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Pointwise restriction is not defined on Lp equivalence classes

Statement refuted

Assume Countable Choice and $n \geq 2$ . Statement refuted: for $1 < p < 2$ the pointwise restriction $\hat{f} ∣_{S^{n - 1}}$ is well defined by the ambient $L^{p^{'}}$ class of the Hausdorff-Young transform. Data: let $f$ be a nonzero Schwartz function on $R^{n}$ and let $F = F_{p} f \in L^{p^{'}} (R^{n})$ be its transform class, with representative $\hat{f}$ ; put $G : = \hat{f} + 1_{S^{n - 1}}$ pointwise. Since $λ_{n} (S^{n - 1}) = 0$ , $G$ represents the same $L^{p^{'}}$ class, but $G ∣_{S^{n - 1}} = \hat{f} ∣_{S^{n - 1}} + 1$ differs from $\hat{f} ∣_{S^{n - 1}}$ everywhere on the sphere. Hence restriction cannot be read off an ambient $L^{p^{'}}$ representative; it begins on Schwartz functions and, when a restriction estimate holds at the chosen exponent, extends by density as in Fourier restriction and adjoint extension operators.

Facts & Assumptions

[F1]

Hausdorff–Young: for $1 \leq p \leq 2$ the transform extends to a bounded map $F_{p} : L^{p} \to L^{p^{'}}$ that agrees almost everywhere with the integral transform on $L^{1} \cap L^{p}$ ; in particular $\hat{f}$ is a representative of $F_{p} f$ when $f$ is Schwartz. (Hausdorff–Young for the Euclidean Fourier transform)

[F2]

Complex $L^{p}$ classes are quotients of measurable functions by almost-everywhere equality, with representative-independent norm; $S (R^{n})$ consists of actual smooth functions and $1_{S^{n - 1}}$ is Borel measurable. (Complex Lp classes and Euclidean test-function conventions, Schwartz space and its seminorms)

[F3]

The unit sphere is Lebesgue null: $λ_{n} (S^{n - 1}) = 0$ , and a nonnegative measurable function with finite integral is finite almost everywhere; adding an indicator of a null set changes a function only on a null set. (The unit sphere is Lebesgue null, A nonnegative measurable function with finite integral is finite almost everywhere)

[F4]

For $1 < p < \infty$ , a restriction bound on Schwartz data yields the unique bounded extension $R : L^{p} \to L^{2} (σ)$ ; its existence is conditional on that bound. (Restriction and extension estimates are dual)

Counterexample

Given: Countable Choice, $n \geq 2$ , $1 < p < 2$ , a nonzero Schwartz function $f \in S (R^{n})$ , its integral transform $\hat{f}$ , the Hausdorff–Young class $F = F_{p} f \in L^{p^{'}} (R^{n})$ with representative $\hat{f}$ , and $G : = \hat{f} + 1_{S^{n - 1}}$ .

1.1F1F2F3

The two representatives coincide almost everywhere. The function $1_{S^{n - 1}}$ is Borel measurable by [F2], and $G = \hat{f} + 1_{S^{n - 1}}$ is measurable. Since $S^{n - 1}$ is Lebesgue null by [F3], $G = \hat{f}$ almost everywhere. Hence $\int ∣ G ∣^{p^{'}} = \int ∣ \hat{f} ∣^{p^{'}} < \infty$ by the Hausdorff–Young membership in [F1], and $G$ belongs to $L^{p^{'}} (R^{n})$ and represents the same class as $\hat{f}$ , namely $F = F_{p} f$ : two almost-everywhere equal integrable functions define the same quotient class by [F2].

1.2givenalgebra

The restrictions differ at every point of the sphere. By construction $G (ω) = \hat{f} (ω) + 1$ for every $ω \in S^{n - 1}$ , so the pointwise restrictions satisfy $G ∣_{S^{n - 1}} = \hat{f} ∣_{S^{n - 1}} + 1_{S^{n - 1}}$ . The difference is $1$ at every point of the nonempty sphere $S^{n - 1}$ , so the two restrictions are different functions on $S^{n - 1}$ .

2.1step 1.1step 1.2

The refutation. Suppose that the pointwise restriction were well defined by the ambient $L^{p^{'}}$ class, that is, that two representatives of one class always have equal restrictions to $S^{n - 1}$ . Steps 1.1 and 1.2 exhibit two representatives $\hat{f}$ and $G$ of the same class $F = F_{p} f$ whose restrictions differ everywhere on $S^{n - 1}$ ; this contradicts the supposition. Therefore pointwise restriction is not well defined on $L^{p^{'}}$ classes.

3.1F2F4given∎

The correct convention. The restriction operator $R_{0}$ of Fourier restriction and adjoint extension operators is defined on the actual functions $\hat{f}$ with $f \in S (R^{n})$ , where the pointwise values exist. If a bound $∥ R_{0} f ∥_{L^{2} (σ)} \leq C ∥ f ∥_{L^{p}}$ holds on Schwartz data, density gives its unique bounded extension $R : L^{p} (R^{n}) \to L^{2} (σ)$ , as in Restriction and extension estimates are dual; no pointwise restriction of a general ambient class is asserted.

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Sources

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