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Compactly supported smooth functions are dense in W^{k,p}(R^n)

Statement

Assume Countable Choice. Let $n \geq 1$ , $k \in N_{0}$ , $1 \leq p < \infty$ and $K \in {R, C}$ . Then the compactly supported smooth functions $C_{c}^{\infty} (R^{n}; K)$ are dense in $W^{k, p} (R^{n}; K)$ : for every $u \in W^{k, p} (R^{n}; K)$ and every $δ > 0$ there is $φ \in C_{c}^{\infty} (R^{n}; K)$ with $∥ φ - u ∥_{W^{k, p} (R^{n})} < δ .$ No such norm-density assertion is made for $p = \infty$ .

Facts & Assumptions

Given: Countable Choice; $k \in N_{0}$ ; $1 \leq p < \infty$ ; $K \in {R, C}$ ; a class $u \in W^{k, p} (R^{n}; K)$ ; and a tolerance $δ > 0$ .

[F1]

Cutoff bumps: for $0 < r < R$ there is $χ \in C_{c}^{\infty} (R^{n}; [0, 1])$ with $χ = 1$ on ${\overline{B}}_{r} (0)$ and $supp χ \subseteq B_{R} (0)$ (A smooth bump between concentric Euclidean balls); fix such a $χ$ with $r = 1$ and outer radius $2$ . For $χ_{R} (x) : = χ (x / R)$ one has $χ_{R} = 1$ on ${\overline{B}}_{R} (0)$ , $supp χ_{R} \subseteq B_{2 R} (0)$ , and, by the chain rule, $∥ D^{β} χ_{R} ∥_{L^{\infty} (R^{n})} \leq C_{β} R^{- ∣ β ∣}$ for $1 \leq ∣ β ∣ \leq k$ with constants depending only on $k$ and the fixed bump.

[F2]

Smooth-factor Leibniz rule: $χ_{R} u \in W^{k, p} (R^{n}; K)$ for every $∣ α ∣ \leq k$ , with $D^{α} (χ_{R} u) = \sum_{β \leq α} (\binom{α}{β}) (D^{β} χ_{R}) D^{α - β} u$ almost everywhere (Weak Leibniz rule with a smooth factor).

[F3]

Dominated convergence: if $g_{R} \to g$ almost everywhere and $∣ g_{R} ∣ \leq G$ for a single integrable $G$ , then $\int g_{R} \to \int g$ (Dominated convergence).

[F4]

Interior mollification: for $f \in W^{k, p} (R^{n}; K)$ and a nonnegative unit-mass bump $ρ$ supported in ${\overline{B}}_{1} (0)$ , the mollifications $f_{ε} = ρ_{ε} * f$ are smooth on $R^{n}$ with $D^{α} f_{ε} = ρ_{ε} * (D^{α} f)$ for every $∣ α ∣ \leq k$ ; if $f$ is compactly supported then so is $f_{ε}$ (Interior mollification commutes with weak derivatives, The mollifier family generated by a unit-mass smooth bump).

[F5]

Approximate identity convergence for finite $p$ : with $(ρ_{ε})$ a mollifier family, $ρ_{ε} * f \to f$ in $L^{p} (R^{n}; K)$ for every $f \in L^{p} (R^{n}; K)$ and $1 \leq p < \infty$ , for real and complex scalars alike (A unit-mass smooth bump generates an $L^{1}$ approximate identity, Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$ , Complex translation, convolution, approximate identities, and mollification).

[F6]

The norm of $W^{k, p} (R^{n}; K)$ is the $ℓ^{p}$ sum of the $L^{p}$ norms of $D^{α} u$ , $∣ α ∣ \leq k$ (Integer-order Sobolev spaces and their norms).

Choice use. Countable Choice is used through the mollification and approximate-identity interfaces of [F4]–[F5]; the cutoffs of [F1] and the dominated-convergence argument of step 2.1 are explicit.

