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The cutoff commutator in the local $W^{2, p}$ estimates

Statement

Assume Countable Choice. Let $n \geq 1$ , $1 < p < \infty$ , $R > 0$ , and let $L = a^{i j} \partial_{i} \partial_{j} + b^{i} \partial_{i} + c$ have bounded coefficients on $B_{R} (x_{0})$ with $∣ A ∣ \leq Λ$ , $∣ b ∣ \leq M_{b}$ and $∣ c ∣ \leq M_{c}$ , where $∣ A ∣ : = \max_{i, j} ∣ a^{i j} ∣$ . Let $η \in C_{c}^{\infty} (B_{R} (x_{0}))$ satisfy $∣ D η ∣ \leq C_{1} R^{- 1}$ and $∣ D^{2} η ∣ \leq C_{2} R^{- 2}$ , and let $u \in W^{2, p} (B_{R} (x_{0}))$ . Then almost everywhere $L (η u) = η L u + (a^{i j} + a^{j i}) (\partial_{i} η) \partial_{j} u + (a^{i j} \partial_{i} \partial_{j} η + b^{i} \partial_{i} η) u,$ and consequently $∥ L (η u) - η L u ∥_{L^{p}} \leq C_{n} (Λ C_{1} R^{- 1} ∥ D u ∥_{L^{p}} + (Λ C_{2} R^{- 2} + M_{b} C_{1} R^{- 1}) ∥ u ∥_{L^{p}}) .$ There is no second derivative of $u$ in the commutator. No symmetry assumption on the principal coefficient matrix $A = (a^{i j})$ is needed.

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 1$ , $1 < p < \infty$ , $R > 0$ , coefficients $a^{i j}, b^{i}, c \in L^{\infty} (B_{R} (x_{0}))$ with $∣ A ∣ \leq Λ$ , $∣ b ∣ \leq M_{b}$ , $∣ c ∣ \leq M_{c}$ , a cutoff $η$ as in the statement, and $u \in W^{2, p} (B_{R} (x_{0}))$ .

[A1]

The only choice assumption is Countable Choice ${A C}_{ω}$ ; it enters through the Sobolev interfaces below. No full Axiom of Choice is used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

A function of class $W^{2, p}$ has weak derivatives $D_{i} u$ and $D_{i} D_{j} u$ in $L^{p}$ , and the Sobolev norm is the $p$ -sum of their $L^{p}$ norms; the $L^{p}$ norm obeys the triangle inequality. (Integer-order Sobolev spaces and their norms)

[F2]

The operator is $L u = \sum_{i, j} a^{i j} \partial_{i} \partial_{j} u + \sum_{i} b^{i} \partial_{i} u + c u$ , with the stated entrywise bounds $∣ a^{i j} ∣ \leq Λ$ , $∣ b^{i} ∣ \leq M_{b}$ , and $∣ c ∣ \leq M_{c}$ almost everywhere. No symmetry of $A = (a^{i j})$ is assumed; in particular $∣ a^{i j} + a^{j i} ∣ \leq 2 Λ$ .

[F3]

On a set of finite measure, the $L^{p}$ norm of a product is at most the sup-norm of one factor times the $L^{p}$ norm of the other. (Holder's inequality for integrals, including the endpoint cases)

