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✓ 14 results · all verified · 7 also independently AI-judged

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Smooth Approximation and Sobolev Extension

1 · Prerequisites

2 · Summary

This page develops the approximation and extension theory of integer-order Sobolev spaces. Mollification is first shown to commute with weak derivatives on the interior of a domain, and the resulting interior mollifications of a class in $W^{k, p}$ converge to it in $W^{k, p}$ on every compactly contained open subset, for every $1 \leq p < \infty$ . Meyers–Serrin density on an arbitrary open set is then obtained by exhausting the domain with compact pieces and mollifying each piece with a dyadic error budget, summing the errors rather than the pieces; it requires no regularity of the boundary and no extension of the class beyond the domain, and the endpoint $p = \infty$ is excluded, as the one-dimensional corner $∣ x ∣$ witnesses. On the whole space the argument upgrades to compactly supported smooth approximation.

The second half introduces zero extension and extension operators. The space $W_{0}^{k, p}$ is defined as a closure, zero extension is proved for classes supported in a compact subset of the open set in every order and exponent, and the one-dimensional case $W_{0}^{1, p}$ is treated through test-function approximants. A $W^{k, p}$ -extension domain is an open set for which restriction admits a bounded linear right inverse. The half-space operator is constructed by reflection with Vandermonde-matched moments, the flattening of $C^{k}$ boundary charts is shown to preserve $W^{k, p}$ locally, and the two are combined into a bounded extension operator on every bounded $C^{k}$ domain whose output is supported in any prescribed neighbourhood of the closure. On such domains, restrictions of globally smooth compactly supported functions are dense for $1 \leq p < \infty$ , and whole-space Sobolev inequalities transfer through the extension operator with its operator norm. A closing remark records the exact boundary regularity each construction uses, together with the endpoints and strengthenings the page does not claim.

Conventions: $Ω \subseteq R^{n}$ is open with $n \geq 1$ , $1 \leq p \leq \infty$ , $k \in N_{0}$ , and the scalar field $K$ is $R$ or $C$ ; Sobolev spaces are almost-everywhere classes and restriction is a contraction. Countable Choice is declared through the published measure, convolution, weak-derivative and localization interfaces that carry it, and the Axiom of Choice is declared on the extension and boundary-density items exactly where their chart, partition and ACL interfaces invoke it. The scalar field is complex where the cited convolution interface is complex; no trace operator, no boundary point values and no $W^{k, \infty}$ norm density are asserted on this page.

3 · Logical flowchart

4 · Definitions, theorems and proofs

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Interior mollification commutes with weak derivatives

Statement

Assume Countable Choice. Let $Ω \subseteq R^{n}$ be open with $n \geq 1$ , let $u \in W^{k, p} (Ω; K)$ with $k \in N_{0}$ , $1 \leq p \leq \infty$ and $K \in {R, C}$ , and let $ρ \in C_{c}^{\infty} (R^{n})$ be nonnegative with $\int_{R^{n}} ρ = 1$ and $supp ρ \subseteq {\overline{B}}_{1} (0)$ . For $ε > 0$ put $ρ_{ε} (x) = ε^{- n} ρ (x / ε)$ and $Ω_{ε} = {x \in Ω : dist (x, R^{n} ∖ Ω) > ε}, Ω_{ε} = R^{n} when Ω = R^{n} .$ Then $ρ_{ε} * u$ is defined and smooth on $Ω_{ε}$ , and for every multi-index $α$ with $∣ α ∣ \leq k$ , $D^{α} (ρ_{ε} * u) = ρ_{ε} * (D^{α} u) on Ω_{ε},$ as pointwise smooth functions for the constructed representatives and as almost-everywhere classes. Here $D^{α} u$ is extended by zero off $Ω$ in the convolution.

Facts & Assumptions

Given: Countable Choice; an open set $Ω \subseteq R^{n}$ ; $k \in N_{0}$ ; $1 \leq p \leq \infty$ ; $K \in {R, C}$ ; $u \in W^{k, p} (Ω; K)$ ; a nonnegative unit-mass $ρ \in C_{c}^{\infty} (R^{n})$ with support in ${\overline{B}}_{1} (0)$ ; $ε > 0$ ; and a multi-index $α$ with $∣ α ∣ \leq k$ .

[F1]

A class $u$ lies in $W^{k, p} (Ω; K)$ exactly when $u \in L^{p} (Ω; K)$ and for each $∣ α ∣ \leq k$ there is an $L^{p}$ class $D^{α} u$ whose locally integrable representative satisfies the weak test identity on $Ω$ ; the derivatives are unique as almost-everywhere classes (Integer-order Sobolev spaces and their norms).

[F2]

The defining weak identity is $\int_{Ω} u D^{α} ψ = (- 1)^{∣ α ∣} \int_{Ω} (D^{α} u) ψ$ for every $ψ \in C_{c}^{\infty} (Ω; K)$ , with a bilinear pairing (Weak derivative of a locally integrable function).

[F3]

The mollifier family is $ρ_{ε} (x) = ε^{- n} ρ (x / ε)$ , so $\int ρ_{ε} = 1$ and $supp ρ_{ε} \subseteq {\overline{B}}_{ε} (0)$ (The mollifier family generated by a unit-mass smooth bump).

[F4]

If $f \in L_{l o c}^{1} (R^{n}; C)$ and $φ \in C_{c}^{\infty} (R^{n})$ has mass one, then $f * φ_{ε}$ is smooth on $R^{n}$ and $\partial^{α} (f * φ_{ε}) = f * (\partial^{α} φ_{ε})$ for every multi-index $α$ (Convolution with a mollifier is smooth, and derivatives pass under the integral sign).

[F5]

The zero extension of a representative of an $L^{p} (Ω)$ class is locally integrable on $R^{n}$ , and changing a representative on a null set changes neither the weak-derivative identities nor the $L^{p}$ classes (Weak differentiation ignores null-set changes).

[F6]

For the smooth map $y \mapsto ρ_{ε} (x - y)$ the chain rule gives $\partial_{y_{i}} ρ_{ε} (x - y) = - (\partial_{i} ρ_{ε}) (x - y)$ , equivalently $D_{y}^{α} ρ_{ε} (x - y) = (- 1)^{∣ α ∣} D_{x}^{α} ρ_{ε} (x - y)$ for every multi-index $α$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F7]

A function with continuous classical derivatives through order $k$ on an open set has those classical derivatives as its weak derivatives, and the weak derivative class is unique (Classical derivatives agree with weak derivatives).

Choice use. Countable Choice is used through the well-definedness, local-integrability and uniqueness interfaces of [F1], [F2] and [F5]; the differentiation-under-the-integral-sign theorem of [F4] also declares it. The support computation and the sign substitution [F6] are choice-free.

Proof

technique · direct

1.1F3F4F5given

Fix a representative of $u$ and let $\tilde{u}$ be its extension by zero to $R^{n}$ ; by [F5], $\tilde{u} \in L_{l o c}^{1} (R^{n})$ , and we define $u_{ε} (x) : = (\tilde{u} * ρ_{ε}) (x) = \int_{R^{n}} \tilde{u} (y) ρ_{ε} (x - y) d y$ . By [F3] and [F4] applied to $\tilde{u}$ , the function $u_{ε}$ is smooth on $R^{n}$ and for every $∣ α ∣ \leq k$ , $\partial^{α} u_{ε} (x) = \int_{R^{n}} \tilde{u} (y) \partial_{x}^{α} ρ_{ε} (x - y) d y .$

2.1F3step 1.1

Let $x \in Ω_{ε}$ . If $Ω = R^{n}$ this is every point; otherwise $dist (x, R^{n} ∖ Ω) > ε$ , so $\overline{B} (x, ε) \subseteq Ω$ . Since $supp ρ_{ε} \subseteq {\overline{B}}_{ε} (0)$ , the integrand $y \mapsto \tilde{u} (y) \partial_{x}^{α} ρ_{ε} (x - y)$ is supported in $\overline{B} (x, ε) \subseteq Ω$ , where $\tilde{u} = u$ almost everywhere; hence $\partial^{α} u_{ε} (x) = \int_{Ω} u (y) \partial_{x}^{α} ρ_{ε} (x - y) d y .$

3.1F6step 2.1

Substitute the sign identity of [F6] in step 2.1: for every $x \in Ω_{ε}$ , $\partial^{α} u_{ε} (x) = (- 1)^{∣ α ∣} \int_{Ω} u (y) D_{y}^{α} ρ_{ε} (x - y) d y .$

3.2F1F2step 2.1

For fixed $x \in Ω_{ε}$ the function $ψ (y) : = ρ_{ε} (x - y)$ lies in $C_{c}^{\infty} (Ω; R)$ by step 2.1, so the weak identity of [F2] applies with this $ψ$ : $\int_{Ω} u (y) D_{y}^{α} ρ_{ε} (x - y) d y = (- 1)^{∣ α ∣} \int_{Ω} (D^{α} u) (y) ρ_{ε} (x - y) d y .$

4.1step 3.1step 3.2

Combining steps 3.1 and 3.2 and cancelling the two signs, which multiply to $+ 1$ , gives for every $x \in Ω_{ε}$ $\partial^{α} u_{ε} (x) = \int_{Ω} (D^{α} u) (y) ρ_{ε} (x - y) d y = (ρ_{ε} * E_{0} D^{α} u) (x),$ the last expression being the convolution of the zero-extended derivative class with $ρ_{ε}$ .

5.1F1F4F5F7step 4.1∎

The right-hand side of step 4.1 is a smooth function of $x$ on $Ω_{ε}$ by [F4] applied to the locally integrable extension $E_{0} D^{α} u$ ; thus $\partial^{α} u_{ε}$ extends the smooth function $ρ_{ε} * E_{0} D^{α} u$ restricted to $Ω_{ε}$ . By [F7] applied on the open set $Ω_{ε}$ , this classical derivative is the weak derivative of $u_{ε}$ there, for every $∣ α ∣ \leq k$ ; in particular $D^{α} (ρ_{ε} * u) = ρ_{ε} * (D^{α} u) on Ω_{ε} .$ Finally, the construction does not depend on the chosen representative of $u$ : changing it on a null set changes $\tilde{u}$ on a null set only, hence leaves both sides unchanged as almost-everywhere classes by [F5]. The case $k = 0$ is the single identity $D^{0} (ρ_{ε} * u) = ρ_{ε} * u$ , which is smoothness of $u_{ε}$ ; the case $Ω = R^{n}$ has $Ω_{ε} = R^{n}$ by definition; and complex scalars are handled by the bilinear pairing componentwise.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Local smooth approximation in integer-order Sobolev spaces

Statement

Assume Countable Choice. Let $Ω \subseteq R^{n}$ be open with $n \geq 1$ , let $k \in N_{0}$ , $1 \leq p < \infty$ and $K \in {R, C}$ , and let $u \in W^{k, p} (Ω; K)$ . Fix a nonnegative $ρ \in C_{c}^{\infty} (R^{n})$ of unit mass with $supp ρ \subseteq {\overline{B}}_{1} (0)$ , put $ρ_{ε} (x) = ε^{- n} ρ (x / ε)$ , and define the interior mollification $u_{ε} (x) = (ρ_{ε} * \tilde{u}) (x) = \int_{R^{n}} \tilde{u} (y) ρ_{ε} (x - y) d y,$ where $\tilde{u}$ is a representative of $u$ extended by zero to $R^{n}$ . Then $u_{ε} \to u$ in $W^{k, p} (U; K)$ for every open $U \subset \subset Ω$ ; equivalently $u_{ε} \to u$ in $W_{l o c}^{k, p} (Ω; K)$ . The exponent $p = 1$ is included, and no assertion about density or convergence in the $W^{k, \infty}$ norm is made.

Facts & Assumptions

Given: Countable Choice; an open set $Ω \subseteq R^{n}$ with $n \geq 1$ ; $k \in N_{0}$ ; $1 \leq p < \infty$ ; $K \in {R, C}$ ; a class $u \in W^{k, p} (Ω; K)$ ; a nonnegative unit-mass $ρ \in C_{c}^{\infty} (R^{n})$ with support in ${\overline{B}}_{1} (0)$ ; and an open set $U \subset \subset Ω$ , so that $\overline{U}$ is a compact subset of $Ω$ .

[F1]

Interior mollification commutes with weak derivatives: with $Ω_{ε} = {x \in Ω : dist (x, R^{n} ∖ Ω) > ε}$ and $D^{α} u$ extended by zero, one has $D^{α} u_{ε} = ρ_{ε} * (D^{α} u)$ on $Ω_{ε}$ for every $∣ α ∣ \leq k$ , and $u_{ε}$ is smooth there (Interior mollification commutes with weak derivatives).

[F2]

The family $ρ_{ε}$ is the mollifier family generated by the unit-mass bump $ρ$ (The mollifier family generated by a unit-mass smooth bump).

