PomeProposition 11.23.2.0B0003A3A873 For every smooth real $v$,
Moreover, for smooth real $h$, the equation $L_ uv=h$ has a smooth solution if and only if $\int _ Xh\, \mathrm{d}\mu _ u=0$. When it exists, the solution is unique modulo constants.
Proposition 11.23.20B0003A3A873
Proof. The integral in (11.23.3) is
\[ n\int _ X\mathrm{dd}^{\mathrm{c}}v\wedge \rho _ u\eta ^{n-1} =n\int _ Xv\, \mathrm{dd}^{\mathrm{c}}(\rho _ u\eta ^{n-1})=0. \]
This proves that the compatibility condition is necessary.
By the Fredholm argument in the proof of theorem 11.4.2, the range of $L_ u:C^{k+2,\alpha }(X)\to C^{k,\alpha }(X)$ has codimension one for every integer $k\geq 0$ and $0<\alpha <1$. The nonzero functional $h\mapsto \int _ Xh\, \mathrm{d}\mu _ u$ annihilates that range, so its kernel is the range. Elliptic regularity gives a smooth solution, and since the kernel of $L_ u$ consists of the constants, the solution is unique modulo constants. $\square$
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