Weak Leibniz rule with a smooth factor

For every $\gamma \le \alpha$, the weak identity and [F2] give ${\mathrm{\partial }}^{\gamma }{T}_u=T_{v_{\gamma }}\mathrm{.}$ In particular, the derivative on the left is represented by a locally integrable function for each such $\gamma$.

Set $w_{\alpha }={\sum }_{\beta \le \alpha }\left(\genfrac{}{}{0px}{}{\alpha }{\beta }\right)(D^{\beta }\eta )v_{\alpha -\beta }\mathrm{.}$ The sum is finite. On each compact subset of $\mathrm{\Omega }$, every smooth $D^{\beta }\eta$ is bounded and every $v_{\alpha -\beta }$ is integrable, so $w_{\alpha }\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega };\mathbb{K})$. Changing any selected representative on a null set changes each product only on a null set; hence $w_{\alpha }$ defines a single almost-everywhere class.

Apply [F3] to $\eta$ and $T_u$, then substitute the identities of step 1.1. By [F4], the left side is the $\alpha$th distributional derivative of $T_{\eta u}$, while each term on the right is the regular distribution of the corresponding summand in $w_{\alpha }$. Thus ${\mathrm{\partial }}^{\alpha }{T}_{\eta u}=T_{w_{\alpha }}\mathrm{.}$ By the weak-derivative identity in [F2], $w_{\alpha }$ is a locally integrable weak $\alpha$-derivative of $\eta u$. By [F5] it is the unique value class, which proves the asserted local formula.

Now suppose the global $W^{k,p}$ hypotheses hold. For each $\mathrm{\mid }\alpha \mathrm{\mid }\le k$, the local result applies because all weak derivatives $D^{\gamma }u$ with $\gamma \le \alpha$ are supplied by [F6]. For every $\beta \le \alpha$, the factor $D^{\beta }\eta$ is bounded, so $(D^{\beta }\eta )D^{\alpha -\beta }u$ is an $L^p$ class: for finite $p$ its modulus is bounded by $\mathrm{\parallel }{D}^{\beta }\eta {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }{D}^{\alpha -\beta }u\mathrm{\mid }$ almost everywhere, and for $p=\mathrm{\infty }$ the same estimate bounds the essential supremum. The finite sum is therefore in $L^p$. Taking $\alpha =0$ also shows $\eta u\in L^p$; the local result for every $\mathrm{\mid }\alpha \mathrm{\mid }\le k$ now gives $\eta u\in W^{k,p}$ by its definition, and its weak derivative classes are exactly the displayed sums.

If $\eta \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{K})$, its support is compact in $\mathrm{\Omega }$. Each derivative is continuous and vanishes off that compact support, so every derivative through order $k$ is bounded. Thus the preceding global argument applies. For $k=0$ the formula has only $\beta =0$ and says $D^0(\eta u)=\eta u$. When $\alpha =0$ the same order-zero identity holds in the local assertion. For the zero input class, zero is a weak derivative at every order by the defining test identity, and [F5] makes each supplied derivative class zero; if $\eta =0$, every summand is zero. On the empty domain every term is the zero class.