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Morrey--Rellich compactness loses the endpoint H"older exponent

Statement refuted

Refuted claim. In the Morrey range $p > n$ the compactness $W^{1, p} (Ω) ↪ C^{0, α} (\overline{Ω})$ holds at the endpoint exponent $α = 1 - \frac{n}{p}$ itself.

The witness rescales a fixed smooth bump; all $W^{1, p}$ norms stay bounded and the functions converge uniformly to $0$ , but the endpoint H"older seminorm is scale invariant and stays bounded away from zero.

Facts & Assumptions

Given: the Axiom of Choice, $n \geq 2$ , $p > n$ , $α = 1 - \frac{n}{p} \in (0, 1)$ , $Ω = B (0, 1)$ , a nonzero $φ \in C_{c}^{\infty} (R^{n})$ with $φ (0) = 1$ and $supp φ \subseteq B (0, 1)$ (for instance a normalised smooth bump), and $u_{k} (x) : = (k + 1)^{- α} φ ((k + 1) x)$ on $Ω$ for $k \geq 0$ .

[F1]

Scaling of norms. By the change of variables $y = (k + 1) x$ , $∥ u_{k} ∥_{L^{p} (Ω)} = (k + 1)^{- α - n / p} ∥ φ ∥_{L^{p}} = (k + 1)^{- 1} ∥ φ ∥_{L^{p}} \to 0$ and $∥ D u_{k} ∥_{L^{p} (Ω)} = (k + 1)^{1 - α - n / p} ∥ D φ ∥_{L^{p}} = ∥ D φ ∥_{L^{p}}$ , because $α + \frac{n}{p} = 1$ ; in particular $(u_{k})$ is bounded in $W^{1, p} (Ω)$ . (A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not, Integer-order Sobolev spaces and their norms)

[F2]

H"older norms. The $C^{0, α}$ seminorm is $[g]_{C^{0, α}} = \sup_{x \neq y} ∣ g (x) - g (y) ∣ / ∣ x - y ∣^{α}$ , and the $C^{0, α}$ norm is the sum of the supremum norm and the seminorm. (Local Hölder and scaled C-two-alpha norms on balls)

[F3]

The reference bump. $φ$ is smooth with compact support and $φ (0) = 1$ , so $[φ]_{C^{0, α} (R^{n})} > 0$ : the pair $x = 0$ and any $y$ with $∣ y ∣ = 1$ gives $∣ φ (0) - φ (y) ∣ / ∣ y ∣^{α} = 1$ . (A Euclidean bump for a compact set inside an open set, The space $L^{p} (μ)$ as the quotient by null functions)

Counterexample

technique · direct

1.1F1given

By [F1], $∥ u_{k} ∥_{W^{1, p} (Ω)} \leq (k + 1)^{- 1} ∥ φ ∥_{L^{p}} + ∥ D φ ∥_{L^{p}} \leq ∥ φ ∥_{L^{p}} + ∥ D φ ∥_{L^{p}}$ is bounded, and $∥ u_{k} ∥_{\infty} = (k + 1)^{- α} ∥ φ ∥_{\infty} \to 0$ , so the representatives converge uniformly to $0$ on $Ω$ .

1.2F2F3given

For $x = 0$ and $y = e_{1} / (k + 1)$ in $\overline{Ω}$ , one has $u_{k} (x) = (k + 1)^{- α}$ and $u_{k} (y) = (k + 1)^{- α} φ (e_{1}) = 0$ because the bump support is inside the unit ball. Therefore $[u_{k}]_{C^{0, α} (\overline{Ω})} \geq (k + 1)^{- α} / ∣ e_{1} / (k + 1) ∣^{α} = 1$ for every $k$ .

2.1F2step 1.1step 1.2∎

If a subsequence converged in $C^{0, α} (\overline{Ω})$ to some $w$ , then it would converge uniformly, hence $w = 0$ by step 1.1, and by [F2] the seminorms would converge: $∣ [u_{k_{j}}]_{C^{0, α}} - [w]_{C^{0, α}} ∣ \leq [u_{k_{j}} - w]_{C^{0, α}} \leq ∥ u_{k_{j}} - w ∥_{C^{0, α}} \to 0$ , forcing the seminorms of step 1.2 to tend to $0$ — a contradiction, since they are at least $1$ . Hence the endpoint exponent in Morrey--Rellich compactness for $p > n$ cannot yield compactness, and the refuted claim is false.

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Sources

John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter notes) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (archived 2025 author manuscript) (standard reference, not scraped)