Peak functions at strongly pseudoconvex boundary points, by a dbar correction

Put $w=z-p$ and define the holomorphic Levi polynomial $f(z)=2{\sum }_{j<n}{\rho }_{z_j}(p)w_j+{\sum }_{j,k<n}{\rho }_{z_j{z}_k}(p)w_j{w}_k\mathrm{.}$ By [F1], $\rho (p+w)=\mathrm{Re}f(p+w)+{\sum }_{j,k<n}{\rho }_{z_j{\bar{z}}_k}(p)w_j{\bar{w}}_k+o(\mathrm{\mid }w{\mathrm{\mid }}^2)\mathrm{.}$ The Hermitian form is bounded below by $\lambda \mathrm{\mid }w{\mathrm{\mid }}^2$ for some $\lambda >0$. Choose $\delta >0$ with $B(p,\delta )\subset U$ and remainder at most $\lambda \mathrm{\mid }w{\mathrm{\mid }}^2\mathrm{/}2$. Since $\rho \le 0$ on $\bar{D}$, $\mathrm{Re}f(z)\le -\frac{\lambda }{2}\mathrm{\mid }z-p{\mathrm{\mid }}^2(z\in \bar{D}\cap B(p,\delta ))\mathrm{.}$ Thus $f(p)=0$ and $f$ is zero-free on this part of $\bar{D}\setminus \{p\}$. The argument includes $d\rho (p)=0$: the linear term then vanishes and the same quadratic estimate applies.

Fix $0<r<R<\delta$. By [F6] take $\chi \in C_c^{\mathrm{\infty }}(B(p,R),[0,1])$ equal to $1$ on a neighborhood of $\bar{B}(p,r)$. Its derivative support lies in the compact annulus $A=\{r\le \mathrm{\mid }z-p\mathrm{\mid }\le R\}$. The compact set $Z=A\cap \{f=0\}$ is disjoint from $\bar{D}$ by step 1.1. Apply [F2] with the prescribed open neighborhood $O=U\setminus Z$ to get $G={\bigcup }_{i=1}^N{G}_i\supset \bar{D}$, with $\bar{G}\subset O$. Hence $f$ is bounded away from zero on $A\cap \bar{G}$ when this set is nonempty. On $G$ define $\alpha =(\bar{\mathrm{\partial }}\chi )\mathrm{/}f$ in the annulus and zero outside it. More precisely, use the quotient on the open zero-free neighborhood of $\mathrm{supp}(\bar{\mathrm{\partial }}\chi )\cap \bar{G}$ and zero wherever $\chi$ is locally constant. These definitions agree, so $\alpha$ is smooth, bounded on $G$, and $\bar{\mathrm{\partial }}\alpha =0$ by [F7]. It vanishes near $p$ and satisfies $f\alpha =\bar{\mathrm{\partial }}\chi$ everywhere on $G$. No sublevel of $\mathrm{\mid }f\mathrm{\mid }$ is asserted to lie in a ball.

Each $G_i$ has a continuous psh exhaustion by [F2]. Using [F9], apply [F4] to regularize it and then conclude that $G_i$ is Hartogs pseudoconvex in the actual polydisc-radius convention. Take $\phi (z)=\mathrm{\mid }z{\mathrm{\mid }}^2$: its Levi eigenvalues are all $1$. The energy ${\int }_{G_i}\mathrm{\mid }\alpha {\mathrm{\mid }}^2{e}^{-\mathrm{\mid }z{\mathrm{\mid }}^2}$ is finite because $G_i$ is bounded and $\alpha$ is bounded. Thus [F5] gives a smooth scalar $v_i$ on $G_i$ with $\bar{\mathrm{\partial }}{v}_i=\alpha$. Define $v=v_i$ on each of the finitely many disjoint components. Then $v\in C^{\mathrm{\infty }}(G)$ solves $\bar{\mathrm{\partial }}v=\alpha$, is holomorphic near $p$, and is bounded on the compact set $\bar{D}\subset G$. The data need not have compact support in each $G_i$: boundedness on the bounded domain proves the required finite energy.

Choose $c>1+{\max }_{\bar{D}}\mathrm{\mid }v\mathrm{\mid }$ and put $Q=(c+v)f-\chi$ on $G$. Since $f\alpha =\bar{\mathrm{\partial }}\chi$, $\bar{\mathrm{\partial }}Q=f\bar{\mathrm{\partial }}v-\bar{\mathrm{\partial }}\chi =0,$ so $Q$ is holomorphic by [F7]. Where $\chi =0$ in a neighborhood, $v$ is holomorphic and the expression $g=1\mathrm{/}(c+v)$ is holomorphic wherever $c+v\mathrm{\ne }0$. Where $Q\mathrm{\ne }0$, the expression $g=f\mathrm{/}Q$ is holomorphic. The two expressions agree on their common domain where $\chi$ is locally zero: $Q=(c+v)f$ and $Q\mathrm{\ne }0$ there forces $f\mathrm{\ne }0$. They therefore glue on the union of these open sets.

This union contains $\bar{D}$. At a point of $\bar{D}$ outside $\mathrm{supp}\chi$, $\chi$ is locally zero and $\mathrm{Re}(c+v)>0$. At a point $z\in \bar{D}\cap \mathrm{supp}\chi$ other than $p$, step 1.1 gives $f(z)\mathrm{\ne }0$ and $\mathrm{Re}(1\mathrm{/}f(z))<0$, whence $\mathrm{Re}(c+v(z)-\frac{\chi (z)}{f(z)})\ge c-\mathrm{\mid }v(z)\mathrm{\mid }>0.$ Thus $Q(z)\mathrm{\ne }0$. At $p$ one has $Q(p)=-1$, so $f\mathrm{/}Q$ is holomorphic on a full neighborhood of $p$ and $g(p)=0$. Every point of $D$ has $\mathrm{Re}g>0$: where $\chi =0$ this follows from $g=1\mathrm{/}(c+v)$, and where $\chi \mathrm{\ne }0$ from the same displayed inequality and $g=1\mathrm{/}(c+v-\chi \mathrm{/}f)$. Therefore $g$ is holomorphic on an open neighborhood of $\bar{D}$, vanishes at $p$, and has positive real part on $D$.

Set $h=e^{-g}$ on this neighborhood. It is holomorphic, $h(p)=1$, and $\mathrm{\mid }h(z)\mathrm{\mid }=e^{-\mathrm{Re}g(z)}<1$ for every $z\in D$. This proves branch 1 under exactly its stated smooth negative-set hypotheses, including critical boundary points and disconnected $D$.

Under the smooth strongly pseudoconvex boundary hypotheses of branch 2, [F3] constructs a $C^{\mathrm{\infty }}$ defining function on a neighborhood of $\bar{D}$, strictly psh near $\mathrm{\partial }D$. Its proof glues the given smooth local defining functions with a finite partition near the compact boundary, extends with a sign-constant interior/exterior term, and applies $e^{Cr}-1$ after a tangent/normal Levi estimate. Thus it supplies the smooth data required by branch 1 without upgrading a merely $C^2$ function. Step 6.1 now gives the same $h$ for branch 2.

Both branches of the Statement hold with all their original hypotheses, under the ambient AC.