Green correctors are smooth at analytic boundaries

Complexifying the chart: by [F5] the parametrization satisfies $\gamma (t)={\sum }_{n\ge 0}{c}_n{t}^n$ for $\mathrm{\mid }t\mathrm{\mid }$ small, with $c_0=\zeta$ and $c_1={\gamma }^{\mathrm{\prime }}(0)\mathrm{\ne }0$. The complex power series $\mathrm{\Gamma }(w):={\sum }_{n\ge 0}{c}_n{w}^n$ converges on a disc $D(0,r_0)$ and defines a holomorphic function there with $\mathrm{\Gamma }(t)=\gamma (t)$ for real $\mathrm{\mid }t\mathrm{\mid }<r_0$ and ${\mathrm{\Gamma }}^{\mathrm{\prime }}(0)=c_1\mathrm{\ne }0$. By [F4] the map $\mathrm{\Gamma }$ restricts to a biholomorphism from some disc $D(0,r)\subseteq D(0,r_0)$ onto an open neighbourhood $V$ of $\zeta$, and the two components of $D(0,r)\setminus (-r,r)$ map onto the two components of $V\setminus \gamma ((-r,r))$, the latter being an arc of $\mathrm{\partial }D$ when $r\le \epsilon$. Replacing $\gamma$ by $t\mapsto \gamma (-t)$ if necessary, we may assume $\mathrm{\Gamma }$ maps the upper half-disc ${\mathbb{D}}_r^{+}:=\{w:\mathrm{\mid }w\mathrm{\mid }<r,\mathrm{Im}w>0\}$ onto $D\cap V$.

A local peak on the domain: define, for $w\in {\mathbb{D}}_r^{+}$, $q(w):=-\mathrm{Re}\sqrt{-iw},$ with the principal square root. Since $w\mapsto \sqrt{-iw}$ is holomorphic on the half-disc --- there $-iw$ has positive real part, so it avoids $(-\mathrm{\infty },0]$ --- both components of that holomorphic square root are smooth by [F6], so the $C^2$ components theorem makes $q$ harmonic and the Laplacian criterion makes it subharmonic on ${\mathbb{D}}_r^{+}$. Writing $w=\rho e^{i\phi }$ with $0<\phi <\pi$ gives $-iw=\rho e^{i(\phi -\pi \mathrm{/}2)}$ with $\phi -\pi \mathrm{/}2\in (-\pi \mathrm{/}2,\pi \mathrm{/}2)$, so $\sqrt{-iw}=\sqrt{\rho }{e}^{i(\phi -\pi \mathrm{/}2)\mathrm{/}2}$ and $q(w)=-\sqrt{\rho }\cos (\frac{\phi }{2}-\frac{\pi }{4})\le -\frac{\sqrt{\rho }}{\sqrt{2}}<0,$ because $\mathrm{\mid }\phi \mathrm{/}2-\pi \mathrm{/}4\mathrm{\mid }<\pi \mathrm{/}4$; moreover $q(w)\to 0$ as $w\to 0$. Hence $q\circ {\mathrm{\Gamma }}^{-1}$ is subharmonic (by [F6], applied to the holomorphic ${\mathrm{\Gamma }}^{-1}$) and negative on $D\cap V$, and it tends to $0$ at $\zeta$.

The peak is bounded away from $0$ on the boundary of a smaller neighbourhood. Let $W:=\mathrm{\Gamma }(D(0,r\mathrm{/}2))$. Then $W$ is a neighbourhood of $\zeta$ with $W⋐V$, and $D\cap \mathrm{\partial }W=\mathrm{\Gamma }({\mathbb{D}}_r^{+}\cap \{\mathrm{\mid }w\mathrm{\mid }=r\mathrm{/}2\})$ by step 1.1; on that set $\rho =r\mathrm{/}2$, so step 2.1 gives $\sup \{q({\mathrm{\Gamma }}^{-1}(z)):z\in D\cap \mathrm{\partial }W\}\le -(r\mathrm{/}2{)}^{1\mathrm{/}2}\mathrm{/}\sqrt{2}<0$.