[F3]

Approximate identity convergence: the family $(ρ_{ε})$ is an $L^{1}$ approximate identity (A unit-mass smooth bump generates an $L^{1}$ approximate identity); if $f \in L^{p} (R^{n})$ and $1 \leq p < \infty$ , then $∥ f * ρ_{ε} - f ∥_{L^{p} (R^{n})} \to 0$ as $ε \to 0^{+}$ (Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$ ), and the same holds for complex-valued $f$ with the complex convolution conventions, including the case of a complex scalar field (Complex translation, convolution, approximate identities, and mollification).

[F4]

For $w \in L^{p} (Ω; K)$ the extension by zero $E_{0} w$ lies in $L^{p} (R^{n}; K)$ with $∥ E_{0} w ∥_{L^{p} (R^{n})} = ∥ w ∥_{L^{p} (Ω)}$ , because the integral over a measurable set is the integral of the indicator product (Integral over a measurable subset, Complex Lp classes and Euclidean test-function conventions).

[F5]

The $W^{k, p}$ norm is the $ℓ^{p}$ sum of the $L^{p}$ norms of all derivative classes $D^{α} u$ with $∣ α ∣ \leq k$ ; for $p = \infty$ it is the maximum of the essential bounds (Integer-order Sobolev spaces and their norms).

[F6]

Compact containment: $\overline{U} \subseteq Ω$ compact with $Ω$ open implies $d : = dist (\overline{U}, R^{n} ∖ Ω) > 0$ , and for $0 < ε < d$ every $x \in U$ satisfies $dist (x, R^{n} ∖ Ω) \geq d > ε$ , i.e. $U \subseteq Ω_{ε}$ ; Here distance to the empty set is $+ \infty$ , including when $U = \emptyset$ or $Ω = R^{n}$ ; the containment still holds. This is elementary metric topology and uses no choice.

Choice use. Countable Choice is used through the approximate-identity and mollification interfaces of [F1] and [F3]; the compact-containment constant of [F6] is explicit and choice-free.

Proof

technique · direct

1.1F4F6given

Fix $U \subset \subset Ω$ and put $d : = dist (\overline{U}, R^{n} ∖ Ω) > 0$ as in [F6]; then $U \subseteq Ω_{ε}$ for every $0 < ε < d$ . Also, for each $∣ α ∣ \leq k$ the zero extension $E_{0} (D^{α} u)$ belongs to $L^{p} (R^{n}; K)$ with $∥ E_{0} (D^{α} u) ∥_{L^{p} (R^{n})} = ∥ D^{α} u ∥_{L^{p} (Ω)}$ by [F4], since $D^{α} u \in L^{p} (Ω; K)$ .

2.1F1step 1.1

By [F1], for every $0 < ε < d$ and every $∣ α ∣ \leq k$ the identity $D^{α} u_{ε} = ρ_{ε} * (E_{0} D^{α} u) on U$ holds as an identity of smooth functions, since $U \subseteq Ω_{ε}$ .

3.1F2F3step 1.1step 2.1

For each $∣ α ∣ \leq k$ the right-hand side of step 2.1 converges to $E_{0} D^{α} u$ in $L^{p} (R^{n}; K)$ : by [F2] and [F3] applied to the $L^{p}$ class $f = E_{0} D^{α} u$ , $∥ ρ_{ε} * f - f ∥_{L^{p} (R^{n})} \to 0$ as $ε \to 0^{+}$ . Hence $∥ D^{α} u_{ε} - D^{α} u ∥_{L^{p} (U)} \leq ∥ ρ_{ε} * (E_{0} D^{α} u) - E_{0} D^{α} u ∥_{L^{p} (R^{n})} ⟶ 0.$

4.1F3F5step 3.1given∎

Summing the finitely many convergences of step 3.1 over $∣ α ∣ \leq k$ , the norm formula of [F5] gives $∥ u_{ε} - u ∥_{W^{k, p} (U)} \to 0$ ; since $U \subset \subset Ω$ was arbitrary, this is convergence in $W_{l o c}^{k, p} (Ω)$ . The proof uses $p < \infty$ exactly through the approximate-identity convergence in step 3.1, makes no claim for $p = \infty$ , and the case $k = 0$ is the single multi-index $α = 0$ , namely convergence in $L_{l o c}^{p}$ .

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Meyers–Serrin density on an arbitrary open set

Statement

Assume Countable Choice. Let $Ω \subseteq R^{n}$ be open with $n \geq 1$ , let $k \in N_{0}$ , $1 \leq p < \infty$ and $K \in {R, C}$ . Then the intersection $C^{\infty} (Ω; K) \cap W^{k, p} (Ω; K)$ is dense in $W^{k, p} (Ω; K)$ : for every $u \in W^{k, p} (Ω; K)$ and every $δ > 0$ there is a function $v \in C^{\infty} (Ω; K)$ with $v \in W^{k, p} (Ω; K)$ and $∥ v - u ∥_{W^{k, p} (Ω)} < δ$ . No regularity of $\partial Ω$ and no extension of $u$ beyond $Ω$ is assumed. The exponent $p = \infty$ is excluded, as the companion remark records.

Facts & Assumptions

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

[F1]

Compact exhaustion. Every nonempty open $Ω \subseteq R^{n}$ admits compact sets $K_{1} \subseteq K_{2} \subseteq \dots$ with $K_{j} \subset int K_{j + 1}$ and $⋃_{j} K_{j} = Ω$ ; the sets $K_{j} = {x \in Ω : ∣ x ∣ \leq j, dist (x, R^{n} ∖ Ω) \geq 1 / j}$ are closed and bounded, hence compact, and satisfy these inclusions (elementary closed-and-bounded compactness in $R^{n}$ ; the sets may be enlarged by finite unions, and for bounded $Ω$ the radius clause is inactive for large $j$ ; when $Ω = R^{n}$ , interpret the distance to the empty complement as $+ \infty$ ).

[F2]

Smooth cutoffs: for compact $K \subseteq Ω$ with $Ω$ open there is $χ \in C_{c}^{\infty} (Ω)$ with $0 \leq χ \leq 1$ and $χ = 1$ on a neighbourhood of $K$ ; this uses no choice (Test function cutoffs and euclidean localization).

[F3]

Local smooth approximation: for every open $U \subset \subset Ω$ , $u_{ε} \to u$ in $W^{k, p} (U; K)$ as $ε \to 0^{+}$ , where $u_{ε}$ is the interior mollification of $u$ ; in particular, for any $θ > 0$ there is $ε > 0$ with $∥ u_{ε} - u ∥_{W^{k, p} (U)} < θ$ (Local smooth approximation in integer-order Sobolev spaces).

[F4]

Smooth-factor Leibniz rule: for $η \in C_{c}^{\infty} (Ω; K)$ and $u \in W^{k, p} (Ω; K)$ one has $η u \in W^{k, p} (Ω; K)$ with the Leibniz formula for $D^{α} (η u)$ , $∣ α ∣ \leq k$ (Weak Leibniz rule with a smooth factor).

[F5]

Classical smooth compactly supported functions have their classical derivatives as weak derivatives, hence lie in $W^{k, p} (Ω; K)$ (Classical derivatives agree with weak derivatives).

[F6]

Norm and linearity: the $W^{k, p}$ norm of Integer-order Sobolev spaces and their norms is well defined and definite on classes (The Sobolev norm descends to equivalence classes), weak differentiation is linear on classes (Linearity, locality, and commutation of weak derivatives), and $∥ S ∥_{W^{k, p}}^{p} = \sum_{∣ α ∣ \leq k} ∥ D^{α} S ∥_{L^{p}}^{p}$ for $1 \leq p < \infty$ .

[F7]

Fatou's lemma: for nonnegative measurable functions $g_{J}$ , $\int {lim inf}_{J} g_{J} \leq {lim inf}_{J} \int g_{J}$ (Fatou's lemma).

[F8]

Mollification on a compactly supported piece stays compactly supported: if $η u$ is supported in a compact set $M \subset Ω$ and $ε < dist (M, R^{n} ∖ Ω)$ , then the mollification $ρ_{ε} * (η u)$ (zero extension outside $Ω$ ) is supported in the closed $ε$ -neighbourhood of $M$ , which is a compact subset of $Ω$ (Local smooth approximation in integer-order Sobolev spaces).

Choice use. Countable Choice selects the exhaustion cutoffs of [F2] and the dyadic mollification radii below; the published weak, $L^{p}$ and mollification interfaces of [F3]–[F6] also declare it. All selections are countable and can be made by a least-index rule.

Proof

technique · direct

1.1F1F2given

Fix the compact exhaustion $K_{j}$ of [F1] and, using [F2] and Countable Choice, cutoffs $ψ_{j} \in C_{c}^{\infty} (Ω)$ with $0 \leq ψ_{j} \leq 1$ , $ψ_{j} = 1$ on a neighbourhood of $K_{j}$ and $supp ψ_{j} \subset int K_{j + 1}$ ; set $χ_{j} : = 1 - \prod_{i \leq j} (1 - ψ_{i})$ and $η_{1} : = χ_{1}$ , $η_{j} : = χ_{j} - χ_{j - 1}$ for $j \geq 2$ . Then each $η_{j} \in C_{c}^{\infty} (Ω)$ is nonnegative with $0 \leq η_{j} \leq 1$ , satisfies $supp η_{j} \subset int K_{j + 1}$ and $η_{j} = 0$ on a neighbourhood of $K_{j - 1}$ , the supports are locally finite, and $\sum_{j} η_{j} = 1$ on $Ω$ ; hence $\sum_{j} η_{j} u = u$ as a locally finite sum of classes. The empty case $Ω = \emptyset$ is trivial because the only class is $0$ , so assume $Ω \neq \emptyset$ .

2.1F3F4F8step 1.1

For each $j$ , [F4] gives $η_{j} u \in W^{k, p} (Ω; K)$ , supported in the compact set $supp η_{j} \subset int K_{j + 1}$ ; using [F3] on the open set $U_{j} : = int K_{j + 1} \subset \subset Ω$ and [F8] to keep the support inside $Ω$ , choose $ε_{j} > 0$ so small that $v_{j} : = ρ_{ε_{j}} * (η_{j} u)$ is a smooth function compactly supported in $U_{j}$ and $∥ v_{j} - η_{j} u ∥_{W^{k, p} (Ω)} = ∥ v_{j} - η_{j} u ∥_{W^{k, p} (U_{j})} < δ 2^{- j - 1} .$ For $j \geq 2$ , also take $ε_{j} < \frac{1}{2} dist (supp η_{j}, K_{j - 1}) > 0$ ; then $v_{j}$ vanishes near $K_{j - 1}$ , so the mollified pieces remain locally finite.

3.1F4F5step 1.1step 2.1

Define $v : = \sum_{j} v_{j}$ and $e_{j} : = v_{j} - η_{j} u$ . Since the $v_{j}$ have locally finite supports, $v$ is a locally finite sum of smooth compactly supported functions on $Ω$ , hence $v \in C^{\infty} (Ω; K)$ ; and $v - u = \sum_{j} (v_{j} - η_{j} u) = \sum_{j} e_{j}$ as a locally finite sum, with each $e_{j} \in W^{k, p} (Ω; K)$ and $∥ e_{j} ∥_{W^{k, p} (Ω)} < δ 2^{- j - 1}$ by step 2.1.

4.1F6step 2.1step 3.1

For every $∣ α ∣ \leq k$ one has $D^{α} (v - u) = \sum_{j} D^{α} e_{j}$ as locally integrable classes: near any point of $Ω$ only finitely many $η_{j}$ , hence only finitely many $e_{j}$ , are nonzero, and on that neighbourhood the identity follows from the linearity and locality of weak differentiation applied to the finite sum; the identity therefore holds as an $L_{l o c}^{p} (Ω)$ identity.

5.1F6F7step 2.1step 4.1

Let $S_{J} : = \sum_{j \leq J} e_{j}$ , so that $D^{α} S_{J} \to D^{α} (v - u)$ pointwise for every $∣ α ∣ \leq k$ by the local finiteness of step 4.1. Fatou's lemma applied to the nonnegative functions $g_{J} : = \sum_{∣ α ∣ \leq k} ∣ D^{α} S_{J} ∣^{p}$ gives $\int_{Ω} \sum_{∣ α ∣ \leq k} ∣ D^{α} (v - u) ∣^{p} \leq {lim inf}_{J \to \infty} \int_{Ω} g_{J} = {lim inf}_{J \to \infty} ∥ S_{J} ∥_{W^{k, p} (Ω)}^{p} \leq (\sum_{j = 1}^{\infty} ∥ e_{j} ∥_{W^{k, p} (Ω)})^{p} < δ^{p},$ where the middle equality is the norm formula of [F6] and the last inequality is the triangle inequality for the norm together with $\sum_{j} ∥ e_{j} ∥_{W^{k, p}} < \sum_{j} δ 2^{- j - 1} \leq δ$ .

6.1F6step 3.1step 5.1given∎

By step 5.1, $∥ v - u ∥_{W^{k, p} (Ω)} < δ$ ; by step 3.1, $v \in C^{\infty} (Ω; K)$ ; and $v = (v - u) + u \in W^{k, p} (Ω; K)$ because $v - u$ and $u$ both are. Since $u \in W^{k, p} (Ω; K)$ and $δ > 0$ were arbitrary, $C^{\infty} (Ω; K) \cap W^{k, p} (Ω; K)$ is dense.

CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

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 $p = \infty$ .

RemarkRemark: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Meyers–Serrin excludes the W^{k,∞} norm endpoint

Remark

The local mollification theorem Local smooth approximation in integer-order Sobolev spaces and the Meyers–Serrin density theorem Meyers–Serrin density on an arbitrary open set both require $1 \leq p < \infty$ ; neither asserts that smooth functions are dense in $W^{k, \infty} (Ω)$ in the Sobolev norm. The exclusion is not a defect of the proofs but a genuine endpoint failure, and the one-dimensional example $u (x) = ∣ x ∣$ on $(- 1, 1)$ exhibits it.

First, $u \in W^{1, \infty} (- 1, 1)$ with weak derivative the sign function $sgn (x) = 1_{(0, 1)} - 1_{(- 1, 0)}$ . Indeed $x \mapsto x$ is smooth with classical derivative $1$ , which is its weak derivative by Classical derivatives agree with weak derivatives, so the truncation calculus Positive, negative, and truncated Sobolev functions applied with $p = \infty$ gives $∣ x ∣ \in W^{1, \infty} (- 1, 1)$ and $D ∣ x ∣ = sgn (x) \cdot 1$ almost everywhere; the norm of Integer-order Sobolev spaces and their norms then computes $∥ ∣ x ∣ ∥_{W^{1, \infty} (- 1, 1)} = \max {1, 1} = 1$ .

Second, no sequence of smooth functions converges to $∣ x ∣$ in $W^{1, \infty} (- 1, 1)$ -norm, hence the finite- $p$ conclusion cannot be extended to $p = \infty$ . Suppose $g_{m} \in C^{\infty} (- 1, 1)$ satisfied $∥ g_{m} - ∣ x ∣ ∥_{W^{1, \infty}} < 1 / (2 m)$ or merely $∥ g_{m}^{'} - sgn ∥_{L^{\infty} (- 1, 1)} < 1 / 2$ for some $m$ . On $(0, 1)$ the sign equals $1$ , so $g_{m}^{'} > 1 / 2$ almost everywhere there; since $g_{m}^{'}$ is continuous, $g_{m}^{'} \geq 1 / 2$ at every point of $(0, 1)$ , for otherwise continuity would give a whole interval on which $g_{m}^{'} < 1 / 2$ , a set of positive measure contradicting the essential bound. Applying the same reasoning on $(- 1, 0)$ , where the sign equals $- 1$ , gives $g_{m}^{'} \leq - 1 / 2$ on $(- 1, 0)$ . Continuity of $g_{m}^{'}$ at $0$ then forces the two incompatible limits $g_{m}^{'} (0) = \lim_{x \to 0^{+}} g_{m}^{'} (x) \geq 1 / 2$ and $g_{m}^{'} (0) = \lim_{x \to 0^{-}} g_{m}^{'} (x) \leq - 1 / 2$ , a contradiction.

The same phenomenon separates the exponents for local approximation: the mollifications of $∣ x ∣$ converge to $∣ x ∣$ in $W_{l o c}^{1, q} (- 1, 1)$ for every finite $q$ by Local smooth approximation in integer-order Sobolev spaces, while the derivative error at the corner stays of size at least one half in $L^{\infty}$ for every mollification scale. Local convergence in every finite $W^{1, q}$ therefore does not imply convergence at the $W^{1, \infty}$ endpoint norm; the companion examples page computes the mollifications of $∣ x ∣$ explicitly.

DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Zero-boundary Sobolev space as a norm closure

Definition

Assume Countable Choice. Let $Ω \subseteq R^{n}$ be open with $n \geq 1$ , let $k \in N_{0}$ , let $1 \leq p \leq \infty$ , and let $K \in {R, C}$ .

Every $φ \in C_{c}^{\infty} (Ω; K)$ belongs to $W^{k, p} (Ω; K)$ . Indeed $φ$ is bounded with compact support, hence lies in $L^{p} (Ω; K)$ for every exponent in range, and each of its classical partial derivatives is again smooth and compactly supported, hence lies in $L^{p} (Ω; K)$ and is the corresponding weak derivative by Classical derivatives agree with weak derivatives.

Define the zero-boundary Sobolev space $W_{0}^{k, p} (Ω; K) : = \overline{C_{c}^{\infty} (Ω; K)},$ the closure taken in the normed space $(W^{k, p} (Ω; K), ∥ \cdot ∥_{W^{k, p} (Ω)})$ of Integer-order Sobolev spaces and their norms. The norm is a genuine norm on classes by The Sobolev norm descends to equivalence classes, so this is the usual metric closure of a subset of a normed space: $u \in W_{0}^{k, p} (Ω; K)$ if and only if for every $δ > 0$ there is $φ \in C_{c}^{\infty} (Ω; K)$ with $∥ u - φ ∥_{W^{k, p} (Ω)} < δ$ . Write $H_{0}^{k} (Ω) : = W_{0}^{k, 2} (Ω; K)$ when the scalar field is fixed by context.

Three warnings are part of the definition. First, this is a closure of almost-everywhere classes, and all its equalities are equalities of Sobolev classes. Second, no pointwise boundary values and no trace characterization are asserted: a description of $W_{0}^{k, p}$ by vanishing boundary data belongs to the later trace theory and is not used here. Third, the closure is defined for every $1 \leq p \leq \infty$ , but for $p = \infty$ it is not claimed that every compactly supported $W^{k, \infty} (Ω)$ function lies in $W_{0}^{k, \infty} (Ω)$ ; the definition merely names the closure of the test functions.

Source notes

Kinnunen, Definition 1.23 and Remarks 1.24, printed pp. 21–22, defines $W_{0}^{k, p}$ as the closure of the test functions and warns that this is not yet a boundary-value statement. Laugesen, Definition 3.11, printed p. 59, uses the same closure convention for $W_{0}^{k, p}$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Zero extension of W_0^{1,p} has no boundary derivative

Statement

Assume Countable Choice. For any open $Ω \subseteq R^{n}$ , $n \geq 1$ , any $1 \leq p \leq \infty$ and any $K \in {R, C}$ , extension by zero $E_{0} u (x) = u (x) (x \in Ω), E_{0} u (x) = 0 (x \notin Ω),$ acted on representatives, sends $W_{0}^{1, p} (Ω; K)$ linearly into $W^{1, p} (R^{n}; K)$ . For every $u \in W_{0}^{1, p} (Ω; K)$ and every coordinate direction $i$ , the class $D_{i} (E_{0} u)$ is the zero extension of the class $D_{i} u$ , and every $L^{p}$ component norm is preserved: $∥ E_{0} u ∥_{L^{p} (R^{n})} = ∥ u ∥_{L^{p} (Ω)}, ∥ D_{i} (E_{0} u) ∥_{L^{p} (R^{n})} = ∥ D_{i} u ∥_{L^{p} (Ω)},$ so that $∥ E_{0} u ∥_{W^{1, p} (R^{n})} = ∥ u ∥_{W^{1, p} (Ω)}$ .

Facts & Assumptions

Given: Countable Choice; an open set $Ω \subseteq R^{n}$ with $n \geq 1$ ; $1 \leq p \leq \infty$ ; $K \in {R, C}$ ; a class $u \in W_{0}^{1, p} (Ω; K)$ ; and a test function $φ \in C_{c}^{\infty} (R^{n})$ .

[F1]

Membership $u \in W_{0}^{1, p} (Ω; K)$ means that for every $δ > 0$ there is $ψ \in C_{c}^{\infty} (Ω; K)$ with $∥ u - ψ ∥_{W^{1, p} (Ω)} < δ$ , and $W_{0}^{1, p} \subseteq W^{1, p}$ ; the norm is the Sobolev norm of Integer-order Sobolev spaces and their norms (Zero-boundary Sobolev space as a norm closure).

[F2]

For $w \in L^{p} (Ω; K)$ and its zero extension $E_{0} w$ : the extension is measurable, $∣ E_{0} w ∣ = ∣ w ∣$ on $Ω$ and $E_{0} w = 0$ off $Ω$ , so $∥ E_{0} w ∥_{L^{p} (R^{n})} = ∥ w ∥_{L^{p} (Ω)}$ for $1 \leq p \leq \infty$ because the integral over a measurable set is the integral of the indicator product; the map $E_{0}$ is linear on classes; and for $v \in C_{c}^{\infty} (Ω; K)$ the extension $E_{0} v$ lies in $C_{c}^{\infty} (R^{n})$ with $\partial_{i} (E_{0} v) = E_{0} (\partial_{i} v)$ for every $i$ (Integral over a measurable subset, Complex Lp classes and Euclidean test-function conventions).

[F3]

The defining weak identity on an open set $V$ is $\int_{V} w D^{α} ψ = (- 1)^{∣ α ∣} \int_{V} (D^{α} w) ψ$ for all $ψ \in C_{c}^{\infty} (V; K)$ (Weak derivative of a locally integrable function).

[F4]

A $C^{1}$ (indeed $C^{\infty}$ ) function on an open set has each of its classical partial derivatives as its weak derivative there (Classical derivatives agree with weak derivatives).

[F5]

Hölder's inequality: for conjugate exponents $p, q$ and measurable $f, g$ , $\int ∣ f g ∣ \leq ∥ f ∥_{p} ∥ g ∥_{q}$ whenever the norms on the right are finite (Holder's inequality for integrals, including the endpoint cases).

[F6]

A class in $W^{1, p} (R^{n}; K)$ is exactly an $L^{p}$ class whose coordinate weak derivatives exist as $L^{p}$ classes, with the norm of Integer-order Sobolev spaces and their norms; weak derivatives are unique up to null sets, and changing representatives does not change the classes (Uniqueness of a weak derivative as an almost-everywhere class, Weak differentiation ignores null-set changes).

Choice use. Countable Choice is used once, in step 1.3, to select a single test-function approximant for each precision $1 / j$ ; the rest of the argument is explicit and choice-free.

Proof

technique · direct

1.1F2given

The zero extension $E_{0}$ is linear on $L^{p}$ classes, preserves $L^{p}$ norms, and sends $C_{c}^{\infty} (Ω; K)$ into $C_{c}^{\infty} (R^{n})$ with $\partial_{i} (E_{0} v) = E_{0} (\partial_{i} v)$ ; in particular $E_{0} v_{j} - E_{0} v_{k} = E_{0} (v_{j} - v_{k})$ and $∥ E_{0} w ∥_{L^{p} (R^{n})} = ∥ w ∥_{L^{p} (Ω)}$ for every $L^{p}$ class $w$ .

1.2F2F3F4given

For $v \in C_{c}^{\infty} (Ω; K)$ the classical partial derivative $\partial_{i} (E_{0} v)$ is the weak $i$ -derivative of $E_{0} v$ on $R^{n}$ , so for every test $φ \in C_{c}^{\infty} (R^{n})$ one has $\int_{R^{n}} E_{0} v \partial_{i} φ = - \int_{R^{n}} E_{0} (\partial_{i} v) φ .$

1.3F1given

Since $u \in W_{0}^{1, p} (Ω; K)$ , [F1] with $δ = 1 / j$ provides for each integer $j \geq 1$ some $v_{j} \in C_{c}^{\infty} (Ω; K)$ with $∥ u - v_{j} ∥_{W^{1, p} (Ω)} < 1 / j$ ; Countable Choice selects one such sequence $(v_{j})_{j \geq 1}$ .

2.1F1step 1.1step 1.3

The linearity and isometry of $E_{0}$ give $E_{0} v_{j} \to E_{0} u$ and $E_{0} (\partial_{i} v_{j}) \to E_{0} (D_{i} u)$ in $L^{p} (R^{n})$ as $j \to \infty$ , because $v_{j} \to u$ and $\partial_{i} v_{j} \to D_{i} u$ in $L^{p} (Ω)$ by step 1.3 and the definition of the Sobolev norm.

3.1F5step 1.2step 2.1

For the fixed test $φ$ , step 1.2 gives $\int E_{0} v_{j} \partial_{i} φ = - \int E_{0} (\partial_{i} v_{j}) φ$ for every $j$ . Hölder's inequality on the compact support of $φ$ turns the $L^{p}$ convergences of step 2.1 into convergence of both integrals: $\int_{R^{n}} E_{0} v_{j} \partial_{i} φ \to \int_{R^{n}} E_{0} u \partial_{i} φ$ and $\int_{R^{n}} E_{0} (\partial_{i} v_{j}) φ \to \int_{R^{n}} E_{0} (D_{i} u) φ$ ; hence $\int_{R^{n}} E_{0} u \partial_{i} φ = - \int_{R^{n}} E_{0} (D_{i} u) φ .$

4.1F6step 1.1step 3.1∎

Since $φ$ was an arbitrary test function, step 3.1 exhibits $E_{0} (D_{i} u)$ as a weak $i$ -derivative of the $L^{p}$ class $E_{0} u$ on $R^{n}$ ; by [F6] therefore $E_{0} u \in W^{1, p} (R^{n}; K)$ with $D_{i} (E_{0} u) = E_{0} (D_{i} u)$ almost everywhere, the component norms agree by step 1.1, and $u \mapsto E_{0} u$ is linear because $E_{0}$ is. Changing representatives on null sets changes no class by [F6], the case $k = 0$ does not arise here, and complex scalars are covered by the same bilinear pairing.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

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.

DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Sobolev extension domains and extension operators

Definition

Assume Countable Choice. Fix $k \in N_{0}$ , $1 \leq p \leq \infty$ , an open set $Ω \subseteq R^{n}$ with $n \geq 1$ , and $K \in {R, C}$ .

Restriction. Let $F \in W^{k, p} (R^{n}; K)$ and let $α \in N_{0}^{n}$ with $∣ α ∣ \leq k$ . By the restriction and locality clause of Linearity, locality, and commutation of weak derivatives, applied with $V = Ω$ , the class $F ∣_{Ω}$ lies in $W^{k, p} (Ω; K)$ and $D^{α} (F ∣_{Ω}) = (D^{α} F) ∣_{Ω}$ almost everywhere on $Ω$ ; by the uniqueness of weak derivatives the derivative class of the restriction is determined by the class of $F$ alone. So restriction is a well-defined operation on Sobolev classes, and the norm formula of Integer-order Sobolev spaces and their norms gives $∥ F ∣_{Ω} ∥_{W^{k, p} (Ω)} \leq ∥ F ∥_{W^{k, p} (R^{n})}$ , since each restricted derivative has no larger $L^{p}$ norm.

Extension domain. For fixed $k, p, Ω$ and scalar field $K$ , call $Ω$ a $W^{k, p}$ -extension domain if there is a bounded linear operator $E : W^{k, p} (Ω; K) ⟶ W^{k, p} (R^{n}; K)$ such that, for every $u \in W^{k, p} (Ω; K)$ , $(E u) ∣_{Ω} = u as an almost-everywhere class on Ω .$ Equivalently, the restriction map in the first paragraph possesses a bounded linear right inverse. Boundedness of $E$ is the finiteness of the operator norm $∥ E ∥ = \sup {∥ E u ∥_{W^{k, p} (R^{n})} : u \in W^{k, p} (Ω; K), ∥ u ∥_{W^{k, p} (Ω)} \leq 1};$ the displayed right-inverse identity is an identity of $L^{p} (Ω)$ classes, not of pointwise values.

The operator, and the numerical bound $∥ E ∥$ , may depend on $k$ , $p$ , $Ω$ and $K$ . This definition asserts no common operator for all indices at once, no linearity of some canonically selected extension, and no control of pointwise values on $\partial Ω$ ; in particular it does not define a trace operator. No claim is made here that zero extension is an extension operator for a general open set, and none that any particular open set fails to be an extension domain; both assertions belong to later items of this page and to its companion.

Source notes

Kinnunen, Definition 3.42, printed p. 84, introduces the extension domain and the bounded right inverse of the restriction map; Theorem 3.43 there transfers whole-space Sobolev inequalities through such an operator. Laugesen, Theorem 3.12, printed pp. 60–62, constructs such an operator for bounded $C^{1}$ graph domains and finite $p$ , and Corollary 3.13 extends the first-order result to $p = \infty$ by direct local bounds.

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Integer-order Sobolev extension from a half-space

Statement

Assume the Axiom of Choice and $n \geq 1$ . In the proof write $e_{n}$ for the last canonical basis vector $e_{n - 1}$ . Let $H = R^{n - 1} \times (0, \infty) \subseteq R^{n}$ be the upper half-space, written $x = (x^{'}, t)$ with $x^{'} \in R^{n - 1}$ and $t > 0$ . For every $k \in N_{0}$ , every $1 \leq p \leq \infty$ and every $K \in {R, C}$ there is a bounded linear extension operator $E_{k, p} : W^{k, p} (H; K) ⟶ W^{k, p} (R^{n}; K), (E_{k, p} u) ∣_{H} = u almost everywhere on H .$ For $k \geq 1$ one may take, with the unique coefficients $(a_{1}, \dots, a_{k}) \in R^{k}$ satisfying $\sum_{j = 1}^{k} a_{j} (- j)^{m} = 1$ for $m = 0, 1, \dots, k - 1$ , $E_{k, p} u (x^{'}, t) = u (x^{'}, t) (t > 0), E_{k, p} u (x^{'}, t) = \sum_{j = 1}^{k} a_{j} u (x^{'}, - j t) (t < 0),$ and for $k = 0$ one may take the even reflection $E_{0, p} u (x^{'}, t) = u (x^{'}, ∣ t ∣)$ . In particular $H$ is a $W^{k, p}$ -extension domain in the sense of Sobolev extension domains and extension operators for every $k$ and every $p$ , and no density assertion is made or needed.

Facts & Assumptions

Given: the Axiom of Choice; the half-space $H = R^{n - 1} \times (0, \infty)$ ; $k \in N_{0}$ ; $1 \leq p \leq \infty$ ; $K \in {R, C}$ ; a class $u \in W^{k, p} (H; K)$ ; and a test function $φ \in C_{c}^{\infty} (R^{n})$ .

[F1]

Sobolev classes on an open set: $u \in W^{k, p} (H; K)$ means that for every multi-index $α$ with $∣ α ∣ \leq k$ there is a class $D^{α} u \in L^{p} (H; K)$ satisfying the weak identity $\int_{H} u D^{α} ψ = (- 1)^{∣ α ∣} \int_{H} (D^{α} u) ψ$ for all $ψ \in C_{c}^{\infty} (H; K)$ , with $D^{0} u = u$ ; the norm is the $ℓ^{p}$ sum over $∣ α ∣ \leq k$ , and the maximum of essential bounds for $p = \infty$ (Integer-order Sobolev spaces and their norms).

[F2]

Weak differentiation is local and linear: for open $V \subseteq R^{n}$ , $v = D^{α} u$ weakly on $R^{n}$ implies $v ∣_{V} = D^{α} (u ∣_{V})$ weakly on $V$ , and the sum of two weakly differentiable classes is weakly differentiable with the sum of the derivatives (Linearity, locality, and commutation of weak derivatives).

[F3]

Linear changes of variables: for the invertible linear map $T_{j} (x^{'}, t) = (x^{'}, - j t)$ , whose determinant has absolute value $j$ , and every nonnegative measurable $f$ , $\int_{R^{n}} f d λ_{n} = j \int_{R^{n}} f \circ T_{j} d λ_{n}$ ; in particular $\int_{{t < 0}} ∣ w (x^{'}, - j t) ∣^{p} d x^{'} d t = j^{- 1} \int_{H} ∣ w (x^{'}, τ) ∣^{p} d x^{'} d τ$ for every measurable $w$ (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).

[F4]

Traces of normal sections. Let $u \in W^{k, p} (H; K)$ with $k \geq 1$ and $1 \leq p \leq \infty$ . For every multi-index $γ$ with $∣ γ ∣ \leq k - 1$ and almost every $x^{'} \in R^{n - 1}$ the section $s \mapsto D^{γ} u (x^{'}, s)$ has a representative that is absolutely continuous on every compact interval $[0, T]$ , extends continuously to $s = 0$ , and satisfies $\frac{d}{d s} D^{γ} u (x^{'}, s) = D^{γ + e_{n}} u (x^{'}, s)$ for almost every $s$ , where $e_{n}$ is the last unit vector; at $p = \infty$ one first restricts to a finite exponent on compact sets, since $W_{l o c}^{1, \infty} \subseteq W_{l o c}^{1, q}$ . The trace ${tr}_{γ} (x^{'}) : = \lim_{s \to 0^{+}} D^{γ} u (x^{'}, s)$ exists for almost every $x^{'}$ , is measurable, and satisfies the estimate $∣ {tr}_{γ} (x^{'}) ∣ \leq \frac{2}{T} \int_{0}^{T} ∣ D^{γ} u (x^{'}, s) ∣ d s + \int_{0}^{T} ∣ D^{γ + e_{n}} u (x^{'}, s) ∣ d s$ for every $0 < T < \infty$ , whose right-hand side is a.e. finite and integrable in $x^{'}$ over compact sets (The ACL characterisation of $W^{1, p}$ , One-dimensional $W^{1, p}$ functions have unique absolutely continuous representatives, Tonelli and Fubini for the completed product, with only almost-everywhere section measurability, The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures, Holder's inequality for integrals, including the endpoint cases).

[F5]

Half-space integration by parts with traces. Let $u \in W^{k, p} (H; K)$ , let $α = (β, m)$ with $∣ α ∣ \leq k$ , and let $φ \in C_{c}^{\infty} (R^{n})$ . Then $\int_{H} u D^{α} φ = (- 1)^{∣ α ∣} \int_{H} (D^{α} u) φ + \sum_{r = 0}^{m - 1} (- 1)^{r + 1} \int_{R^{n - 1}} {tr}_{r e_{n}} (x^{'}) \partial_{t}^{m - 1 - r} D^{β} φ (x^{'}, 0) d x^{'} .$ For the reflected function $f (x^{'}, t) = u (x^{'}, - j t)$ on the lower half-space, whose $α$ -derivative is $D^{α} f (x^{'}, t) = (- j)^{m} D^{α} u (x^{'}, - j t)$ and whose traces of the normal derivatives at $t = 0^{-}$ are $(- j)^{r} {tr}_{r e_{n}} (x^{'})$ , and the same formula holds with the boundary signs $(- 1)^{r}$ in place of $(- 1)^{r + 1}$ . In both formulas the boundary term keeps the tangential derivatives on the test function; they are not also applied to the trace. This follows by integrating in the normal variable first, then moving the tangential derivatives in the interior term onto $u$ . The one-dimensional integrations are justified by the absolutely continuous representatives of [F4] and assembled with Fubini's theorem (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability, A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[F6]

Vandermonde systems. For $k \geq 1$ the matrix $((- j)^{m})_{0 \leq m \leq k - 1, 1 \leq j \leq k}$ is invertible, because its determinant is the Vandermonde product $\prod_{i < i^{'}} ((- i^{'}) - (- i)) \neq 0$ over the distinct nodes $- 1, - 2, \dots, - k$ ; hence the moment system $\sum_{j = 1}^{k} a_{j} (- j)^{m} = 1$ , $m = 0, \dots, k - 1$ , has exactly one solution. This is finite linear algebra over $R$ and uses no choice.

[F7]

Extension domain and operator: $Ω$ is a $W^{k, p}$ -extension domain when there is a bounded linear $E : W^{k, p} (Ω; K) \to W^{k, p} (R^{n}; K)$ with $(E u) ∣_{Ω} = u$ almost everywhere for every class $u$ (Sobolev extension domains and extension operators).

Choice use. The Axiom of Choice is invoked only through the ACL, one-dimensional absolutely-continuous and Fubini interfaces cited in [F4]–[F5]; the Vandermonde coefficients of [F6] and the reflection formula are explicit.

Proof

technique · direct

1.1F6given

For $k \geq 1$ set $J_{k} = {1, \dots, k}$ and let $(a_{j})_{j \in J_{k}}$ be the unique solution of the moment system supplied by [F6]. For $k = 0$ set $J_{0} = {1}$ and $a_{1} = 1$ , with no moment condition. In either case define $E u : = u$ on $H$ and $E u (x^{'}, t) : = \sum_{j \in J_{k}} a_{j} u (x^{'}, - j t) (t < 0) .$ For $k = 0$ this is exactly the even reflection $u (x^{'}, ∣ t ∣)$ .

1.2F4

Traces exist as in [F4]: for every $∣ γ ∣ \leq k - 1$ the trace ${tr}_{γ} (x^{'})$ of $D^{γ} u$ at $t = 0$ exists for almost every $x^{'}$ , is measurable, and obeys the displayed estimate, so each trace is integrable against the compactly supported traces of $D^{β} φ$ that appear below.

2.1step 1.1

Candidate derivatives. For every multi-index $α = (β, m)$ with $∣ α ∣ \leq k$ define $g_{α} : = D^{α} u$ on $H$ and $g_{α} (x^{'}, t) : = \sum_{j \in J_{k}} a_{j} (- j)^{m} D^{α} u (x^{'}, - j t) (t < 0) .$ When $k = 0$ , this gives $g_{0} = E u$ on the lower half-space as well.

3.1F1F3step 2.1

Membership and bounds. By [F3] each reflected summand satisfies $\int_{{t < 0}} ∣ D^{α} u (x^{'}, - j t) ∣^{p} = j^{- 1} \int_{H} ∣ D^{α} u ∣^{p}$ for $p < \infty$ , so $∥ g_{α} ∥_{L^{p} (R^{n})} \leq (1 + \sum_{j \in J_{k}} ∣ a_{j} ∣ j^{m - 1 / p}) ∥ D^{α} u ∥_{L^{p} (H)}, ∥ E u ∥_{L^{p} (R^{n})} \leq (1 + \sum_{j \in J_{k}} ∣ a_{j} ∣ j^{- 1 / p}) ∥ u ∥_{L^{p} (H)},$ with the corresponding essential-supremum bounds $(1 + \sum_{j \in J_{k}} ∣ a_{j} ∣ j^{m})$ and $(1 + \sum_{j \in J_{k}} ∣ a_{j} ∣)$ when $p = \infty$ ; in particular $E u$ and all $g_{α}$ lie in $L^{p} (R^{n}; K)$ .

