Lemma 6.4.2F03962A0AC31

Proof. Step 1: cutting off. Let $f\in \operatorname{Dom}T^*\cap \operatorname{Dom}S$, and write $\chi =\chi _\nu $. Since $\chi $ is real, $|\partial _{z_ j}\chi |=|\partial _{\bar z_ j}\chi |$, so the Cauchy–Schwarz inequality gives $|\sum _ j(\partial _{z_ j}\chi )f_ j|^2\le |\bar\partial \chi |^2|f|^2$. Lagrange's identity $\sum _{j<k}|a_ jb_ k-a_ kb_ j|^2=|a|^2|b|^2-|\sum _ ja_ j\bar b_ j|^2$ gives likewise $|\bar\partial \chi \wedge f|^2\le |\bar\partial \chi |^2|f|^2$. Since $\varphi _1-2\varphi _2=-\varphi =-\varphi _3$ and $|\bar\partial \chi |^2\mathrm e^{-\varphi }\le \mathrm e^{-\varphi _2}$, the form $\bar\partial \chi \wedge f$ lies in $\mathcal H_2$, the function $\mathrm e^{\varphi _1-\varphi _2}\sum _ j(\partial _{z_ j}\chi )f_ j$ lies in $\mathcal H_0$, and both of their squared norms are at most

Let $u\in \operatorname{Dom}T$. Then $\chi u\in \operatorname{Dom}T$ and $T(\chi u)=\chi Tu+u\, \bar\partial \chi $, because $u\, \bar\partial \chi \in \mathcal H_1$, the weights being bounded on the support of $\chi $. As $\chi $ is real and $\overline{\partial _{\bar z_ j}\chi }=\partial _{z_ j}\chi $, we get

Hence $\chi f\in \operatorname{Dom}T^*$, and the distributional product rule shows that $\chi f\in \operatorname{Dom}S$, with

By (6.4.3), the squared norms of both differences are bounded by $\int |f|^2|\bar\partial \chi _\nu |^2\mathrm e^{-\varphi }\, \mathrm dV$. As $\nu \to \infty $, $\bar\partial \chi _\nu $ tends to zero pointwise, since it vanishes near each point for large $\nu $, and the integrand is dominated by the integrable function $|f|^2\mathrm e^{-\varphi _2}$. By dominated convergence, both differences tend to zero. Again by dominated convergence, $\chi _\nu f\to f$, $\chi _\nu T^*f\to T^*f$ and $\chi _\nu Sf\to Sf$. Therefore $\chi _\nu f\to f$ in the graph norm.

Step 2: smoothing. It remains to approximate a form $f\in \operatorname{Dom}T^*\cap \operatorname{Dom}S$ with compact support $K\subset \Omega $. By (6.4.1),

and $\bar\partial f=Sf\in L^2_{\mathrm{loc}}(\Omega )$. Let $f^\varepsilon =\sum _ j(f_ j*\rho _\varepsilon )\, \mathrm d\bar z_ j$, with $\rho _\varepsilon $ the mollifier of section 2.5; for small $\varepsilon $ these forms are smooth and supported in a fixed compact neighborhood $K'$ of $K$ in $\Omega $. Convolution commutes with derivatives with constant coefficients (proposition 2.5.3), so $\bar\partial f^\varepsilon =(\bar\partial f)*\rho _\varepsilon $ and $\sum _ j\partial _{z_ j}f^\varepsilon _ j=(\sum _ j\partial _{z_ j}f_ j)*\rho _\varepsilon $. By lemma 2.5.1, $f^\varepsilon $, $\sum _ j\partial _{z_ j}f^\varepsilon _ j$ and $\bar\partial f^\varepsilon $ converge in $L^2(K')$ to $f$, $\sum _ j\partial _{z_ j}f_ j$ and $\bar\partial f$. Since (6.4.1) applies to both $f^\varepsilon $ and $f$, and its remaining coefficients are smooth, $T^*f^\varepsilon \to T^*f$ in $L^2(K')$. The weights are bounded above and below on $K'$, so $f^\varepsilon \to f$ in the graph norm. $\square$