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Compactly supported Sobolev functions extend by zero in every integer order

Statement

Assume Countable Choice. Let $Ω \subseteq R^{n}$ be open with $n \geq 1$ , let $k \in N_{0}$ , $1 \leq p \leq \infty$ and $K \in {R, C}$ . Suppose $u \in W^{k, p} (Ω; K)$ vanishes almost everywhere outside a compact set $K_{0} \subset Ω$ . Let $E_{0} u$ denote the extension of a representative of $u$ by zero to $R^{n}$ , and for $∣ α ∣ \leq k$ let $E_{0} (D^{α} u)$ denote the extension by zero of the corresponding derivative representative. Then $E_{0} u \in W^{k, p} (R^{n}; K), D^{α} (E_{0} u) = E_{0} (D^{α} u) almost everywhere, ∣ α ∣ \leq k,$ and every $L^{p}$ component norm is preserved: $∥ E_{0} (D^{α} u) ∥_{L^{p} (R^{n})} = ∥ D^{α} u ∥_{L^{p} (Ω)}, ∥ E_{0} u ∥_{L^{p} (R^{n})} = ∥ u ∥_{L^{p} (Ω)},$ so that $∥ E_{0} u ∥_{W^{k, p} (R^{n})} = ∥ u ∥_{W^{k, p} (Ω)}$ . The case $k = 0$ is included, and complex scalars are handled by the same bilinear pairing.

Facts & Assumptions

Given: Countable Choice; an open set $Ω \subseteq R^{n}$ ; $k \in N_{0}$ ; $1 \leq p \leq \infty$ ; $K \in {R, C}$ ; a class $u \in W^{k, p} (Ω; K)$ with a representative that vanishes almost everywhere outside a compact $K_{0} \subset Ω$ ; a multi-index $α$ with $∣ α ∣ \leq k$ ; and a test function $φ \in C_{c}^{\infty} (R^{n})$ .

[F1]

A class $u$ lies in $W^{k, p} (Ω; K)$ exactly when $u \in L^{p} (Ω; K)$ and, for every $∣ α ∣ \leq k$ , there is a class $D^{α} u \in L^{p} (Ω; K)$ with a locally integrable representative satisfying the weak test identity on $Ω$ ; the norm is the derivative sum for finite $p$ and the maximum of the essential bounds for $p = \infty$ (Integer-order Sobolev spaces and their norms).

[F2]

The defining weak identity reads $\int_{Ω} u D^{α} ψ = (- 1)^{∣ α ∣} \int_{Ω} (D^{α} u) ψ$ for every $ψ \in C_{c}^{\infty} (Ω; K)$ , with a bilinear pairing and no conjugation; any two locally integrable weak $α$ -derivatives agree almost everywhere (Weak derivative of a locally integrable function, Uniqueness of a weak derivative as an almost-everywhere class).

[F3]

If $V \subseteq Ω$ is open and $v = D^{α} u$ weakly on $Ω$ , then $v ∣_{V} = D^{α} (u ∣_{V})$ weakly on $V$ ; consequently, if $u = 0$ almost everywhere on an open $V \subseteq Ω$ , then $D^{α} u = 0$ almost everywhere on $V$ (Linearity, locality, and commutation of weak derivatives, Uniqueness of a weak derivative as an almost-everywhere class).

[F4]

For compact $K_{0} \subseteq Ω$ with $Ω$ open there is $χ \in C_{c}^{\infty} (Ω)$ with $0 \leq χ \leq 1$ and $χ = 1$ on an open neighbourhood $U$ of $K_{0}$ ; this cutoff exists in ZF (Test function cutoffs and euclidean localization).

[F5]

If a $C^{\infty}$ function vanishes identically on an open set, then all of its partial derivatives vanish there: along each coordinate direction the function is constant on a small interval, and induction on the order of differentiation gives the assertion.

[F6]

The integral over a measurable set is the integral of the product with its indicator, and a function that vanishes off $Ω$ has the same integral over $R^{n}$ as over $Ω$ ; this applies to $∣ u ∣^{p}$ and to $∣ D^{α} u ∣^{p}$ for finite $p$ (Integral over a measurable subset, Complex Lp classes and Euclidean test-function conventions).

[F7]

Changing a locally integrable representative on a null set does not change a weak derivative class or an $L^{p}$ class (Weak differentiation ignores null-set changes).

Choice use. The declared principle is Countable Choice, used through the locality and uniqueness interfaces of [F2]–[F3] and through the Sobolev well-definedness recorded in [F1]; the cutoff of [F4] is choice-free, and the computation below is otherwise explicit.

