Lemma 10.7.46217A93C596F

Proof. On a compact subset of a coordinate chart over which $F$ has a holomorphic frame, the pointwise norms of $\sigma $ and $h_ k$ are comparable with the Euclidean norms of the coefficients, and $\mathrm dV_\sigma $ with Lebesgue measure. Hence convergence in $L^2_{(n,q)}(\sigma ,h_ k)$ implies $L^2$ convergence of the coefficients on such sets, and therefore convergence in the sense of distributions; since $\bar\partial $ is continuous for distributional convergence, $T$ and $S$ are closed. Compactly supported smooth forms belong to $\operatorname{Dom}T$ and $\operatorname{Dom}S$, and they are dense in $L^2_{(n,0)}(\sigma ,h_0)$ and in $L^2_{(n,1)}(\sigma ,h_1)$: if $(\psi _ a)$ is a partition of unity with compact supports in such charts (proposition 5.2.6), then an $L^2$ form $u$ is the $L^2$ limit of the finite partial sums of $\sum _ a\psi _ au$, by dominated convergence, and each $\psi _ au$ is the $L^2$ limit of the forms whose coefficients are the mollifications of those of $\psi _ au$, by lemma 2.5.1. Thus $T$ and $S$ are densely defined, and $ST=0$ because $\bar\partial \bar\partial =0$ on distributions.

If $v$ is a compactly supported smooth $(n,1)$-form and $u\in \operatorname{Dom}T$, then $\langle Tu,v\rangle $ is the value of the distribution $\bar\partial u$ on a smooth compactly supported form built from $v$, and the definition of the distributional derivative gives $\langle Tu,v\rangle =\langle u,\delta v\rangle $; hence $v\in \operatorname{Dom}T^*$ and $T^*v=\delta v$. For an arbitrary $v\in \operatorname{Dom}T^*$, testing with compactly supported smooth $u$ shows in the same way that $T^*v=\delta v$ in the sense of distributions, and $Sv=\bar\partial v$ in the sense of distributions for $v\in \operatorname{Dom}S$ by definition.

For the density, multiplication by the cutoffs $\chi _ R$ first reduces the problem to forms with compact support. Let $v\in \operatorname{Dom}T^*\cap \operatorname{Dom}S$ and $\chi =\chi _ R$. For $u\in \operatorname{Dom}T$, the Leibniz rule for distributions gives $\chi u\in \operatorname{Dom}T$ and $T(\chi u)=\chi Tu+\bar\partial \chi \wedge u$, so that $\langle Tu,\chi v\rangle =\langle T(\chi u),v\rangle -\langle \bar\partial \chi \wedge u,v\rangle =\langle u,\chi T^*v-(\bar\partial \chi \wedge )^*v\rangle $, where $(\bar\partial \chi \wedge )^*$ is the adjoint of the bounded operator $\bar\partial \chi \wedge $ from $L^2_{(n,0)}(\sigma ,h_0)$ to $L^2_{(n,1)}(\sigma ,h_1)$. Hence $\chi v\in \operatorname{Dom}T^*$ and $T^*(\chi v)=\chi T^*v-(\bar\partial \chi \wedge )^*v$; and the Leibniz rule gives $\chi v\in \operatorname{Dom}S$ with $S(\chi v)=\chi Sv+\bar\partial \chi \wedge v$. Since $|\bar\partial \chi _ R|_\sigma \leq |\mathrm d\chi _ R|_\sigma \leq C/R$ and the ratios $h_1/h_0$ and $h_2/h_1$ are bounded, the error terms have norm at most $C'\| v\| /R$, and $\chi _ Rv\to v$, $\chi _ RT^*v\to T^*v$, $\chi _ RSv\to Sv$ by dominated convergence.

Second, a compactly supported form is approximated by local convolutions, after a finite partition of unity (proposition 5.2.6). For a first-order operator $P=\sum _ la_ l(x)\partial _ l+b(x)$ with smooth matrix coefficients and a mollifier $\rho _ t$, the commutator $[P,\rho _ t*\, \cdot \, ]$ applied to $v$ equals

On a fixed compact set, the mean-value bound $|a_ l(x)-a_ l(y)|\leq C|x-y|$ and the scaling of $\rho _ t$ bound the kernels of these integral operators by $k_ t(x-y)$, where $\int k_ t$ is bounded independently of $t$. An integral operator with a kernel bounded by $k(x-y)$, $k\geq 0$ integrable, has norm at most $\int k$ on $L^2$, since by the Cauchy–Schwarz inequality $|\int k(x-y)v(y)\, \mathrm dy|^2\leq \int k\cdot \int k(x-y)|v(y)|^2\, \mathrm dy$; so these operators are bounded on $L^2$ uniformly in $t$. For smooth $v$ the commutator tends to zero in $L^2$, and by the uniform bound and the density of smooth functions in $L^2$ it tends to zero for every $v\in L^2$.

Now let $v\in \operatorname{Dom}T^*\cap \operatorname{Dom}S$ have compact support, and let $(\psi _ a)$ be a finite partition of unity near $\operatorname{Supp}v$ by functions with compact supports in charts over which $F$ has frames. By the product rules above, with $\psi _ a$ in place of $\chi $, each $w=\psi _ av$ lies in $\operatorname{Dom}T^*\cap \operatorname{Dom}S$. In the chart and frame, $\delta $ and $\bar\partial $ are first-order operators with smooth matrix coefficients, and $T^*w=\delta w$ and $Sw=\bar\partial w$ lie in $L^2$. The forms $w_ t$ whose coefficients are the mollifications of those of $w$ are smooth with compact support, so $T^*w_ t=\delta w_ t=\rho _ t*(\delta w)+[\delta ,\rho _ t*\, \cdot \, ]w$ and $Sw_ t=\rho _ t*(\bar\partial w)+[\bar\partial ,\rho _ t*\, \cdot \, ]w$. By lemma 2.5.1 and the commutator bound, $w_ t\to w$ in the graph norm as $t\to 0$, which proves the density. $\square$