g_{δ} ∥_{q} > C ∥ g_{δ} ∥_{2}$ , which refutes the existence of any finite extension constant $∥ E ∥_{L^{2} (σ) \to L^{q}}$ for $q$ below the Tomas exponent.

3.1F1F2F3F4step 2.1

The restriction form. If a restriction estimate at some finite $p > 1$ held with constant $R$ , then by the duality of restriction and extension estimates the extension estimate would hold at $q_{0} = p^{'}$ with the same constant; for $p > 2 (n + 1) / (n + 3)$ one has $p^{'} < 2 (n + 1) / (n - 1)$ , which step 2.1 rules out. At $p = \infty$ , the Knapp theorem [F3] also rules out the estimate by its compact norm tests. Hence no restriction estimate exists for $p \in (2 (n + 1) / (n + 3), \infty]$ , as asserted.

4.1step 1.1step 2.1step 3.1∎

Conclusion. Steps 1.1–3.1 exhibit the cap family $g_{δ}$ whose extension norms exceed any proposed constant below the exponent $2 (n + 1) / (n - 1)$ , and step 3.1 transfers the failure to the restriction side for $p > 2 (n + 1) / (n + 3)$ .

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Sources

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