Proof

technique · direct

1.1F3F4given

Fix $α$ with $∣ α ∣ \leq k$ and $φ \in C_{c}^{\infty} (R^{n})$ , and choose $χ$ as in [F4], with $χ = 1$ on a neighbourhood $U \supseteq K_{0}$ . Since $Ω ∖ K_{0}$ is open and $u = 0$ almost everywhere on it, [F3] gives $D^{α} u = 0$ almost everywhere on $Ω ∖ K_{0}$ ; in particular $u$ and $D^{α} u$ vanish almost everywhere off $K_{0}$ .

2.1F1F2step 1.1

The function $χ φ$ lies in $C_{c}^{\infty} (Ω; K)$ , so the weak identity of [F2] on $Ω$ applies to it: $\int_{Ω} u D^{α} (χ φ) = (- 1)^{∣ α ∣} \int_{Ω} (D^{α} u) χ φ .$

3.1F5step 1.1step 2.1

Compare the left side of step 2.1 with $\int_{Ω} u D^{α} φ$ . The difference is $\int_{Ω} u D^{α} ((χ - 1) φ)$ ; the smooth function $(χ - 1) φ$ vanishes identically on the open set $U$ , so by [F5] all its partial derivatives vanish on $U \supseteq K_{0}$ , while off $K_{0}$ the factor $u$ vanishes almost everywhere. Hence the integrand vanishes almost everywhere on $Ω$ and $\int_{Ω} u D^{α} (χ φ) = \int_{Ω} u D^{α} φ = \int_{R^{n}} (E_{0} u) D^{α} φ .$

3.2F3step 1.1step 2.1

Compare the right side of step 2.1 with $(- 1)^{∣ α ∣} \int_{Ω} (D^{α} u) φ$ . The difference is $(- 1)^{∣ α ∣} \int_{Ω} (D^{α} u) (χ - 1) φ$ ; here $(χ - 1) φ$ vanishes on $U$ and $D^{α} u$ vanishes almost everywhere off $K_{0}$ , so the integrand vanishes almost everywhere. Therefore $\int_{Ω} (D^{α} u) χ φ = \int_{Ω} (D^{α} u) φ = \int_{R^{n}} E_{0} (D^{α} u) φ .$

4.1F2step 3.1step 3.2

Steps 3.1 and 3.2 together with step 2.1 give $\int_{R^{n}} (E_{0} u) D^{α} φ = (- 1)^{∣ α ∣} \int_{R^{n}} E_{0} (D^{α} u) φ$ for the arbitrary test $φ \in C_{c}^{\infty} (R^{n})$ fixed in step 1.1. Since $φ$ was arbitrary, $E_{0} (D^{α} u)$ is a weak $α$ -derivative of $E_{0} u$ on $R^{n}$ , and it is the unique locally integrable class by [F2].

5.1F6step 4.1

Membership and norms. The extension $E_{0} u$ is measurable with $∣ E_{0} u ∣ = ∣ u ∣$ on $Ω$ and $E_{0} u = 0$ off $Ω$ ; by [F6], for finite $p$ , $\int_{R^{n}} ∣ E_{0} u ∣^{p} = \int_{Ω} ∣ u ∣^{p} < \infty,$ while for $p = \infty$ the two essential suprema agree because the two functions agree almost everywhere on $Ω$ and the extension vanishes off $Ω$ . Thus $E_{0} u \in L^{p} (R^{n}; K)$ with equal norm, and the same computation applied to each $E_{0} (D^{α} u)$ , $∣ α ∣ \leq k$ , gives $E_{0} (D^{α} u) \in L^{p} (R^{n}; K)$ with $∥ E_{0} (D^{α} u) ∥_{L^{p} (R^{n})} = ∥ D^{α} u ∥_{L^{p} (Ω)}$ .

6.1F1F7step 4.1step 5.1∎

By step 4.1 and step 5.1 the class $E_{0} u$ has, for every $∣ α ∣ \leq k$ , an $L^{p}$ weak $α$ -derivative on $R^{n}$ ; [F1] therefore gives $E_{0} u \in W^{k, p} (R^{n}; K)$ with $D^{α} (E_{0} u) = E_{0} (D^{α} u)$ almost everywhere, and the norm formula of [F1] together with the component equalities of step 5.1 gives $∥ E_{0} u ∥_{W^{k, p} (R^{n})} = ∥ u ∥_{W^{k, p} (Ω)}$ . Changing representatives on null sets changes nothing by [F7]; $k = 0$ is the case of the single multi-index $α = 0$ ; complex scalars use the same bilinear pairing componentwise.

Depends on

Used by

Dependency tree · two levels

41 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Juha Kinnunen, Sobolev Spaces (2026), Lemma 1.14(4)–(5) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (2014), §3.4 (standard reference, not scraped)