3.2F2F5step 1.1step 1.2step 2.1

Interface cancellation. Fix $φ \in C_{c}^{\infty} (R^{n})$ and $α = (β, m)$ . Apply [F5] on $H$ and to each reflected summand on ${t < 0}$ . The tangential derivatives of $φ$ remain in the boundary terms; the tangential weak-derivative identity moves them onto the interior terms, giving $\int_{R^{n}} E u D^{α} φ = (- 1)^{∣ α ∣} \int_{R^{n}} g_{α} φ + \sum_{r = 0}^{m - 1} (- 1)^{r} \int_{R^{n - 1}} {tr}_{r e_{n}} (x^{'}) \partial_{t}^{m - 1 - r} D^{β} φ (x^{'}, 0) [(- 1) + \sum_{j \in J_{k}} a_{j} (- j)^{r}] d x^{'} .$ If $k \geq 1$ , every index $r \leq m - 1 \leq k - 1$ satisfies $\sum_{j \in J_{k}} a_{j} (- j)^{r} = 1$ by step 1.1, so each bracket vanishes; if $k = 0$ , then $m = 0$ and the boundary sum is empty. In either case $\int_{R^{n}} E u D^{α} φ = (- 1)^{∣ α ∣} \int_{R^{n}} g_{α} φ$ . The traces in the sum are integrable by step 1.2.

4.1F1F2F7step 1.1step 3.1step 3.2∎

Since $φ$ was an arbitrary test function, step 3.2 exhibits $g_{α}$ as the weak $α$ -derivative of the $L^{p}$ class $E u$ for every $∣ α ∣ \leq k$ ; with $E u \in L^{p} (R^{n})$ and $g_{α} \in L^{p} (R^{n})$ from step 3.1 and the norm formula of [F1], this gives $E u \in W^{k, p} (R^{n}; K)$ and $∥ E u ∥_{W^{k, p} (R^{n})} \leq C_{k, p} ∥ u ∥_{W^{k, p} (H)}$ for the finite constant determined by the coefficients. The map $u \mapsto E u$ is linear because the reflection formula is linear in $u$ on each half-space, and $(E u) ∣_{H} = u$ holds by construction; hence $E = E_{k, p}$ is a bounded linear extension operator and $H$ is a $W^{k, p}$ -extension domain in the sense of [F7]. The case $k = 0$ is the even reflection with no interface terms, and the case $n = 1$ is the same argument with $R^{n - 1}$ a single point and the traces taken at $0$ .

DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Bounded C^k domains and boundary charts

Definition

Assume Countable Choice for the Sobolev interfaces used by consumers of this definition. Let $k \geq 1$ and $n \geq 2$ . A bounded $C^{k}$ domain in $R^{n}$ is a nonempty bounded open set $Ω \subset R^{n}$ with the following local graph property. For every boundary point $x \in \partial Ω$ there are an open neighbourhood $W \subseteq R^{n}$ of $x$ , a rigid motion $R (p) = Q p + b$ with orthogonal $Q$ and $b \in R^{n}$ , an open ball $B \subseteq R^{n - 1}$ , and a function $h \in C^{k} (B; R)$ such that, after shrinking $W$ so that $R (W) \subseteq B \times R$ , $R (Ω \cap W) = R (W) \cap {(y, s) \in B \times R : s < h (y)} .$ In the coordinates $z = R (p)$ the domain therefore lies locally strictly below the graph $s = h (y)$ , and the boundary is that graph. The graph convention, including the requirement that the domain occupy exactly the one-sided subgraph, is the one of Bounded C1 domains and their outward normals; the regularity $h \in C^{k}$ is the only strengthening here, and connectedness is not required.

Write $R (p) = (y, s)$ . The flattening chart is $Φ : W ⟶ Φ (W) \subseteq B \times R, Φ (p) = (y, s - h (y)),$ with inverse $Φ^{- 1} (y, t) = R^{- 1} (y, t + h (y))$ on $Φ (W)$ . Thus $Φ (Ω \cap W) = Φ (W) \cap {t < 0}$ . Both maps are of class $C^{k}$ ; the coordinate shear $(y, s) \mapsto (y, s - h (y))$ has Jacobian determinant one. If $B^{'} \subset \overline{B^{'}} \subset B$ is a compactly contained concentric ball, then every derivative $D^{β} h$ with $∣ β ∣ \leq k$ is continuous on $\overline{B^{'}}$ , hence bounded there, and consequently the derivatives through order $k$ of $Φ$ and of $Φ^{- 1}$ are bounded on the corresponding compact patch. This is the precise sense in which a boundary chart is said to have bounded derivatives through order $k$ on the compact patches used; it does not assert any bound uniform in the chart, the point, or a boundary atlas.

In dimension $n = 1$ , the bounded sets satisfying this local one-sided condition are finite disjoint unions of bounded open intervals. Each endpoint has an interval neighbourhood on which the domain occupies one side; the coordinate is the identity or its reflection, and the graph function and derivative bounds are vacuous. General bounded open subsets of $R$ need not have this property. A bounded $C^{1}$ domain in the sense of Bounded C1 domains and their outward normals is a bounded $C^{1}$ domain in this sense when $n \geq 2$ .

This definition asserts no extension theorem, no trace operator, and no uniformity of chart constants: it fixes only the regularity of the boundary and the exact one-sided graph convention that later local statements use.

Source notes

Laugesen, Definition 3.10, printed pp. 58–59, fixes the $C^{m}$ boundary-graph convention and the local one-sided subgraph placement. Oh, Proposition 11.13 and Remark 11.14, printed pp. 157–159, use $C^{k}$ boundary charts and record that the extension construction depends on the order $k$ through the chart regularity. The shear determinant and the compact derivative bounds recorded here are immediate from the definition of $C^{k}$ and are used by C^k boundary flattening preserves local W^{k,p}.

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C^k boundary flattening preserves local W^{k,p}

Statement

Assume Countable Choice. Let $k \geq 1$ , $1 \leq p \leq \infty$ and $K \in {R, C}$ . Let $U, V \subseteq R^{n}$ be open and let $Φ : U \to V$ be a $C^{k}$ diffeomorphism with inverse $Ψ : = Φ^{- 1} : V \to U$ . Fix open sets $U_{0} \subset \subset U$ and $V_{0} \subset \subset V$ with $Φ (U_{0}) \subseteq V_{0}$ , and suppose that on $\overline{U_{0}}$ the derivatives of $Φ$ through order $k$ are bounded and that on $\overline{V_{0}}$ the derivatives of $Ψ$ through order $k$ are bounded; in the situation of Bounded C^k domains and boundary charts these are exactly the compact patches on which the flattening chart and its inverse have bounded derivatives through order $k$ . Then:

for every $u \in W^{k, p} (V_{0}; K)$ the composition $u \circ Φ$ belongs to $W^{k, p} (U_{0}; K)$ , and for every $1 \leq ∣ α ∣ \leq k$ its weak derivatives satisfy, almost everywhere on $U_{0}$ , $D^{α} (u \circ Φ) = \sum_{1 \leq ∣ β ∣ \leq ∣ α ∣} ((D^{β} u) \circ Φ) P_{α β} (D Φ, \dots, D^{k} Φ),$ where $P_{α β}$ is a universal polynomial with integer coefficients, whose values on $\overline{U_{0}}$ are bounded by a constant depending only on $n, k$ and the stated bounds for $Φ$ ;
there is a constant $C$ , depending only on $n$ , $k$ , $p$ and the two sets of chart bounds, with $∥ u \circ Φ ∥_{W^{k, p} (U_{0})} \leq C ∥ u ∥_{W^{k, p} (V_{0})} for all u \in W^{k, p} (V_{0}; K) .$ If in addition $Ψ (V_{0}) \subseteq U_{0}$ (equivalently, under the stated $Φ (U_{0}) \subseteq V_{0}$ , $Φ (U_{0}) = V_{0}$ ), the same assertion holds for composition with $Ψ$ from $W^{k, p} (U_{0})$ to $W^{k, p} (V_{0})$ .

For $k = 1$ only the first derivatives of $Φ$ and $Ψ$ enter, so bounded $C^{1}$ chart and inverse data suffice; no $C^{1}$ -only claim is made for $k > 1$ .

Facts & Assumptions

Given: Countable Choice; $k \geq 1$ ; $1 \leq p \leq \infty$ ; $K \in {R, C}$ ; the $C^{k}$ diffeomorphism $Φ : U \to V$ with inverse $Ψ$ ; open sets $U_{0} \subset \subset U$ , $V_{0} \subset \subset V$ with $Φ (U_{0}) \subseteq V_{0}$ ; and bounded derivatives through order $k$ of $Φ$ on $\overline{U_{0}}$ and of $Ψ$ on $\overline{V_{0}}$ .

[F1]

The flattening charts of Bounded C^k domains and boundary charts are built from a rigid motion and the graph function $h \in C^{k}$ and have Jacobian determinant $\det Q \in {- 1, 1}$ , hence absolute determinant $1$ ; only the coordinate shear has determinant $1$ ; their derivatives through order $k$ are bounded on every compactly contained patch, and this boundedness is exactly the hypothesis used below.

[F2]

$C^{1}$ change of variables: for a $C^{1}$ diffeomorphism $T : U^{'} \to V^{'}$ of open sets and every nonnegative measurable $f$ , $\int_{V^{'}} f (y) d y = \int_{U^{'}} f (T (x)) ∣ \det D T (x) ∣ d x$ (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions).

[F3]

A $C^{1}$ diffeomorphism maps Lebesgue null sets to Lebesgue null sets, so composition of almost-everywhere classes with $Φ$ or $Ψ$ is well defined independently of representatives (A C^1 diffeomorphism maps Lebesgue null sets to Lebesgue null sets).

[F4]

Chain rule: iterating The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ gives, for $u \in C^{k} (V; K)$ , the classical identity for $1 \leq ∣ α ∣ \leq k$ $D^{α} (u \circ Φ) (x) = \sum_{1 \leq ∣ β ∣ \leq ∣ α ∣} (D^{β} u) (Φ (x)) P_{α β} (x)$ , where $P_{α β} (x)$ is a universal integer-coefficient polynomial in the partial derivatives $D^{γ} Φ (x)$ , $1 \leq ∣ γ ∣ \leq ∣ α ∣$ , and in particular $∣ P_{α β} (x) ∣ \leq C_{α}$ on $\overline{U_{0}}$ with $C_{α}$ determined by the bounds on $Φ$ ; for $k = 1$ this is $D_{i} (u \circ Φ) = \sum_{j} (D_{j} u) \circ Φ \cdot D_{i} Φ_{j}$ . At order zero, $D^{0} (u \circ Φ) = u \circ Φ$ directly.

[F5]

Meyers--Serrin density on an arbitrary open set: for $u \in W^{k, q} (V_{0}; K)$ and $1 \leq q < \infty$ there are $u_{m} \in C^{\infty} (V_{0}; K) \cap W^{k, q} (V_{0}; K)$ with $u_{m} \to u$ in $W^{k, q} (V_{0}; K)$ (Meyers–Serrin density on an arbitrary open set).

[F6]

Classical derivatives of a $C^{k}$ function are its weak derivatives (Classical derivatives agree with weak derivatives).

[F7]

Weak stability: if $w_{m} \to w$ in $L_{l o c}^{p} (U^{'})$ and $z_{m} \to z$ in $L_{l o c}^{q} (U^{'})$ with $z_{m} = D^{α} w_{m}$ weakly and $1 \leq p, q \leq \infty$ , then $z = D^{α} w$ weakly on $U^{'}$ (Weak derivatives persist under local Lp limits).

[F8]

Bounded open sets have finite Lebesgue measure, and on a finite measure space every essentially bounded function is in every $L^{q}$ , with $∥ f ∥_{L^{q}} \leq ∣ V_{0} ∣^{1 / q} ∥ f ∥_{L^{\infty}}$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included, Holder's inequality for integrals, including the endpoint cases).

[F9]

Norm conventions: $∥ w ∥_{W^{k, p}}^{p} = \sum_{∣ α ∣ \leq k} ∥ D^{α} w ∥_{L^{p}}^{p}$ for $p < \infty$ and $∥ w ∥_{W^{k, \infty}} = \max_{∣ α ∣ \leq k} ∥ D^{α} w ∥_{L^{\infty}}$ (Integer-order Sobolev spaces and their norms).

Choice use. Countable Choice is used through the density interface [F5] and the weak-derivative interface [F7]; the chart bounds are given.

