Proposition 6.4.4.866FA0174B79 Let $\varphi $ be a smooth real function on $\Omega $, and let $\delta _ j$ be the differential operator $\delta _ jg=-\partial _{z_ j}g+(\partial _{z_ j}\varphi )g$, $1\le j\le n$. Then every smooth compactly supported $(0,1)$-form $f=\sum _ jf_ j\, \mathrm d\bar z_ j$ on $\Omega $ satisfies
\[ \begin{split} & \int \Bigl |\sum _ j\delta _ j f_ j\Bigr |^2\mathrm e^{-\varphi }\, \mathrm dV +\int |\bar\partial f|^2\mathrm e^{-\varphi }\, \mathrm dV\\ & \qquad =\int \sum _{j,k}\varphi _{j\bar k}f_ j\bar f_ k\, \mathrm e^{-\varphi }\, \mathrm dV +\int \sum _{j,k}|\partial _{\bar z_ k}f_ j|^2\mathrm e^{-\varphi }\, \mathrm dV. \end{split} \]
Consequently, if $\psi $ is a smooth real function and $T$ and $S$ are the operators defined by the weights $\varphi _ k=\varphi +(k-3)\psi $ of lemma 6.4.2, then for every $t>0$
038AC5695B6F
6.4.5
\begin{equation} \int \sum _{j,k}\varphi _{j\bar k}f_ j\bar f_ k\mathrm e^{-\varphi }\, \mathrm dV \le (1+t^{-1})\| T^*f\| _{\mathcal H_0}^2+\| Sf\| _{\mathcal H_2}^2 +(1+t)\int |f|^2|\bar\partial \psi |^2\mathrm e^{-\varphi }\, \mathrm dV. \end{equation}
Proof.
For smooth compactly supported functions $g$ and $w$, integration by parts (Fubini's theorem and the fundamental theorem of calculus on lines) gives
\[ \langle \partial _{\bar z_ j}g,w\rangle _\varphi =-\int g\, \partial _{\bar z_ j}\bigl (\bar w\mathrm e^{-\varphi }\bigr )\, \mathrm dV=\int g\, \overline{\bigl (-\partial _{z_ j}w+(\partial _{z_ j}\varphi )w\bigr )}\, \mathrm e^{-\varphi }\, \mathrm dV=\langle g,\delta _ jw\rangle _\varphi , \]
because $\varphi $ is real; thus $\delta _ j$ is the formal adjoint of $\partial _{\bar z_ j}$ for the inner product $\langle \cdot ,\cdot \rangle _\varphi $, and taking complex conjugates we also have $\langle \delta _ jw,g\rangle _\varphi =\langle w,\partial _{\bar z_ j}g\rangle _\varphi $. Since $\partial _{\bar z_ k}\bigl ((\partial _{z_ j}\varphi )g\bigr )=\varphi _{j\bar k}g+(\partial _{z_ j}\varphi )\partial _{\bar z_ k}g$ and the derivatives $\partial _{z_ j}$ and $\partial _{\bar z_ k}$ commute, we obtain the commutation relation
\[ \partial _{\bar z_ k}(\delta _ jg)-\delta _ j(\partial _{\bar z_ k}g)=\varphi _{j\bar k}\, g. \]
By the adjoint relation, then the commutation relation, and then the conjugate adjoint relation,
\[ \begin{split} \Bigl \| \sum _ j\delta _ jf_ j\Bigr \| _\varphi ^2& =\sum _{j,k}\langle \delta _ jf_ j,\delta _ kf_ k\rangle _\varphi =\sum _{j,k}\langle \partial _{\bar z_ k}\delta _ jf_ j,f_ k\rangle _\varphi \\ & =\sum _{j,k}\langle \varphi _{j\bar k}f_ j,f_ k\rangle _\varphi +\sum _{j,k}\langle \delta _ j\partial _{\bar z_ k}f_ j,f_ k\rangle _\varphi \\ & =\int \sum _{j,k}\varphi _{j\bar k}f_ j\bar f_ k\mathrm e^{-\varphi }\, \mathrm dV+\sum _{j,k}\langle \partial _{\bar z_ k}f_ j,\partial _{\bar z_ j}f_ k\rangle _\varphi . \end{split} \]
On the other hand, $\bar\partial f=\sum _{j<k}(\partial _{\bar z_ j}f_ k-\partial _{\bar z_ k}f_ j)\, \mathrm d\bar z_ j\wedge \mathrm d\bar z_ k$, and expanding the squares gives
\[ \| \bar\partial f\| _\varphi ^2=\sum _{j\ne k}\| \partial _{\bar z_ k}f_ j\| _\varphi ^2-\sum _{j\ne k}\langle \partial _{\bar z_ k}f_ j,\partial _{\bar z_ j}f_ k\rangle _\varphi . \]
When we add the two identities, the terms with $j\ne k$ in the last sums cancel, and the remaining terms $\| \partial _{\bar z_ j}f_ j\| _\varphi ^2$ combine with $\sum _{j\ne k}\| \partial _{\bar z_ k}f_ j\| _\varphi ^2$ to give $\sum _{j,k}\| \partial _{\bar z_ k}f_ j\| _\varphi ^2$. This proves the identity.
For the estimate, we have $\varphi _1-\varphi _2=-\psi $ and $\partial _{z_ j}\varphi _2=\partial _{z_ j}\varphi -\partial _{z_ j}\psi $, so (6.4.1) reads
\[ \sum _ j\delta _ jf_ j=\mathrm e^\psi T^*f+\sum _ j(\partial _{z_ j}\psi )f_ j. \]
By the elementary inequality $|x+y|^2\le (1+t^{-1})|x|^2+(1+t)|y|^2$ for $x,y\in \mathbb C$,
\[ \Bigl \| \sum _ j\delta _ jf_ j\Bigr \| _\varphi ^2\le (1+t^{-1})\int |T^*f|^2\mathrm e^{2\psi -\varphi }\, \mathrm dV+(1+t)\int \Bigl |\sum _ j(\partial _{z_ j}\psi )f_ j\Bigr |^2\mathrm e^{-\varphi }\, \mathrm dV. \]
The first integral is $\| T^*f\| _{\mathcal H_0}^2$, because $\varphi _1=\varphi -2\psi $, and $|\sum _ j(\partial _{z_ j}\psi )f_ j|^2\le |\bar\partial \psi |^2|f|^2$ by the Cauchy–Schwarz inequality, since $|\partial _{z_ j}\psi |=|\partial _{\bar z_ j}\psi |$ for real $\psi $. Moreover $\| \bar\partial f\| _\varphi ^2=\| Sf\| _{\mathcal H_2}^2$, because $\varphi _3=\varphi $. Inserting these bounds into the identity and dropping its last, nonnegative, term, we obtain (6.4.5).
$\square$
Pome
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