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Relative compactness forces uniform translation continuity in $L^{p}$

Statement

Assume the Axiom of Countable Choice. Let $1 \leq p < \infty$ and let $F \subseteq L^{p} (R^{n})$ be relatively compact, that is, its closure in $L^{p} (R^{n})$ is compact. In the displayed nonnegative supremum, take the value $0$ if $F = \emptyset$ . Then $\sup_{f \in F} ∥ τ_{h} f - f ∥_{L^{p} (R^{n})} ⟶ 0 (∣ h ∣ \to 0) .$ This is the necessity of the translation hypothesis in the Fr'echet--Kolmogorov criterion.

Facts & Assumptions

Given: the Axiom of Countable Choice, $1 \leq p < \infty$ , and a relatively compact family $F \subseteq L^{p} (R^{n})$ with closure $\overline{F}$ .

[F1]

Compact metric spaces are totally bounded. A compact metric space is totally bounded and complete. (A compact metric space is complete and totally bounded, and neither implication uses any choice principle)

[F2]

Total boundedness is finite-net covering. A metric space $(X, d)$ is totally bounded when for every $δ > 0$ there are finitely many points $x_{1}, \dots, x_{N} \in X$ with $X = ⋃_{i = 1}^{N} B (x_{i}, δ)$ ; a subset of a totally bounded space which is itself totally bounded as a subspace has the same property for every $δ > 0$ . (Finite $ε$ -net and totally bounded metric space, Open ball, closed ball and sphere in a metric space, A totally bounded metric space is bounded, every subspace of a totally bounded space is totally bounded, and the closure of a totally bounded subset is totally bounded)

[F3]

Continuity of translation in $L^{p}$ . For every $g \in L^{p} (R^{n})$ , $∥ τ_{h} g - g ∥_{p} \to 0$ as $∣ h ∣ \to 0$ , where $τ_{h} g = g (\cdot - h)$ acts on almost-everywhere classes. ( $∥ τ_{h} f - f ∥_{p} \to 0$ in $L^{p} (R^{n})$ as $h \to 0$ , for $1 \leq p < \infty$ , Translation of a function on $R^{n}$ )

[F4]

Translation is an $L^{p}$ -isometry. $∥ τ_{h} g ∥_{p} = ∥ g ∥_{p}$ for every $g \in L^{p} (R^{n})$ and every $h$ , because Lebesgue measure is translation invariant. (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, The space $L^{p} (μ)$ as the quotient by null functions)

Proof

technique · Cover the compact closure by finitely many small balls, use continuity of translation at the finitely many centres, and propagate to the whole family by the isometry property and the triangle inequality

1.1F1F2given

If $F = \emptyset$ the supremum is $0$ and the claim holds. Otherwise the closure $\overline{F}$ is a compact subset of $L^{p} (R^{n})$ , so $(\overline{F}, d_{p})$ is a compact metric space and [F1] makes it totally bounded. Given $ε > 0$ , [F2] provides finitely many centres $f_{1}, \dots, f_{N} \in \overline{F}$ with $\overline{F} \subseteq ⋃_{i = 1}^{N} B (f_{i}, ε / 3)$ .

2.1F3F4step 1.1

For each $i$ the centre $f_{i}$ is an element of $L^{p} (R^{n})$ , so [F3] gives $δ_{i} > 0$ with $∥ τ_{h} f_{i} - f_{i} ∥_{p} < ε / 3$ whenever $∣ h ∣ < δ_{i}$ ; set $δ : = \min_{i} δ_{i} > 0$ , a minimum over the nonempty finite set of indices. By [F4], $∥ τ_{h} (f - f_{i}) ∥_{p} = ∥ f - f_{i} ∥_{p}$ for every $f \in L^{p} (R^{n})$ and every $h$ .

3.1F3F4step 1.1step 2.1∎

Fix $f \in F$ and $∣ h ∣ < δ$ . By step 1.1 there is $i$ with $∥ f - f_{i} ∥_{p} < ε / 3$ , and the triangle inequality together with steps 2.1 and 1.1 gives $∥ τ_{h} f - f ∥_{p} \leq ∥ τ_{h} (f - f_{i}) ∥_{p} + ∥ τ_{h} f_{i} - f_{i} ∥_{p} + ∥ f_{i} - f ∥_{p} < ε / 3 + ε / 3 + ε / 3 = ε$ . Hence $\sup_{f \in F} ∥ τ_{h} f - f ∥_{p} \leq ε$ for all $∣ h ∣ < δ$ , and since $ε > 0$ was arbitrary the supremum tends to $0$ . Countable Choice enters only through the continuity-of-translation interface [F3].

Depends on

Used by

High frequencies destroy uniform translation control Counterexample

Dependency tree · two levels

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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)

Relative compactness forces uniform translation continuity in LpLp