Proof

technique · direct

1.1F1F2F3given

Since $Ψ$ has bounded first derivatives on $\overline{V_{0}}$ , [F2] gives, for every nonnegative measurable $f$ on $V_{0}$ and $1 \leq p < \infty$ , $\int_{U_{0}} ∣ f (Φ (x)) ∣^{p} d x = \int_{Φ (U_{0})} ∣ f (y) ∣^{p} ∣ \det D Ψ (y) ∣ d y \leq C \int_{V_{0}} ∣ f (y) ∣^{p} d y .$ For $p = \infty$ , [F3] makes composition well defined on a.e. classes and gives $∥ f \circ Φ ∥_{L^{\infty} (U_{0})} \leq ∥ f ∥_{L^{\infty} (V_{0})}$ . Thus pullback by $Φ$ is bounded on the stated $L^{p}$ spaces. The analogous estimate for $Ψ$ holds when $Ψ (V_{0}) \subseteq U_{0}$ .

1.2F4given

Classical composition formula: if $u \in C^{k} (V_{0}; K)$ , then $u \circ Φ \in C^{k} (U_{0}; K)$ ; for $1 \leq ∣ α ∣ \leq k$ , [F4] gives $D^{α} (u \circ Φ) = \sum_{1 \leq ∣ β ∣ \leq ∣ α ∣} ((D^{β} u) \circ Φ) P_{α β}$ on $U_{0}$ , with $∣ P_{α β} ∣ \leq C_{α}$ determined by the bounds on $Φ$ . For $α = 0$ the identity is $D^{0} (u \circ Φ) = u \circ Φ$ .

2.1F9step 1.1step 1.2

Smooth-case estimate. Let $u \in C^{k} (V_{0}; K) \cap W^{k, p} (V_{0}; K)$ . For $α = 0$ , step 1.1 bounds $u \circ Φ$ . For $1 \leq ∣ α ∣ \leq k$ , step 1.2 and the bounded coefficients give $∥ D^{α} (u \circ Φ) ∥_{L^{p} (U_{0})} \leq C_{α} \sum_{1 \leq ∣ β ∣ \leq ∣ α ∣} ∥ (D^{β} u) \circ Φ ∥_{L^{p} (U_{0})} \leq C \sum_{1 \leq ∣ β ∣ \leq ∣ α ∣} ∥ D^{β} u ∥_{L^{p} (V_{0})}$ for finite $p$ by step 1.1, and the same estimate with essential suprema for $p = \infty$ . The Sobolev norm formula [F9] and the finiteness of the index sets then give $∥ u \circ Φ ∥_{W^{k, p} (U_{0})} \leq C ∥ u ∥_{W^{k, p} (V_{0})}$ .

3.1F5F6F7step 1.1step 1.2step 2.1

Finite exponent, general class. Let $1 \leq p < \infty$ and $u \in W^{k, p} (V_{0}; K)$ . By [F5] choose $u_{m} \in C^{\infty} (V_{0}; K) \cap W^{k, p} (V_{0}; K)$ with $u_{m} \to u$ in $W^{k, p} (V_{0})$ . For each $m$ , step 1.2 gives the classical derivative formulas, and step 2.1 gives $∥ u_{m} \circ Φ ∥_{W^{k, p} (U_{0})} \leq C ∥ u_{m} ∥_{W^{k, p} (V_{0})}$ . By step 1.1, $u_{m} \circ Φ \to u \circ Φ$ in $L^{p} (U_{0})$ and $(D^{β} u_{m}) \circ Φ \to (D^{β} u) \circ Φ$ in $L^{p} (U_{0})$ for every $∣ β ∣ \leq k$ . Since each $P_{α β}$ is bounded, the derivative fields converge to $h_{α} : = \sum_{1 \leq ∣ β ∣ \leq ∣ α ∣} ((D^{β} u) \circ Φ) P_{α β}$ for $1 \leq ∣ α ∣ \leq k$ . By [F7], each $h_{α}$ is the weak derivative $D^{α} (u \circ Φ)$ ; the order-zero derivative is $u \circ Φ$ . Thus $u \circ Φ \in W^{k, p} (U_{0})$ with the stated formulas, and the bound follows by passing the smooth estimates to the limit.

4.1F3F8F9step 3.1

Exponent $p = \infty$ . Let $u \in W^{k, \infty} (V_{0}; K)$ . Since $V_{0}$ has finite measure, [F8] gives $u \in W^{k, q} (V_{0})$ for any finite $q \geq 1$ , so step 3.1 yields the same weak derivative formulas for $u \circ Φ$ in one such $W^{k, q} (U_{0})$ . Each formula field $h_{α}$ is in $L^{\infty} (U_{0})$ because its factors $(D^{β} u) \circ Φ$ are essentially bounded by [F3] and its coefficients are bounded; also $u \circ Φ \in L^{\infty} (U_{0})$ by [F3]. Hence these weak derivatives lie in $L^{\infty}$ , giving $u \circ Φ \in W^{k, \infty} (U_{0})$ and the claimed norm bound.

5.1F4step 1.2step 2.1step 3.1step 4.1∎

If $Ψ (V_{0}) \subseteq U_{0}$ , then the two patch inclusions force $Φ (U_{0}) = V_{0}$ . Applying steps 1.1–4.1 with the roles of $Φ$ and $Ψ$ interchanged gives the asserted inverse estimate. For $k = 1$ the formula of [F4] involves only first derivatives, so bounded $C^{1}$ data for $Φ$ and $Ψ$ suffice; at order $k > 1$ the polynomials $P_{α β}$ involve derivatives of the chart through order $∣ α ∣$ , and no $C^{1}$ -only statement is claimed.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Bounded C^k domains admit integer-order Sobolev extension

Statement

Assume the Axiom of Choice. Let $k \geq 1$ , $1 \leq p \leq \infty$ , $K \in {R, C}$ , and let $Ω \subset R^{n}$ be a bounded $C^{k}$ domain in the graph sense of Bounded C^k domains and boundary charts. Then for every open set $V \subseteq R^{n}$ with $\overline{Ω} \subseteq V$ there is a bounded linear extension operator $E : W^{k, p} (Ω; K) ⟶ W^{k, p} (R^{n}; K), (E u) ∣_{Ω} = u a.e. on Ω,$ such that $supp (E u)$ is a compact subset of $V$ for every $u \in W^{k, p} (Ω; K)$ . In particular each bounded $C^{k}$ domain is a $W^{k, p}$ -extension domain, with an operator whose output is supported in any prescribed neighbourhood of $\overline{Ω}$ . For $k = 0$ , extension by zero is an isometric $L^{p}$ extension on any open $Ω$ ; its output is supported in the compact set $\overline{Ω} \subseteq V$ when $Ω$ is bounded.

Facts & Assumptions

Given: the Axiom of Choice; $k \geq 1$ ; $1 \leq p \leq \infty$ ; $K \in {R, C}$ ; a bounded $C^{k}$ domain $Ω \subset R^{n}$ ; an open $V$ with $\overline{Ω} \subseteq V$ ; and a class $u \in W^{k, p} (Ω; K)$ .

[F1]

Chart data: by Bounded C^k domains and boundary charts, at every $x \in \partial Ω$ there are an open neighbourhood $W_{x}$ , a rigid motion and a graph function $h_{x} \in C^{k}$ making $W_{x} \cap Ω$ the one-sided subgraph $t < h_{x} (y)$ ; the flattening chart $Φ_{x}$ and its inverse are $C^{k}$ maps whose derivatives through order $k$ are bounded on compactly contained patches, and $\det D Φ_{x} = \det Q_{x} \in {- 1, 1}$ , so $∣ \det D Φ_{x} ∣ = 1$ . Compactness of $\partial Ω$ is what allows finitely many such charts to cover the boundary.

[F2]

Half-space extension: for $H = R^{n - 1} \times (0, \infty)$ and all $k \geq 0$ , $1 \leq p \leq \infty$ there is a bounded linear extension operator $W^{k, p} (H; K) \to W^{k, p} (R^{n}; K)$ , equal to the input on $H$ , given for $k \geq 1$ by the moment reflection and for $k = 0$ by even reflection (Integer-order Sobolev extension from a half-space).

[F3]

$C^{k}$ flattening is a bounded change of variables between corresponding compactly contained local $W^{k, p}$ spaces when the two patches are images of one another, with constants depending only on $n, k, p$ and the compact chart bounds; for $k = 1$ bounded $C^{1}$ data suffice (C^k boundary flattening preserves local W^{k,p}).

[F4]

Cutoffs and locally finite partitions: every open cover of an open Euclidean set has an at most countable locally finite smooth partition of unity with compact supports, each lying in some cover member; a compact set inside an open set admits a smooth cutoff equal to one nearby (Test function cutoffs and euclidean localization). On a compact neighbourhood of $\overline{Ω}$ , local finiteness leaves only finitely many active pieces, which can be grouped by the finitely many chart labels.

[F5]

Multiplication by a smooth factor with bounded derivatives through order $k$ is bounded on $W^{k, p}$ and satisfies the Leibniz formula (Weak Leibniz rule with a smooth factor).

[F6]

Restriction and cutoff localisation: restriction to an open subset is a contraction, and multiplication by a factor $η \in C_{c}^{\infty} (Ω)$ is bounded, with the explicit constants $C_{α} (η) = \sum_{β \leq α} (\binom{α}{β}) ∥ D^{β} η ∥_{\infty}$ (Bounded restriction and cutoff localisation in Sobolev spaces).

[F7]

Compactly supported Sobolev classes extend by zero in every integer order and every $1 \leq p \leq \infty$ , with equal norms (Compactly supported Sobolev functions extend by zero in every integer order).

[F8]

Extension operator and Sobolev norms: the definition of a bounded linear extension operator as a right inverse of the restriction map (Sobolev extension domains and extension operators), with the norm convention of Integer-order Sobolev spaces and their norms.

[F9]

For an increasing sequence of nonnegative measurable functions, the integrals converge to the integral of the pointwise limit (Monotone convergence for the integral).

Choice use. AC selects a chart from the nonempty chart family at each boundary point before compactness reduces the cover to finitely many charts. Its countable instance is inherited through the Sobolev, cutoff and weak-Leibniz interfaces [F3]–[F6]. The remaining finite cutoffs and reflection formulas use no additional selection.

Proof

technique · direct

1.1F1given

Since $\partial Ω$ is compact and $V$ is an open neighbourhood of $\overline{Ω}$ , choose $ρ > 0$ with ${x : dist (x, \overline{Ω}) < 2 ρ} \subseteq V$ . Choose finitely many boundary charts $Φ_{i}$ , with the larger patches compactly contained in their original chart neighbourhoods, on nested patches $W_{i}^{'} ⋐ W_{i}$ , so that the larger patches have $ρ$ -neighbourhoods in $V$ and the smaller patches cover $\partial Ω$ . The inner patches may be taken thin enough in flattened normal coordinates that reflection by any factor $j \leq k$ keeps the support of a function localized there inside the larger flattened patch. Compactness also gives an open set $Ω_{0} \subset \subset Ω$ with $\overline{Ω} \subseteq Ω_{0} \cup ⋃_{i = 1}^{m} W_{i}^{'}$ .

2.1F4step 1.1

Let $N : = Ω_{0} \cup ⋃_{i = 1}^{m} W_{i}^{'}$ , an open neighbourhood of $\overline{Ω}$ inside $V$ . Apply [F4] on $N$ to this finite cover and choose a compact neighbourhood $K ⋐ N$ of $\overline{Ω}$ . Only finitely many partition supports meet $K$ ; these pieces still sum to one on a neighbourhood of $\overline{Ω}$ . Group them by their assigned cover member and extend them by zero outside $N$ . This gives $ζ_{0}, ζ_{1}, \dots, ζ_{m} \in C_{c}^{\infty} (V; [0, 1])$ with $\sum_{i} ζ_{i} = 1$ near $\overline{Ω}$ , $supp ζ_{0} \subset Ω_{0} ⋐ Ω$ , and $supp ζ_{i} \subset W_{i}^{'}$ for $i \geq 1$ . Each support is compact in $V$ .

2.2F3F6F9step 1.1

To apply [F3] on patches reaching the boundary, let $T : A \to B$ be either direction of a chart restricted to corresponding open half-patches. The derivatives of $T$ and $T^{- 1}$ through order $k$ have uniform bounds inherited from the compact ambient chart. Exhaust $A$ by nested open sets $A_{j} ⋐ A$ , $j \geq 1$ , and put $B_{j} = T (A_{j}) ⋐ B$ . For $v \in W^{k, p} (B)$ , [F3] applied on each matched pair gives the weak composition formulas and $∥ v \circ T ∥_{W^{k, p} (A_{j})} \leq C ∥ v ∥_{W^{k, p} (B_{j})} \leq C ∥ v ∥_{W^{k, p} (B)}$ , with $C$ independent of $j$ . Every test support in $A$ lies in some $A_{j}$ , so these same formula fields are weak derivatives on $A$ . Increasing the integrals by [F9] for finite $p$ , or taking essential bounds on the countable union for $p = \infty$ , proves the identical norm bound on $A$ . Thus no compact-containment hypothesis is being assumed of the entire half-patch.