Conclusion of regularity: step 3.1 verifies hypothesis 3 of [F2] for the subharmonic function $q\circ {\mathrm{\Gamma }}^{-1}$ of step 2.1, whose hypotheses 1 and 2 were also verified there; hence $D$ has a global barrier at $\zeta$ and $\zeta$ is a regular boundary point by [F2]. As $\zeta \in \mathrm{\partial }D$ was arbitrary, every boundary point of $D$ is regular.

Smoothness near the arc by reflection: Countable Choice [F7] licenses the Green kernel and its regular-boundary limits from [F1]. Fix $\zeta$ and keep the chart $\mathrm{\Gamma }$ of step 1.1. Choose $0<R<r$ small enough that $\overline{D(0,R)}$ remains in the chart and $\mathrm{\Gamma }(\overline{{\mathbb{D}}_R^{+}})\subseteq \bar{D}\setminus \{a\}$; this is possible since $\mathrm{\Gamma }(0)=\zeta \mathrm{\ne }a$, while step 1.1 puts the open upper half-disc in $D$ and its real diameter on $\mathrm{\partial }D$. Define $G(w):=g_D(\mathrm{\Gamma }(w),a)$ for $w\in {\mathbb{D}}_R^{+}$. Then $G$ is harmonic there by [F6], since $g_D(\cdot ,a)$ is harmonic on $D\setminus \{a\}$ by [F1] and $\mathrm{\Gamma }$ is holomorphic. At a real $t\in (-R,R)$ define the boundary value $G(t):=0$. This is a continuous extension across the diameter: whenever $w_n$ approaches $t$ from the upper half-disc, $\mathrm{\Gamma }(w_n)\in D$ approaches the regular boundary point $\mathrm{\Gamma }(t)$, so [F1] and step 4.1 give $G(w_n)\to 0$. On the upper semicircle of radius $R$, the image lies in $D$ except at its two real endpoints; the same interior continuity and regular-boundary limits give continuity on the whole closed half-disc. Rescale by $w\mapsto w\mathrm{/}R$ and apply [F3]; the odd reflection is harmonic on $D(0,R)$ and smooth there. Choose $0<r_1<R$; its restriction to $\overline{{\mathbb{D}}_{r_1}^{+}}$ is therefore $C^2$, including the real diameter near $0$.

The corrector near the arc is $C^2$: on the smaller closed half-disc $\overline{{\mathbb{D}}_{r_1}^{+}}$ of step 5.1 one has $h_a(\mathrm{\Gamma }(w))=F_a(\mathrm{\Gamma }(w))-G(w)$ for interior $w$, and this equality extends continuously to its real diameter using the regular boundary values. The reflected extension of $G$ is $C^2$ by step 5.1. Also $F_a\circ \mathrm{\Gamma }$ is smooth on a neighbourhood of this closed half-disc: its compact image lies in $\bar{D}\setminus \{a\}$, hence at positive distance from $a$, and $F_a=-\log \mathrm{\mid }\cdot -a\mathrm{\mid }$ is smooth on the ambient open set $\mathbb{C}\setminus \{a\}$. Thus $F_a\circ \mathrm{\Gamma }-G$ gives a $C^2$ extension of $h_a\circ \mathrm{\Gamma }$ across the real diameter. Transport through the local biholomorphism $\mathrm{\Gamma }$ gives a $C^2$ extension of $h_a$ to an ambient neighbourhood of the boundary arc near $\zeta$.

Since $\zeta$ was arbitrary, step 6.1 supplies a $C^2$ ambient extension near every boundary point; interior harmonicity supplies smoothness at every interior point. On overlaps the restrictions of these extensions to $D$ equal the same $h_a$, and continuity makes their boundary values and one-sided derivatives agree on $\bar{D}$. The extensions need not agree outside $D$; their local existence is exactly the asserted $C^2$ regularity on the closure. Finally $g_D(z,a)=-\log \mathrm{\mid }z-a\mathrm{\mid }-h_a(z)$ for $z\in D\setminus \{a\}$; since $-\log \mathrm{\mid }\cdot -a\mathrm{\mid }$ is smooth away from $a$, $g_D(\cdot ,a)$ is $C^2$ on $\bar{D}\setminus \{a\}$ in the same local-extension sense, and its trace on $\mathrm{\partial }D$ vanishes by the boundary limits of [F1] in step 4.1.