Proof

technique · direct

1.1F1F2given

Fix the cutoff family $χ_{R}$ of [F1]. For each $R > 0$ the function $χ_{R} u$ satisfies the hypotheses of [F2] with $η = χ_{R}$ , so $χ_{R} u \in W^{k, p} (R^{n}; K)$ and $D^{α} (χ_{R} u) = χ_{R} D^{α} u + \sum_{0 \neq β \leq α} (\binom{α}{β}) (D^{β} χ_{R}) D^{α - β} u$ ; in particular $χ_{R} u$ is compactly supported, with support in $B_{2 R} (0)$ .

2.1F1F3F6step 1.1

Large- $R$ convergence. For each $∣ α ∣ \leq k$ , $D^{α} (χ_{R} u) - D^{α} u = (χ_{R} - 1) D^{α} u + \sum_{0 \neq β \leq α} (\binom{α}{β}) (D^{β} χ_{R}) D^{α - β} u .$ The first term tends to $0$ in $L^{p}$ by [F3], since $(χ_{R} - 1) D^{α} u \to 0$ pointwise as $R \to \infty$ and its $p$ -th power is bounded by $2^{p} ∣ D^{α} u ∣^{p} \in L^{1}$ ; each remaining term is bounded in $L^{p}$ by $C_{β} R^{- ∣ β ∣} ∥ D^{α - β} u ∥_{L^{p}}$ , which tends to $0$ . Summing over the finitely many $∣ α ∣ \leq k$ and using [F6], there is $R$ with $∥ χ_{R} u - u ∥_{W^{k, p} (R^{n})} < \frac{δ}{2} .$

3.1step 2.1

Fix such an $R$ and write $f : = χ_{R} u$ , a compactly supported class in $W^{k, p} (R^{n}; K)$ ; then $∥ f - u ∥_{W^{k, p} (R^{n})} < δ / 2$ and $supp f \subseteq B_{2 R} (0)$ .

4.1F1F4step 3.1

Mollification. Fix a nonnegative unit-mass $ρ \in C_{c}^{\infty} (R^{n})$ supported in ${\overline{B}}_{1} (0)$ , obtained by normalizing a bump from [F1] with inner radius $1 / 2$ and outer radius $1$ . For every $0 < ε < 1$ the function $f_{ε} = ρ_{ε} * f$ lies in $C_{c}^{\infty} (R^{n}; K)$ (smoothness and support in $B_{2 R + ε} (0)$ by [F4]), and $D^{α} f_{ε} = ρ_{ε} * (D^{α} f)$ for every $∣ α ∣ \leq k$ .

5.1F5F6step 3.1step 4.1∎

Convergence of the mollified approximants: for each $∣ α ∣ \leq k$ , the class $D^{α} f$ lies in $L^{p} (R^{n}; K)$ and [F5] gives $∥ D^{α} f_{ε} - D^{α} f ∥_{L^{p}} \to 0$ as $ε \to 0^{+}$ ; summing over $∣ α ∣ \leq k$ with [F6], choose $ε > 0$ with $∥ f_{ε} - f ∥_{W^{k, p} (R^{n})} < δ / 2$ . Then $φ : = f_{ε} \in C_{c}^{\infty} (R^{n}; K)$ satisfies $∥ φ - u ∥_{W^{k, p}} \leq ∥ φ - f ∥_{W^{k, p}} + ∥ f - u ∥_{W^{k, p}} < δ$ by steps 3.1 and 4.1. Since $u$ and $δ$ were arbitrary, $C_{c}^{\infty} (R^{n}; K)$ is dense; the hypothesis $p < \infty$ enters exactly here, and no assertion is made for $p = \infty$ .

Depends on

Used by

Boundary regularity required by the constructed extension Remark

Dependency tree · two levels

66 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), Theorem 1.21 and Remark 1.22(1) (standard reference, not scraped)
Richard S. Laugesen, Linear Analysis and Partial Differential Equations (2020), Theorem 3.9 (standard reference, not scraped)