3.1F5F6step 2.1given

Let $u \in W^{k, p} (Ω; K)$ . By [F5], since the ambient cutoff has bounded derivatives through order $k$ , each product $ζ_{i} u ∣_{Ω}$ belongs to $W^{k, p} (Ω; K)$ , is supported in $supp ζ_{i} \cap Ω$ , and satisfies $∥ ζ_{i} u ∥_{W^{k, p} (Ω)} \leq C_{i} ∥ u ∥_{W^{k, p} (Ω)}$ with $C_{i}$ determined by the cutoff; moreover $u = \sum_{i = 0}^{m} ζ_{i} u$ almost everywhere on $Ω$ .

4.1F7step 3.1

Interior piece: $ζ_{0} u$ is supported in the compact set $supp ζ_{0} \subset Ω$ , so by [F7] its extension by zero $E_{0} (ζ_{0} u)$ lies in $W^{k, p} (R^{n}; K)$ , agrees with $ζ_{0} u$ on $Ω$ , is supported in $supp ζ_{0} \subseteq V$ , and satisfies $∥ E_{0} (ζ_{0} u) ∥_{W^{k, p} (R^{n})} = ∥ ζ_{0} u ∥_{W^{k, p} (Ω)}$ .

4.2F2F3F5F7step 1.1step 2.1step 2.2step 3.1

Boundary pieces: fix $i \geq 1$ . The flattened function $w_{i} : = (ζ_{i} u) \circ Φ_{i}^{- 1}$ is defined on $Φ_{i} (W_{i} \cap Ω) = Φ_{i} (W_{i}) \cap {t < 0}$ . The cutoff support is compactly contained laterally in the chart; extending $w_{i}$ by zero across the artificial edges inside this half-space gives a $W^{k, p} (H_{-})$ class, since the cutoff vanishes near those edges and tests in $H_{-}$ stay away from $t = 0$ . Step 2.2 applied to the corresponding half-patches bounds its norm by $C_{i} ∥ ζ_{i} u ∥_{W^{k, p} (Ω)}$ . Conjugating the upper-half-space operator of [F2] by the coordinate flip gives an extension from $H_{-} = {t < 0}$ , so the zero-extended $w_{i}$ has an extension ${\tilde{w}}_{i} \in W^{k, p} (R^{n}; K)$ . Its support remains inside the larger flattened patch by the choice in step 1.1, and its extension formula is linear. Pulling back by $Φ_{i}$ and multiplying by a cutoff $ψ_{i} \in C_{c}^{\infty} (W_{i})$ equal to one near $supp ζ_{i}$ gives $E_{i} (u) : = ψ_{i} \cdot ({\tilde{w}}_{i} \circ Φ_{i}) \in W^{k, p} (R^{n}; K),$ by [F3] on matched ambient patches and [F5], followed by [F7] to extend the compactly supported product from $W_{i}$ by zero. These operations give $∥ E_{i} (u) ∥_{W^{k, p} (R^{n})} \leq C_{i} ∥ ζ_{i} u ∥_{W^{k, p} (Ω)}$ . It agrees with $ζ_{i} u$ almost everywhere on $Ω$ , and its support lies in $supp ψ_{i} \cap Φ_{i}^{- 1} (supp {\tilde{w}}_{i})$ , a compact subset of $V$ .

5.1F8step 3.1step 4.1step 4.2

Define $E (u) : = E_{0} (ζ_{0} u) + \sum_{i = 1}^{m} E_{i} (u)$ . Each step above is linear in $u$ , so $E$ is linear; on $Ω$ the sum equals $\sum_{i = 0}^{m} ζ_{i} u = u$ almost everywhere by step 3.1; the support of $E (u)$ is contained in the union of finitely many compact subsets of $V$ , hence compact in $V$ ; and [F8] together with the bounds of steps 3.1, 4.1 and 4.2 gives $∥ E (u) ∥_{W^{k, p} (R^{n})} \leq C ∥ u ∥_{W^{k, p} (Ω)}$ for a constant $C$ independent of $u$ .

6.1F7F8step 5.1∎

Therefore $E$ is a bounded linear extension operator in the sense of [F8], and $Ω$ is a $W^{k, p}$ -extension domain for every $1 \leq p \leq \infty$ , including $p = \infty$ through the chart and half-space interfaces used above. For $k = 0$ the $L^{p}$ extension property is immediate on any open $Ω$ : extension by zero of an $L^{p} (Ω; K)$ class lies in $L^{p} (R^{n}; K)$ with the same norm, is linear, and restricts back to the class, so zero extension is the required operator; its support is contained in $\overline{Ω}$ , which is compact in $V$ under the bounded-domain hypotheses.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Ambient smooth restrictions are dense on bounded C^k domains

Statement

Assume the Axiom of Choice. Let $k \geq 1$ , $1 \leq p < \infty$ , $K \in {R, C}$ , and let $Ω \subset R^{n}$ be a bounded $C^{k}$ domain in the graph sense of Bounded C^k domains and boundary charts. Then the set of restrictions to $Ω$ of functions in $C_{c}^{\infty} (R^{n}; K)$ is dense in $W^{k, p} (Ω; K)$ : for every $u \in W^{k, p} (Ω; K)$ and every $δ > 0$ there is $φ \in C_{c}^{\infty} (R^{n}; K)$ with $∥ φ ∣_{Ω} - u ∥_{W^{k, p} (Ω)} < δ$ .

The exponent range is $1 \leq p < \infty$ ; the result asserts no density in the $W^{k, \infty}$ norm, and it does not hold on arbitrary open sets, as the companion examples page shows.

Facts & Assumptions

Given: the Axiom of Choice; $k \geq 1$ ; $1 \leq p < \infty$ ; $K \in {R, C}$ ; a bounded $C^{k}$ domain $Ω \subset R^{n}$ ; and a class $u \in W^{k, p} (Ω; K)$ .

[F1]

Extension: for every open $V$ with $\overline{Ω} \subseteq V$ there is a bounded linear extension operator $E : W^{k, p} (Ω; K) \to W^{k, p} (R^{n}; K)$ with $(E u) ∣_{Ω} = u$ almost everywhere and $supp (E u)$ a compact subset of $V$ , for the given $k \geq 1$ and $1 \leq p < \infty$ (Bounded C^k domains admit integer-order Sobolev extension).

[F2]

Smooth bumps: for $0 < r < R$ there is a smooth $η : R^{n} \to [0, 1]$ equal to one on ${\overline{B}}_{r} (0)$ with $supp η \subseteq B_{R} (0)$ (A smooth bump between concentric Euclidean balls).

[F3]

Interior commutation: for $u \in W^{k, p} (Ω; K)$ , a nonnegative unit-mass $ρ \in C_{c}^{\infty} (R^{n})$ with $supp ρ \subseteq {\overline{B}}_{1} (0)$ , and $ρ_{ε} (x) = ε^{- n} ρ (x / ε)$ , the convolution $ρ_{ε} * u$ is defined and smooth on $Ω_{ε} = {x \in Ω : dist (x, R^{n} ∖ Ω) > ε}$ , which equals $R^{n}$ when $Ω = R^{n}$ , and $D^{α} (ρ_{ε} * u) = ρ_{ε} * (D^{α} u)$ there for every $∣ α ∣ \leq k$ , as almost-everywhere classes (Interior mollification commutes with weak derivatives).

[F4]

The family $ρ_{ε}$ generated by a smooth unit-mass $ρ$ with $supp ρ \subseteq {\overline{B}}_{1} (0)$ is an $L^{1}$ approximate identity (A unit-mass smooth bump generates an $L^{1}$ approximate identity).

[F5]

Real convergence: an $L^{1}$ approximate identity $(K_{ε})$ on $R^{n}$ satisfies $∥ f * K_{ε} - f ∥_{p} \to 0$ for $1 \leq p < \infty$ and $f \in L^{p} (R^{n})$ (Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$ ).

[F6]

Complex interface: for complex $K \in L^{1} (R^{n}; C)$ and $f \in L^{p}$ , $∥ K * f ∥_{p} \leq ∥ K ∥_{1} ∥ f ∥_{p}$ , and if $\int K_{ε} = 1$ , $\sup_{ε > 0} ∥ K_{ε} ∥_{1} < \infty$ and $\int_{∣ y ∣ \geq δ} ∣ K_{ε} ∣ \to 0$ for every $δ > 0$ , then $K_{ε} * f \to f$ in $L^{p}$ for $p < \infty$ ; the rescalings $ε^{- n} K (x / ε)$ of a unit-mass $K \in L^{1}$ have these properties (Complex translation, convolution, approximate identities, and mollification).

[F7]

Restriction is a contraction: for open $U \subseteq Ω$ , restriction defines a contraction $W^{k, p} (Ω; K) \to W^{k, p} (U; K)$ (Bounded restriction and cutoff localisation in Sobolev spaces).

[F8]

Sobolev norm: for $1 \leq p < \infty$ , $∥ w ∥_{W^{k, p} (U)} = {(\sum_{∣ α ∣ \leq k} ∥ D^{α} w ∥_{L^{p} (U)}^{p})}^{1 / p}$ (Integer-order Sobolev spaces and their norms).

Choice use. The assumed Axiom of Choice supplies the hypotheses of the extension interface [F1] in step 1.1 and the restriction interface [F7] in step 5.1. It also implies the Countable Choice assumed by [F3]–[F6], [F8] and the bounded $C^{k}$ -domain definition. Fixing one bump from [F2] and normalising it in step 1.2 requires no further choice.

Proof

technique · direct

1.1F1given

Fix $u \in W^{k, p} (Ω; K)$ . Since $Ω$ is bounded, choose a bounded open $V$ with $\overline{Ω} \subseteq V$ , and let $E$ be the extension operator supplied by [F1] for this $V$ ; put $F : = E u \in W^{k, p} (R^{n}; K)$ . Then $F ∣_{Ω} = u$ almost everywhere on $Ω$ and $supp F$ is a compact subset of $V$ .

1.2F2given

Let $η$ be a bump as in [F2] with $r = 1 / 2$ , $R = 1$ , so that $η = 1$ on ${\overline{B}}_{1 / 2} (0)$ and $supp η \subseteq B_{1} (0)$ ; then $0 < \int_{R^{n}} η < \infty$ and $ρ : = η / \int η$ is nonnegative, of class $C_{c}^{\infty} (R^{n})$ , of unit mass, with $supp ρ \subseteq B_{1} (0) \subseteq {\overline{B}}_{1} (0)$ . Put $ρ_{ε} (x) = ε^{- n} ρ (x / ε)$ for $ε > 0$ , so $ρ_{ε}$ is nonnegative with $\int ρ_{ε} = 1$ and $supp ρ_{ε} \subseteq {\overline{B}}_{ε} (0)$ .

2.1F3step 1.1step 1.2

For every multi-index $α$ with $∣ α ∣ \leq k$ , the class $D^{α} F \in L^{p} (R^{n}; K)$ exists, and the commutation clause of [F3] applied with $Ω = R^{n}$ (so that $Ω_{ε} = R^{n}$ for every $ε$ ) gives $D^{α} (ρ_{ε} * F) = ρ_{ε} * (D^{α} F)$ as almost-everywhere classes on $R^{n}$ .

2.2F3step 1.1step 1.2

For each $ε > 0$ the convolution $v_{ε} : = ρ_{ε} * F$ is of class $C^{\infty}$ on $R^{n}$ by the smoothness clause of [F3], and $supp v_{ε} \subseteq supp F + supp ρ_{ε} \subseteq supp F + {\overline{B}}_{ε} (0)$ is compact, first because the support of a convolution is contained in the sum of the supports and then because $supp F$ is compact; hence $v_{ε} \in C_{c}^{\infty} (R^{n}; K)$ and its restriction is an admissible approximant.

3.1F4F5F6step 2.1

For each $α$ with $∣ α ∣ \leq k$ , $∥ ρ_{ε} * (D^{α} F) - D^{α} F ∥_{L^{p} (R^{n})} \to 0$ as $ε \to 0^{+}$ : for $K = R$ this is the real convergence of [F5] applied to the $L^{1}$ approximate identity of [F4] and the class $D^{α} F \in L^{p}$ ; for $K = C$ the complex interface of [F6] applies to $D^{α} F$ directly, a real class being a complex class.

4.1F8step 2.1step 3.1

By the norm formula of [F8] and step 3.1, the smooth convolutions converge in the whole-space Sobolev norm: $∥ ρ_{ε} * F - F ∥_{W^{k, p} (R^{n})}^{p} = \sum_{∣ α ∣ \leq k} ∥ ρ_{ε} * (D^{α} F) - D^{α} F ∥_{L^{p} (R^{n})}^{p} \to 0$ .

5.1F7step 1.1step 4.1step 2.2

By the restriction contraction [F7] applied to $v_{ε} - F \in W^{k, p} (R^{n}; K)$ and the identity $F ∣_{Ω} = u$ of step 1.1, $∥ v_{ε} ∣_{Ω} - u ∥_{W^{k, p} (Ω)} \leq ∥ v_{ε} - F ∥_{W^{k, p} (R^{n})}$ , which tends to zero by step 4.1; given $δ > 0$ choose $ε > 0$ with $∥ v_{ε} ∣_{Ω} - u ∥_{W^{k, p} (Ω)} < δ$ .