Proof

technique · direct

1.1F1givenalgebra

Product rule for the cutoff. Since $η \in C_{c}^{\infty} (B_{R} (x_{0}))$ , for every test function $φ \in C_{c}^{\infty} (B_{R} (x_{0}))$ the product $η φ$ is again a test function; the defining identity for the weak derivative of $u$ therefore gives $\int (η u) \partial_{i} φ = \int u \partial_{i} (η φ) - \int u (\partial_{i} η) φ = - \int (η D_{i} u + u \partial_{i} η) φ$ , where the classical product rule was used on $η φ$ . Hence $η u \in W^{1, p} (B_{R} (x_{0}))$ with $D_{i} (η u) = η D_{i} u + u \partial_{i} η$ almost everywhere, both terms lying in $L^{p}$ . Applying the same argument to the $W^{1, p}$ function $D_{j} u$ in place of $u$ gives $D_{i} (η D_{j} u) = η D_{i} D_{j} u + (\partial_{i} η) D_{j} u$ , and combining the two identities yields $D_{j} D_{i} (η u) = η D_{j} D_{i} u + (\partial_{j} η) D_{i} u + (\partial_{i} η) D_{j} u + u \partial_{i} \partial_{j} η$ almost everywhere, all terms in $L^{p}$ . This direct weak-derivative argument does not require $u$ or $D_{j} u$ to be bounded.

2.1step 1.1F2algebra

Expanding the operator. Multiplying the pointwise equation $L u = a^{i j} \partial_{i} \partial_{j} u + b^{i} \partial_{i} u + c u$ by $η$ and substituting the product rule of step 1.1 gives, almost everywhere, $L (η u) = a^{i j} (η \partial_{i} \partial_{j} u + \partial_{j} η \partial_{i} u + \partial_{i} η \partial_{j} u + \partial_{i} \partial_{j} η u) + b^{i} (η \partial_{i} u + \partial_{i} η u) + c η u = η L u + (a^{i j} + a^{j i}) (\partial_{i} η) \partial_{j} u + (a^{i j} \partial_{i} \partial_{j} η + b^{i} \partial_{i} η) u,$ where relabelling $i$ and $j$ in the first cross term gives its coefficient $a^{j i}$ ; no symmetry assumption is needed.

3.1step 2.1F1F2F3algebra

$L^{p}$ bound. By [F2] and the cutoff bounds, $∣ (a^{i j} + a^{j i}) \partial_{i} η \partial_{j} u ∣ \leq 2 Λ n^{2} C_{1} R^{- 1} ∣ D u ∣$ and $∣ (a^{i j} \partial_{i} \partial_{j} η + b^{i} \partial_{i} η) u ∣ \leq (Λ n^{2} C_{2} R^{- 2} + M_{b} n C_{1} R^{- 1}) ∣ u ∣$ almost everywhere. Taking $L^{p}$ norms, using the triangle inequality of [F1] and the multiplicativity of the norm against bounded factors [F3] gives $∥ L (η u) - η L u ∥_{L^{p}} \leq C_{n} (Λ C_{1} R^{- 1} ∥ D u ∥_{L^{p}} + (Λ C_{2} R^{- 2} + M_{b} C_{1} R^{- 1}) ∥ u ∥_{L^{p}})$ . The dimension constant absorbs the factor $2 n^{2}$ from the nonsymmetric cross coefficient.

4.1step 2.1step 3.1A1given∎

Conclusion. The identity of step 2.1 involves only $u$ , $D u$ and the coefficient fields, never $D^{2} u$ , and the bound of step 3.1 is exactly the commutator estimate of the statement; the constants depend only on $n$ and the coefficient bounds, not on $u, η$ beyond the stated cutoff constants, and no choice beyond [A1] is used.

Remarks

The two first-order terms do not cancel: they are the symmetric pair produced by the product rule, and they are the reason a local $W^{2, p}$ estimate needs the interpolation inequality to absorb $R^{- 1} ∥ D u ∥_{L^{p}}$ .
The cutoff is compactly supported in the ball, so $η u$ extends by zero to a $W^{2, p}$ function on $R^{n}$ ; this is the localization used in the interior estimates below.

Depends on

Used by

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Sources

Armin Schikorra, Partial Differential Equations I & II (version October 1, 2025; complete 281-page graduate lecture notes) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, complete 392 pages) (standard reference, not scraped)
John Villavert, Elementary Theory and Methods for Elliptic Partial Differential Equations (2017; complete 220-page lecture notes) (standard reference, not scraped)

The cutoff commutator in the local W2,pW2,p estimates