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Weak product rule for bounded Sobolev functions

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $Ω \subseteq R^{n}$ be open, $1 \leq p < \infty$ , and let $u, v \in W^{1, p} (Ω; K) \cap L^{\infty} (Ω)$ . Then $u v \in W^{1, p} (Ω)$ and $D_{j} (u v) = (D_{j} u) v + u (D_{j} v)$ a.e. for every $j$ .

Facts & Assumptions

Given: The Axiom of Choice; an open $Ω \subseteq R^{n}$ with $n \geq 1$ ; an exponent $1 \leq p < \infty$ ; and classes $u, v \in W^{1, p} (Ω; K) \cap L^{\infty} (Ω)$ .

[F1]

$W^{1, p} (Ω; K)$ consists of the $L^{p}$ classes whose weak first derivatives exist as $L^{p}$ classes (Integer-order Sobolev spaces and their norms), and $L^{p}$ is the quotient by almost-everywhere null functions (The space $L^{p} (μ)$ as the quotient by null functions).

[F2]

Chain rule for a globally Lipschitz scalar function $F$ : for real $w \in W^{1, p} (Ω; R)$ , $F \circ w \in W^{1, p}$ whenever $F (0) = 0$ , and $D_{i} (F \circ w)$ agrees almost everywhere with $F^{'} (w) D_{i} w$ where $F$ is differentiable at $w$ , with the product defined as $0$ on the preimage of the nondifferentiability set, which need not itself be null (Chain rule for globally Lipschitz scalar maps of Sobolev functions).

[F3]

Weak differentiation is linear and local: weak derivatives of linear combinations are the corresponding linear combinations, and they restrict to open subsets (Linearity, locality, and commutation of weak derivatives).

[F5]

The Axiom of Choice, used through the chain-rule interface of [F2] (The Axiom of Choice).

Proof

technique · direct

1.1F1givenalgebra

Integrability of the products. Since $v \in L^{\infty} (Ω)$ and $u \in L^{p} (Ω)$ , the pointwise bound $∣ u v ∣^{p} \leq ∥ v ∥_{\infty}^{p} ∣ u ∣^{p}$ , followed by integration, gives $∥ u v ∥_{L^{p}} \leq ∥ u ∥_{L^{p}} ∥ v ∥_{L^{\infty}} < \infty$ , and the same argument applies to $u (D_{j} v)$ and $(D_{j} u) v$ because $D_{j} u, D_{j} v \in L^{p}$ and $u, v \in L^{\infty}$ . Thus the three classes $u v$ , $(D_{j} u) v$ and $u (D_{j} v)$ all lie in $L^{p} (Ω; K)$ , and so does their sum $(D_{j} u) v + u (D_{j} v)$ (taken componentwise for $K = C$ ).

1.2F2F5givenalgebra

The real case for a truncated square. Let $w \in W^{1, p} (Ω; R) \cap L^{\infty} (Ω)$ and $M : = ∥ w ∥_{L^{\infty}}$ . Define $G_{M} (t) : = t^{2}$ for $∣ t ∣ \leq M$ and $G_{M} (t) : = 2 M ∣ t ∣ - M^{2}$ for $∣ t ∣ > M$ . Then $G_{M}$ is in $C^{1} (R)$ , with $G_{M}^{'} (t) = 2 t$ for $∣ t ∣ \leq M$ and $G_{M}^{'} (t) = 2 M sgn (t)$ for $∣ t ∣ > M$ , it is globally Lipschitz with constant $2 M$ , $G_{M} (0) = 0$ , and $G_{M} (t) = t^{2}$ for $∣ t ∣ \leq M$ . Since $∣ w ∣ \leq M$ almost everywhere, $G_{M} \circ w = w^{2}$ almost everywhere and $G_{M}^{'} (w) = 2 w$ almost everywhere on $Ω$ ; by [F2] the class $G_{M} \circ w$ lies in $W^{1, p} (Ω)$ with $D_{j} (G_{M} \circ w) = G_{M}^{'} (w) D_{j} w = 2 w D_{j} w$ almost everywhere. In particular $w^{2} = G_{M} \circ w \in W^{1, p} (Ω)$ and $D_{j} (w^{2}) = 2 w D_{j} w$ .

2.1F3step 1.1step 1.2algebra

Polarization in the real case. Suppose first that $K = R$ , and put $w_{\pm} : = u \pm v \in W^{1, p} (Ω; R) \cap L^{\infty} (Ω)$ ; these classes lie in $W^{1, p}$ by the linearity part of [F3]. Applying $G_{M_{\pm}}$ of step 1.2 with $M_{\pm} : = ∥ w_{\pm} ∥_{L^{\infty}}$ gives $u v = \frac{1}{4} (w_{+}^{2} - w_{-}^{2})$ as $L^{p}$ classes and, by linearity of weak derivatives [F3], $D_{j} (u v) = \frac{1}{4} (D_{j} (w_{+}^{2}) - D_{j} (w_{-}^{2})) = \frac{1}{2} (w_{+} D_{j} w_{+} - w_{-} D_{j} w_{-}) = (D_{j} u) v + u (D_{j} v)$ almost everywhere.

3.1F1F3step 2.1algebra∎

Complex case and conclusion. For general $K \in {R, C}$ , write $u = u_{1} + i u_{2}$ and $v = v_{1} + i v_{2}$ with real components; these components lie in $W^{1, p} (Ω; R) \cap L^{\infty} (Ω)$ and $D_{j} u = D_{j} u_{1} + i D_{j} u_{2}$ , $D_{j} v = D_{j} v_{1} + i D_{j} v_{2}$ by the componentwise definition of the weak derivative [F1]. Applying step 2.1 to the four real products and using linearity [F3], $D_{j} (u v) = D_{j} ((u_{1} v_{1} - u_{2} v_{2}) + i (u_{1} v_{2} + u_{2} v_{1})) = (D_{j} u) v + u (D_{j} v)$ almost everywhere, and $u v \in W^{1, p} (Ω; K)$ by step 1.1.

Source notes

The classical route approximates $u$ and $v$ by smooth functions and passes to the limit in a closed graph; the proof above instead polarises the product and applies the published chain rule for globally Lipschitz scalar functions to a truncated square, which is available for all $1 \leq p < \infty$ and avoids any global smooth-approximation theorem. The boundedness of $u$ and $v$ is used through the truncation radius $M$ and in the integrability step.

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Sources

Juha Kinnunen, Sobolev Spaces (Aalto University, 2026, complete graduate lecture notes) (standard reference, not scraped)