Theorem 9.11.66.6EDDAD49D627 For sufficiently small nonzero $\varepsilon $, depending on $n$, $s$ and the fixed forms, there are explicit constants $R,\delta >0$ such that every $g\in H^ s_0$ with $\| g\| _{H^ s}\leq \delta $ admits a unique $w\in H^{s+2}_0$ with $\| w\| _{H^{s+2}}\leq R$ satisfying
\[ (k+\mathrm{dd}^{\mathrm{c}}w)^ n=(f_\varepsilon +g)\Omega , \qquad f_\varepsilon =\frac{k^ n}{\Omega },\qquad k+\mathrm{dd}^{\mathrm{c}}w\geq \tfrac 14K_0. \]
The constants may be chosen as in (9.11.67). In particular, $w$ is of class $C^2$.
Proof.
Set $b_1=\lambda $ and $b_ j=1$ for $j\geq 2$. The operator
\[ L_0w=\frac{nK_0^{n-1}\wedge \mathrm{dd}^{\mathrm{c}}w}{\Omega } =\sum _ j b_ j^{-1}\partial _{z_ j}\partial _{\bar z_ j}w \]
has Fourier eigenvalues $\ell (m)=-\pi ^2\sum _ jb_ j^{-1}(m_{2j-1}^2+m_{2j}^2)$. It has an inverse $H^ s_0\to H^{s+2}_0$ of norm
\[ A_0=\sup _{m\ne 0}\frac{1+|m|^2}{|\ell (m)|} \leq \frac{2\max _ jb_ j}{\pi ^2}. \]
Let $L_\varepsilon w=nk^{n-1}\wedge \mathrm{dd}^{\mathrm{c}}w/\Omega $. It preserves zero mean because $\mathrm{dd}^{\mathrm{c}}(k^{n-1})=0$. Fix $\varepsilon _0\leq 1$ within the positivity range. The finite expansion of $k^{n-1}$ gives
\[ L_\varepsilon -L_0=\sum _{j=1}^{n-1}\varepsilon ^ jP_ j, \qquad P_ jw=n\binom {n-1}{j} \frac{K_0^{n-1-j}\wedge \alpha ^ j\wedge \mathrm{dd}^{\mathrm{c}}w}{\Omega }. \]
Choose bounds $e_ j$ for the norms $\| P_ j\| _{H^{s+2}\to H^ s}$, which are finite since multiplication by smooth coefficients is bounded on $H^ s$, and set $B=\sum _ j\varepsilon _0^{j-1}e_ j$. If $|\varepsilon |\leq \varepsilon _0$ and $|\varepsilon |A_0B\leq 1/2$, the Neumann series gives $\| L_\varepsilon ^{-1}\| \leq A:=2A_0$.
The remainder is
\[ Q(w)=\sum _{r=2}^ n\binom nr\frac{k^{n-r}\wedge (\mathrm{dd}^{\mathrm{c}}w)^ r}{\Omega }. \]
For $s>n$, Fourier convolution and the Cauchy–Schwarz inequality, together with $\sum _ m(1+|m|^2)^{-s}<\infty $, make $H^ s$ an algebra. Multiplication by smooth coefficients and the finite expansion therefore give a constant $C_ Q$ such that, on the unit ball of $H^{s+2}$,
\[ \begin{aligned} \| Q(w)\| _{H^ s}& \leq C_ Q\| w\| _{H^{s+2}}^2,\\ \| Q(w)-Q(z)\| _{H^ s}& \leq C_ Q(\| w\| _{H^{s+2}}+\| z\| _{H^{s+2}})\| w-z\| _{H^{s+2}}. \end{aligned} \]
Each term of $Q(w)$ has zero integral against $\Omega $: for smooth $w$, its integral is $\int w\, \mathrm{dd}^{\mathrm{c}}(k^{n-r})\wedge (\mathrm{dd}^{\mathrm{c}}w)^{r-1}=0$, and Sobolev approximation gives the general case. The Sobolev embedding also gives a constant $D$ such that the pointwise operator norm of $\mathrm{dd}^{\mathrm{c}}w$ with respect to $K_0$ is at most $D\| w\| _{H^{s+2}}$. Choose
E79D47E722CF
9.11.67
\begin{equation} 0<R\leq \min \{ 1,(4AC_ Q)^{-1},(4D)^{-1}\} ,\qquad \delta =R/(2A). \end{equation}
The equation is equivalent to $L_\varepsilon w+Q(w)=g$. The map $w\mapsto L_\varepsilon ^{-1}(g-Q(w))$ takes the closed ball of radius $R$ in $H^{s+2}_0$ into the ball of radius $3R/4$ and has Lipschitz constant at most $1/2$. Its fixed point gives existence and uniqueness in that ball. Since $k\geq K_0/2$ and the operator norm of $\mathrm{dd}^{\mathrm{c}}w$ is at most $1/4$, the form $k+\mathrm{dd}^{\mathrm{c}}w$ is at least $K_0/4$. The embedding $H^{s+2}\subset C^2$ follows from $s\geq n+3$.
$\square$
Pome
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