6.1F5F6step 5.1given∎

Therefore the restrictions of $C_{c}^{\infty} (R^{n}; K)$ functions are dense in $W^{k, p} (Ω; K)$ for $1 \leq p < \infty$ , while nothing is asserted at $p = \infty$ : the convergence inputs [F5] and [F6] are stated for finite exponents only, and step 3.1 fails for the essential-supremum norm.

CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Whole-space inequalities transfer through a Sobolev extension

Statement

Assume the Axiom of Choice. Let $k \in N_{0}$ , $1 \leq p \leq \infty$ , $K \in {R, C}$ , and let $Ω \subseteq R^{n}$ be open. Let $E : W^{k, p} (Ω; K) ⟶ W^{k, p} (R^{n}; K)$ be a bounded linear extension operator, so that $(E u) ∣_{Ω} = u$ almost everywhere on $Ω$ for every class $u$ , and let $∥ E ∥$ be its operator norm. Suppose a whole-space functional $N$ on Sobolev classes, together with its restrictions $N_{Ω}$ to the classes of $Ω$ , satisfies $N_{Ω} (F ∣_{Ω}) \leq N (F) and N (F) \leq C ∥ F ∥_{W^{k, p} (R^{n})}$ for every $F \in W^{k, p} (R^{n}; K)$ and a constant $C$ independent of $F$ . Then $N_{Ω} (u) \leq C ∥ E ∥ ∥ u ∥_{W^{k, p} (Ω)} for every u \in W^{k, p} (Ω; K) .$ In particular, if $1 \leq q \leq \infty$ and a whole-space Sobolev inequality $∥ F ∥_{L^{q} (R^{n})} \leq C ∥ F ∥_{W^{k, p} (R^{n})}$ is available, then $∥ u ∥_{L^{q} (Ω)} \leq C ∥ E ∥ ∥ u ∥_{W^{k, p} (Ω)}$ for every $u$ . The corollary is conditional on that whole-space inequality and asserts no embedding theorem itself; every bounded $C^{k}$ domain supplies an admissible operator $E$ through Bounded C^k domains admit integer-order Sobolev extension.

Facts & Assumptions

Given: the Axiom of Choice; $k \in N_{0}$ ; $1 \leq p \leq \infty$ ; $K \in {R, C}$ ; an open $Ω \subseteq R^{n}$ ; a bounded linear extension operator $E$ with right-inverse property and norm $∥ E ∥$ ; a functional $N$ with restrictions $N_{Ω}$ satisfying the two displayed hypotheses with constant $C$ ; and a class $u \in W^{k, p} (Ω; K)$ .

[F1]

Extension operator: $E : W^{k, p} (Ω; K) \to W^{k, p} (R^{n}; K)$ is bounded and linear with $(E u) ∣_{Ω} = u$ as an almost-everywhere class on $Ω$ for every $u$ , and its operator norm is $∥ E ∥ = \sup {∥ E u ∥ : u \in W^{k, p} (Ω; K), ∥ u ∥_{W^{k, p} (Ω)} \leq 1}$ (Sobolev extension domains and extension operators).

[F2]

Restriction is a well-defined operation on Sobolev classes: $F ∣_{Ω} \in W^{k, p} (Ω; K)$ with $D^{α} (F ∣_{Ω}) = (D^{α} F) ∣_{Ω}$ almost everywhere, and it is a contraction (Bounded restriction and cutoff localisation in Sobolev spaces).

[F3]

Restriction monotonicity of $N$ : $N_{Ω} (F ∣_{Ω}) \leq N (F)$ for every $F \in W^{k, p} (R^{n}; K)$ , and the whole-space bound $N (F) \leq C ∥ F ∥_{W^{k, p} (R^{n})}$ , both hypotheses of the statement.

[F4]

The Sobolev norm is the finite derivative sum of Integer-order Sobolev spaces and their norms, and $∥ E u ∥ \leq ∥ E ∥ ∥ u ∥$ holds for every $u$ by the definition of the operator norm in [F1].

[F5]

Bounded $C^{k}$ domains: for $k \geq 1$ and every $1 \leq p \leq \infty$ there is a bounded linear extension operator $W^{k, p} (Ω; K) \to W^{k, p} (R^{n}; K)$ for every bounded $C^{k}$ domain in the graph sense, and extension by zero supplies the case $k = 0$ on any open set (Bounded C^k domains admit integer-order Sobolev extension).

Choice use. The Axiom of Choice enters only through the published interfaces of [F2] and [F5], which invoke the Countable Choice they require; [F5] also invokes it for the chart and partition-of-unity steps of the extension construction. The three-line norm chain of the proof itself uses no choice.

Proof

technique · direct

1.1F1given

Fix $u \in W^{k, p} (Ω; K)$ and put $F : = E u \in W^{k, p} (R^{n}; K)$ . By the right-inverse property of [F1], $F ∣_{Ω} = u$ as an almost-everywhere class on $Ω$ .

2.1F3step 1.1

Apply the restriction monotonicity of [F3] to the pair $(F, Ω)$ : $N_{Ω} (u) = N_{Ω} (F ∣_{Ω}) \leq N (F) = N (E u)$ .

2.2F3step 1.1

Apply the whole-space bound of [F3] to $F = E u$ : $N (E u) \leq C ∥ E u ∥_{W^{k, p} (R^{n})}$ .

2.3F1F4step 1.1

Apply the operator-norm inequality of [F4] to $u$ : $∥ E u ∥_{W^{k, p} (R^{n})} \leq ∥ E ∥ ∥ u ∥_{W^{k, p} (Ω)}$ .

3.1step 2.1step 2.2step 2.3

Chaining steps 2.1, 2.2 and 2.3 gives $N_{Ω} (u) \leq C ∥ E ∥ ∥ u ∥_{W^{k, p} (Ω)}$ for the fixed class $u$ ; since $u$ was arbitrary, the first assertion holds.

4.1F2F3step 3.1given

$L^{q}$ instance. Let $1 \leq q \leq \infty$ and set $N (F) : = ∥ F ∥_{L^{q} (R^{n})}$ and $N_{Ω} (v) : = ∥ v ∥_{L^{q} (Ω)}$ , with the value $+ \infty$ when the class is not in $L^{q}$ . The inclusion $Ω \subseteq R^{n}$ gives $\int_{Ω} ∣ F ∣^{q} \leq \int_{R^{n}} ∣ F ∣^{q}$ for $q < \infty$ and ${ess sup}_{Ω} ∣ F ∣ \leq {ess sup}_{R^{n}} ∣ F ∣$ for $q = \infty$ ; hence $N_{Ω} (F ∣_{Ω}) \leq N (F)$ , where the left side is interpreted through the class of [F2]. If the whole-space inequality $∥ F ∥_{L^{q} (R^{n})} \leq C ∥ F ∥_{W^{k, p} (R^{n})}$ is available, the remaining hypothesis of [F3] holds with that same constant, and step 3.1 yields $∥ u ∥_{L^{q} (Ω)} \leq C ∥ E ∥ ∥ u ∥_{W^{k, p} (Ω)}$ .

5.1F5step 4.1∎

Domains. If $Ω$ is a bounded $C^{k}$ domain with $k \geq 1$ , [F5] supplies an admissible operator $E$ for every $1 \leq p \leq \infty$ , so step 4.1 transfers any available whole-space $L^{q}$ inequality to $Ω$ with the extension constant of that operator; for $k = 0$ extension by zero supplies the analogous operator on any open set. No embedding is proved here: the implication is conditional on the whole-space inequality, and the conclusion is stated only for the functional $N$ and the operator $E$ that are given.

RemarkRemark: Literature-sourcedProof: Not applicableaudited 2026-10-02Open item page →

Boundary regularity required by the constructed extension

Remark

The four approximation and extension statements on this page carry deliberately different regularity hypotheses on $\partial Ω$ , and none of them may be strengthened by accident. This remark records what each construction actually uses. The comparisons retain the cited Choice hypotheses: Countable Choice for the density and zero-extension interfaces, and the Axiom of Choice for the constructed bounded-domain extension and boundary-density interfaces.

Approximation needs no boundary regularity. Meyers–Serrin density Meyers–Serrin density on an arbitrary open set, under Countable Choice, assumes only that $Ω$ is open and that $1 \leq p < \infty$ : no boundary chart, no extension operator and no unboundedness of the derivatives of a cutoff near $\partial Ω$ enters, because the argument exhausts $Ω$ by compactly contained pieces. Likewise the zero extension of Zero extension of W_0^{1,p} has no boundary derivative is defined by extending a class in $W_{0}^{1, p} (Ω)$ by zero, and the limit definition of the closure Zero-boundary Sobolev space as a norm closure requires no boundary regularity; the price is the membership hypothesis in $W_{0}^{1, p}$ , not a hypothesis on $\partial Ω$ . The interior mollification Interior mollification commutes with weak derivatives is available only on the shrinking sets $Ω_{ε} = {x \in Ω : dist (x, R^{n} ∖ Ω) > ε}$ . For a general class on $Ω$ , this interior construction alone supplies convergence on compactly contained subsets, not convergence in the norm on all of $Ω$ . A licensed extension to the whole space changes that conclusion: for $u \in W_{0}^{1, p} (Ω)$ and $1 \leq p < \infty$ , zero extension lies in $W^{1, p} (R^{n})$ by Zero extension of W_0^{1,p} has no boundary derivative. Whole-space mollification then converges in $W^{1, p} (R^{n})$ , and restriction gives convergence on $Ω$ , including regions arbitrarily close to its boundary. Indeed, the whole-space density corollary Compactly supported smooth functions are dense in W^{k,p}(R^n) supplies ambient compactly supported smooth restrictions converging to $u$ without any boundary regularity. These restrictions need not themselves lie in $C_{c}^{\infty} (Ω)$ or have zero boundary values.

Extension needs exactly the chart regularity of its order. The half-space operator and the $C^{k}$ flattening lemma C^k boundary flattening preserves local W^{k,p} are combined by Bounded C^k domains admit integer-order Sobolev extension under the hypothesis that $Ω$ is a bounded $C^{k}$ domain in the graph sense of Bounded C^k domains and boundary charts: at each boundary point the flattening chart and the graph function $h$ must have bounded derivatives through order $k$ , because the pullback of a $W^{k, p}$ class needs bounded derivatives of the chart through that same order, and the reflected moment formula cancels interface terms involving derivatives up to order $k - 1$ . For fixed $k$ the hypothesis is "boundary of class $C^{k}$ ": a $C^{1}$ boundary suffices for the first-order theorem and gives no control of second or higher derivatives, and a $C^{\infty}$ boundary is needed only when one wants the construction at every order simultaneously. Consequently the smooth-up-to-the-boundary density statement Ambient smooth restrictions are dense on bounded C^k domains, which is obtained by extending first and mollifying afterwards, inherits the same bounded- $C^{k}$ hypothesis and the same fixed $k$ .

Excluded endpoints and unproved strengthenings. The density statements are for $1 \leq p < \infty$ ; at $p = \infty$ the excluded endpoint is recorded in Meyers–Serrin excludes the W^{k,∞} norm endpoint, where the one-dimensional corner $∣ x ∣$ shows that no smooth sequence converges in the $W^{1, \infty}$ norm. Nothing here asserts extension theorems for Lipschitz domains, for domains with less regular boundary, or a single operator simultaneously bounded on all orders $k$ : such results are separate theorems, not consequences of the $C^{k}$ chart-and-reflection argument displayed on this page, and no step of that argument supplies the uniform higher-order estimates they would require.

5 · Examples, counterexamples and false statements

None yet.

Smooth Approximation and Sobolev Extension

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Interior mollification commutes with weak derivatives

Statement

Facts & Assumptions

Proof

Local smooth approximation in integer-order Sobolev spaces

Statement

Facts & Assumptions

Proof

Meyers–Serrin density on an arbitrary open set

Statement

Facts & Assumptions

Proof

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

Statement

Facts & Assumptions

Proof

Meyers–Serrin excludes the W^{k,∞} norm endpoint

Remark

Zero-boundary Sobolev space as a norm closure

Definition

Source notes

Zero extension of W_0^{1,p} has no boundary derivative

Statement

Facts & Assumptions

Proof

Compactly supported Sobolev functions extend by zero in every integer order

Statement

Facts & Assumptions

Proof

Sobolev extension domains and extension operators

Definition

Source notes

Integer-order Sobolev extension from a half-space

Statement

Facts & Assumptions

Proof

Bounded C^k domains and boundary charts

Definition

Source notes

C^k boundary flattening preserves local W^{k,p}

Statement

Facts & Assumptions

Proof

Bounded C^k domains admit integer-order Sobolev extension

Statement

Facts & Assumptions

Proof

Ambient smooth restrictions are dense on bounded C^k domains

Statement

Facts & Assumptions

Proof

Whole-space inequalities transfer through a Sobolev extension

Statement

Facts & Assumptions

Proof

Boundary regularity required by the constructed extension

Remark

5 · Examples, counterexamples and false statements

Sources