Lemma 5.22.40DED015E67EF

Proof. Taking the exterior derivative and multiplying by $\gamma ^{n-2}$ gives

At each point, $\xi \mapsto \xi \wedge \gamma ^{n-1}$ is an isomorphism from real one-forms to real $(2n-1)$-forms, as one sees in a unitary frame. Its inverse depends smoothly on the point. Let $B$ and $C$ be the images of $\mathrm{d}\theta \wedge \gamma ^{n-2}$ and $\mathrm{d}\gamma \wedge \gamma ^{n-2}$ under this inverse. They are smooth real one-forms, and

in the sense of distributions. Since $a\in L^1_{\mathrm{loc}}$, the right-hand side is locally integrable, so $a\in W^{1,1}_{\mathrm{loc}}$. Differentiating the equation and using the smoothness of its coefficients, we obtain inductively $a\in W^{k,1}_{\mathrm{loc}}$ for every integer $k$, and the local Sobolev embedding shows that $a$ is smooth. Consequently $\mathrm{dd}^{\mathrm{c}}u=a\gamma -\theta $ is smooth; taking its trace in local coordinates and applying elliptic regularity shows that $u$ is smooth as well. $\square$