The cutoff difference-quotient commutator estimate

The classes $\eta$ and $u$ lie in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$ and their product $\eta u$ does too, since $\eta$ is bounded with compact support; [F1] with the identifications $u⇝\eta$ and $v⇝u$ (the second displayed form) gives, almost everywhere on ${\mathrm{\Omega }}_{i,h}$, ${\delta }_h^i(\eta u)=\eta {\delta }_h^{i}u+({\delta }_h^i\eta )({\tau }_{-h{e}_i}u),$ which is the first displayed identity after moving the term $\eta {\delta }_h^{i}u$ to the left.

For fixed $x\in {\mathbb{R}}^n$ put $g(t):=\eta (x+t{e}_i)$ for $t$ between $0$ and $h$. The chain rule gives $g^{\mathrm{\prime }}(t)=D\eta (x+t{e}_i)\cdot e_i$, so $\mathrm{\mid }{g}^{\mathrm{\prime }}\mathrm{\mid }\le M_{\eta }$ on that interval, and the mean value inequality [F2] applied on the interval with endpoints $0,h$ gives $\mathrm{\mid }\eta (x+h{e}_i)-\eta (x)\mathrm{\mid }=\mathrm{\mid }g(h)-g(0)\mathrm{\mid }\le M_{\eta }\mathrm{\mid }h\mathrm{\mid },\text{hence}\mathrm{\mid }{\delta }_h^i\eta (x)\mathrm{\mid }\le M_{\eta }\mathrm{.}$

For the second identity apply [F1] with the identifications $u⇝{\eta }^2$ and $v⇝{\delta }_h^{i}u$: both classes are locally integrable by [F5], as is their product, and the second displayed form of [F1] gives, on the domain where the twice-shifted quotient is defined, ${\delta }_h^i\left({\eta }^2{\delta }_h^{i}u\right)={\eta }^2{\delta }_h^i\left({\delta }_h^{i}u\right)+({\delta }_h^i{\eta }^2)({\tau }_{-h{e}_i}{\delta }_h^{i}u)\mathrm{.}$

Taking absolute values in step 1.1 and applying step 1.2 pointwise almost everywhere on ${\mathrm{\Omega }}_{i,h}$ yields $\mathrm{\mid }{\delta }_h^i(\eta u)-\eta {\delta }_h^{i}u\mathrm{\mid }\le M_{\eta }\mathrm{\mid }({\tau }_{-h{e}_i}u)\mathrm{\mid },$ and integrating the $p$-th powers over ${\mathrm{\Omega }}_{i,h}$ (with the essential-supremum reading for $p=\mathrm{\infty }$) gives $\mathrm{\parallel }{\delta }_h^i(\eta u)-\eta {\delta }_h^{i}u{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}_{i,h})}\le M_{\eta }\mathrm{\parallel }{\tau }_{-h{e}_i}u{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}_{i,h})}$; by the change of variables of [F3] the right-hand side is $M_{\eta }\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}_{i,-h})}\le M_{\eta }\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}$.

Steps 1.1--2.1 prove the quotient identities and bounds. To establish the multiplier fact at order one, take $q\in W^{1,\mathrm{\infty }}(U)$, $z\in H^1(U)$ and $\phi \in C_c^{\mathrm{\infty }}(U)$. On a bounded neighbourhood $V⋐U$ of its support, the smooth-factor rule makes $t=z\phi$ a compactly supported $W^{1,1}(V)$ class: its $H^1$ derivatives are $L^1$ there by Cauchy--Schwarz. Applying [F6] and expanding $D_i(z\phi )$ gives ${\int }_{U}qz{D}_i\phi =-{\int }_U((D_{i}q)z+q{D}_{i}z)\phi$. Both proposed derivative terms are $L^2(U)$, proving the first-order product rule. Iterating this rule gives the displayed multi-index formula; each term is bounded in $L^2$ by its bounded coefficient factor times its $L^2$ factor, and a finite sum proves the norm estimate. At order zero this is just multiplication by an $L^{\mathrm{\infty }}$ class.

For the support fact, let $z\in H^m(U)$ vanish outside a compact $K\subset U$. By [F7], $E_{0}z\in H^m({\mathbb{R}}^n)$ and choose ${\phi }_{\nu }\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ converging to $E_{0}z$ in $H^m$. Choose $\chi \in C_c^{\mathrm{\infty }}(U)$ equal to one near $K$. Smooth-factor multiplication is bounded in $H^m$, so $\chi {\phi }_{\nu }\to \chi E_{0}z=E_{0}z$ in $H^m$; restricting gives compactly supported smooth approximations to $z$ in $U$. Thus $z\in H_0^m(U)$, completing all assertions.