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The $W^{1, p} \to L^{q}$ bound fails for $q > p^{*}$ by dilation

Statement refuted

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 2$ , $1 \leq p < n$ and $q > p^{*} = \frac{n p}{n - p}$ . The assertion that there is a finite constant $C$ with $∥ u ∥_{L^{q} (R^{n})} \leq C ∥ D u ∥_{L^{p} (R^{n})} for every u \in C_{c}^{\infty} (R^{n})$ is false. The stronger full-norm assertion $∥ u ∥_{L^{q}} \leq C ∥ u ∥_{W^{1, p}}$ also fails for this family, so the exponent $p^{*}$ is sharp for both estimates.

Facts & Assumptions

Given: Countable Choice; $n \geq 2$ ; $1 \leq p < n$ ; a fixed nonzero $φ \in C_{c}^{\infty} (B (0, 1))$ ; and $0 < λ \leq 1$ .

[F1]

The Sobolev conjugate satisfies $\frac{1}{p^{*}} = \frac{1}{p} - \frac{1}{n}$ , so for finite $q > p^{*}$ one has $\frac{1}{q} < \frac{1}{p} - \frac{1}{n}$ ; the case $q = \infty$ is treated separately with $1 / q = 0$ (The Sobolev conjugate exponent and the scaling identity).

[F2]

An invertible linear map scales Lebesgue measure by $∣ \det ∣$ , so for $a > 0$ the substitution $y = x / a$ gives $\int_{R^{n}} g (x / a) d x = a^{n} \int_{R^{n}} g (y) d y$ ; an $L^{q}$ class is determined by its values almost everywhere (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, The space $L^{p} (μ)$ as the quotient by null functions).

[F3]

The closed unit ball is compact, so a continuous function on it is bounded and the support of $φ$ is compact (For $n \geq 1$ , every Euclidean closed ball and every Euclidean sphere of positive radius is compact).

[F4]

A nonzero smooth bump in $B (0, 1)$ exists (A Euclidean bump for a compact set inside an open set); its gradient scales by the chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ). Its gradient norm is positive: otherwise continuity gives zero gradient everywhere, the fundamental theorem along segments makes it constant, and compact support makes it zero (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz).

Counterexample

technique · direct

1.1F2F3F4givenalgebra

The norm scalings. For $0 < λ \leq 1$ put $φ_{λ} (x) : = φ (x / λ)$ , so $φ_{λ} \in C_{c}^{\infty} (B (0, λ)) ∖ {0}$ . If $q < \infty$ , substituting $y = x / λ$ and using [F2] gives $∥ φ_{λ} ∥_{L^{q}}^{q} = λ^{n} ∥ φ ∥_{L^{q}}^{q}$ , hence $∥ φ_{λ} ∥_{L^{q}} = λ^{n / q} ∥ φ ∥_{L^{q}} > 0$ . If $q = \infty$ , then for each $t > 0$ , [F2] gives $λ_{n} ({∣ φ_{λ} ∣ > t}) = λ^{n} λ_{n} ({∣ φ ∣ > t})$ , so $∥ φ_{λ} ∥_{L^{\infty}} = ∥ φ ∥_{L^{\infty}} > 0$ . In either case $D φ_{λ} (x) = λ^{- 1} (D φ) (x / λ)$ and $∥ D φ_{λ} ∥_{L^{p}} = λ^{n / p - 1} ∥ D φ ∥_{L^{p}} > 0$ ; the support statement uses [F3].

2.1F1step 1.1givenalgebra∎

The ratio diverges above $p^{*}$ . For finite $q > p^{*}$ , dividing the two scalings of step 1.1 gives $∥ φ_{λ} ∥_{L^{q}} / ∥ D φ_{λ} ∥_{L^{p}} = λ^{1 + n / q - n / p} ∥ φ ∥_{L^{q}} / ∥ D φ ∥_{L^{p}}$ ; the exponent is negative exactly when $\frac{1}{q} < \frac{1}{p} - \frac{1}{n} = \frac{1}{p^{*}}$ by [F1], so this ratio tends to $+ \infty$ . For $q = \infty$ , step 1.1 gives $∥ φ_{λ} ∥_{L^{\infty}} / ∥ D φ_{λ} ∥_{L^{p}} = λ^{1 - n / p} ∥ φ ∥_{L^{\infty}} / ∥ D φ ∥_{L^{p}} \to + \infty$ because $p < n$ . In either case any finite $C$ satisfying the proposed inequality for every compactly supported smooth $u$ would have to dominate this unbounded ratio, which is impossible. Also $∥ φ_{λ} ∥_{W^{1, p}}^{p} = λ^{n} ∥ φ ∥_{p}^{p} + λ^{n - p} \sum_{i} ∥ D_{i} φ ∥_{p}^{p} \leq λ^{n - p} (∥ φ ∥_{p}^{p} + \sum_{i} ∥ D_{i} φ ∥_{p}^{p})$ for $λ \leq 1$ . Thus the full-norm ratio has the same divergent lower bound $c λ^{1 + n / q - n / p}$ , with $1 / q = 0$ for $q = \infty$ . Both bounds fail for every $q > p^{*}$ .

Source notes

The dilation computation is Kinnunen's, printed pp. 61-62, and Laugesen's sharpness discussion, printed pp. 65-66; the counterexample is the standard concentrated-bump family.

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Sources

Juha Kinnunen, Sobolev Spaces (Aalto University, 2026, complete graduate lecture notes) (standard reference, not scraped)
Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, complete 158-page graduate notes) (standard reference, not scraped)

The W1,p→LqW1,p→Lq bound fails for q>p∗q>p∗ by dilation