The difference-quotient test function and its commutators

In the setting of (i) write $w:={\eta }^2{\delta }_h^{i}u$ on ${\mathrm{\Omega }}_{i,h}$. The class ${\delta }_h^{i}u$ lies in $H^1({\mathrm{\Omega }}_{i,h})$ with $D_k({\delta }_h^{i}u)={\delta }_h^i(D_{k}u)$ by [F1], so [F2] gives $w\in H^1({\mathrm{\Omega }}_{i,h})$ with $D_{k}w={\eta }^2{\delta }_h^i(D_{k}u)+D_k({\eta }^2){\delta }_h^{i}u\mathrm{.}$ Since $\mathrm{\mid }h\mathrm{\mid }<\mathrm{dist}(\mathrm{supp}\eta ,\mathrm{\partial }\mathrm{\Omega })$, the compact set $\mathrm{supp}\eta$ lies in the interior of ${\mathrm{\Omega }}_{i,h}$, so $w$ is supported in a compact subset of ${\mathrm{\Omega }}_{i,h}$ and its zero extension lies in $H^1(\mathrm{\Omega })$ with the same weak gradient; then form $v=-{\delta }_{-h}^{i}w$ using the whole-space zero extension of $w$. Its support lies in $\mathrm{supp}\eta \cup (\mathrm{supp}\eta +h{e}_i)⋐\mathrm{\Omega }$, so its restriction lies in $H^1(\mathrm{\Omega })$ by [F1], and hence $v\in H_0^1(\mathrm{\Omega })$ by [F3].

In the setting of (ii), for fixed $h\mathrm{\ne }0$ consider the operations $T_{1}w:={\eta }^{2}w$ and $T_{2}w:={\delta }_h^{j}w$, $T_{3}w:={\delta }_{-h}^{j}w$ on $H^1(H)$. Each is bounded: $T_1$ by [F2] with the fixed smooth factor $\eta$ and $T_2,T_3$ by [F4], and each carries $C_c^{\mathrm{\infty }}(H)$ into itself, since multiplication by a smooth compactly supported factor and tangential shifts preserve smoothness and compact support in $H$. If ${\phi }_k\in C_c^{\mathrm{\infty }}(H)$ approximate $u$ in $H^1(H)$, then $v_k:=-T_3{T}_1{T}_2{\phi }_k\in C_c^{\mathrm{\infty }}(H)$ and $v_k\to v$ in $H^1(H)$ by the boundedness, so $v\in H_0^1(H)$ by the definition of the closure; under the bounded-datum-functional hypothesis of (ii), [F3] makes $v$ an admissible test class.

Principal pairing. Fix a quotient direction $k$ and write $w={\eta }^2{\delta }_h^{k}u$, $v=-{\delta }_{-h}^{k}w$. Difference quotients commute with weak derivatives, and discrete integration by parts, applied to the compactly supported test factor in the interior case or by tangential translation on $H$, gives $\int a^{ij}{D}_{j}u\overline{D_{i}v}=\int {\delta }_h^k(a^{ij}{D}_{j}u)\overline{D_{i}w}$. The quotient direction $k$ is fixed throughout and the form indices $i,j$ are summed independently.

Product expansion. Insert $D_{i}w={\eta }^2{\delta }_h^k{D}_{i}u+D_i({\eta }^2){\delta }_h^{k}u$ and ${\delta }_h^k(a^{ij}{D}_{j}u)=a^{ij}(x+h{e}_k){\delta }_h^k{D}_{j}u+({\delta }_h^k{a}^{ij})D_{j}u$. Multiplication produces precisely the displayed shifted principal term and the three remainder terms in the Statement. This algebra uses the correct shifted product rule.

Bounds and lower-order terms. For fixed $h$, the coefficient quotient is bounded by $2M_a\mathrm{/}\mathrm{\mid }h\mathrm{\mid }$ and all translated first derivatives and quotient classes are $L^2$ on the supported patches, so Cauchy--Schwarz makes every displayed remainder finite. If $a\in W^{1,\mathrm{\infty }}$ there, its quotient is uniformly bounded by the corresponding weak gradient bound. Young's inequality then bounds each principal remainder by $\epsilon \mathrm{\parallel }\eta {\delta }_h^{k}Du{\mathrm{\parallel }}_2^2+C_{\epsilon }\mathrm{\parallel }Du{\mathrm{\parallel }}_2^2$, with norms on a slightly enlarged patch in the interior case. The drift and reaction pairings are simply $\int (b^i{D}_{i}u+cu)\bar{v}$ and are finite by boundedness of $b,c$ and $v\in H^1$; they can be estimated directly without taking coefficient quotients.

Conclusion. Steps 1.1 and 1.2 establish the admissible test classes. Steps 2.1--3.1 establish the exact principal decomposition and finite full form pairing, distinguishing the principal commutators from the undifferentiated lower-order terms.