Proposition 11.40.41A47D2E8B6D5

Proof. Fix $k\geq 0$ and $0<\alpha <1$. On the open set where $\eta +\mathrm{dd}^{\mathrm{c}}w>0$, consider the smooth map

defined for $C^{k,\alpha }$ forms $\eta $ and functions $G$, mean-zero $C^{k+2,\alpha }$ potentials $w$ and $a\in \mathbb R$, with values in $C^{k,\alpha }$. At the limiting solution its derivative in $(w,a)$ is $(h,q)\mapsto \Delta _\Omega h-q$, where $\Omega =\beta +\mathrm{dd}^{\mathrm{c}}v$ and $\Delta _\Omega h=n\mathrm{dd}^{\mathrm{c}}h\wedge \Omega ^{n-1}/\Omega ^ n$. Given a target $g$, choose $q=-\int _ Xg\, \mathrm{d}\mu _ v$, where $\mu _ v$ is the adjoint probability of proposition 11.23.2. The Fredholm alternative then gives a unique mean-zero $h$ with $\Delta _\Omega h=g+q$, and the inverse is bounded as in proposition 14.4.1. In dimension one this is the Fredholm alternative for the Poisson equation.

The implicit function theorem gives a local branch of solutions depending continuously on the parameters in these spaces. For smooth parameters, uniqueness of bounded solutions modulo constants, corollary 14.24.17, identifies this branch with the global smooth solution. Thus $v_ j\to v$ in $C^{k+2,\alpha }$ for every $k$, which proves the assertion. $\square$