Weak solutions factor holomorphically in Hölder coordinates

Put $W=\mathrm{\Omega }\cap V$ and $Y=\mathrm{\Phi }(W)$. Since $\mathrm{\Phi }$ is a $C^1$ diffeomorphism of $V$ onto the open set $\mathrm{\Phi }(V)$, its restriction maps the open set $W$ diffeomorphically onto the open set $Y$. Composition of the given a.e. class with ${\mathrm{\Phi }}^{-1}$ is well-defined by [F4]. The rational open boxes whose closures lie in $Y$ form a countable cover by [F16]. For such a box $B$, put $W_B:={\mathrm{\Phi }}^{-1}(B)$. Its closure is compact in $W$: $\bar{B}$ is compact by [F16], and its image under ${\mathrm{\Phi }}^{-1}$ is compact by [F17]. Apply [F3] to ${\mathrm{\Phi }}^{-1}:B\to W_B$ and $f{\mathrm{\mid }}_{W_B}\in W^{1,2}(W_B)$; the chart and inverse derivatives are bounded on these compact patches by [F1]. Thus $h=f\circ {\mathrm{\Phi }}^{-1}$ lies in $W^{1,2}(B)$ with $Dh(w)=Df({\mathrm{\Phi }}^{-1}w)D{\mathrm{\Phi }}^{-1}(w)$ almost everywhere on $B$. The countable cover, locality, and uniqueness of weak derivatives in [F2] give $h\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,2}(Y)$.

On each box $B$ of step 1.1, the chain identity there and $D{\mathrm{\Phi }}^{-1}(\mathrm{\Phi }z)D\mathrm{\Phi }(z)=I$ imply $Df(z)=Dh(\mathrm{\Phi }z)D\mathrm{\Phi }(z)$ for almost every $z\in W_B$: the exceptional null set pulls back to a null set by [F4]. Rewriting this real-linear identity in Wirtinger coordinates [F5] gives $f_z=(h_w\circ \mathrm{\Phi }){\mathrm{\Phi }}_z+(h_{\bar{w}}\circ \mathrm{\Phi })\overline{{\mathrm{\Phi }}_{\stackrel{ˉ}{z}}},f_{\bar{z}}=(h_w\circ \mathrm{\Phi }){\mathrm{\Phi }}_{\bar{z}}+(h_{\bar{w}}\circ \mathrm{\Phi })\overline{{\mathrm{\Phi }}_z}\mathrm{.}$ Subtract $\mu f_z$ from $f_{\bar{z}}$, use the weak equation for $f$ and ${\mathrm{\Phi }}_{\bar{z}}=\mu {\mathrm{\Phi }}_z$, and obtain $0=(h_{\bar{w}}\circ \mathrm{\Phi })(\overline{{\mathrm{\Phi }}_z}-\mu \overline{{\mathrm{\Phi }}_{\stackrel{ˉ}{z}}})=(h_{\bar{w}}\circ \mathrm{\Phi })(1-\mathrm{\mid }\mu {\mathrm{\mid }}^2)\overline{{\mathrm{\Phi }}_z}$ almost everywhere on $W_B$. By [F1], the last factor is nowhere zero, since $J_{\mathrm{\Phi }}=(1-\mathrm{\mid }\mu {\mathrm{\mid }}^2)\mathrm{\mid }{\mathrm{\Phi }}_z{\mathrm{\mid }}^2>0$. Hence $h_{\bar{w}}\circ \mathrm{\Phi }=0$ almost everywhere on each $W_B$. By [F19], choose one exceptional null set for each box in the countable cover. Their images under $\mathrm{\Phi }$ are null by [F4], and [F18] makes their union null, so $h_{\bar{w}}=0$ almost everywhere on all of $Y$.

Write $h=u+iv$ with real locally integrable classes $u,v$; local integrability follows from [F7]. Let $T_q$ denote the regular distribution of each locally integrable class $q$ as in [F10]. The equation $h_{\bar{w}}=0$ says $u_x-v_y=0$ and $v_x+u_y=0$ almost everywhere, hence the same equalities hold for their regular distributions. Using [F6], $\mathrm{\Delta }{T}_u={\mathrm{\partial }}_x{T}_{u_x}+{\mathrm{\partial }}_y{T}_{u_y}={\mathrm{\partial }}_x{T}_{v_y}-{\mathrm{\partial }}_y{T}_{v_x}=0,$ $\mathrm{\Delta }{T}_v={\mathrm{\partial }}_x{T}_{v_x}+{\mathrm{\partial }}_y{T}_{v_y}=-{\mathrm{\partial }}_x{T}_{u_y}+{\mathrm{\partial }}_y{T}_{u_x}=0.$ Apply [F8] separately to these real distributions. There are smooth harmonic functions $U,V$ on $Y$ with $u=U$ and $v=V$ almost everywhere.

Since $T_U=T_u$ and $T_V=T_v$, the distributional identities ${\mathrm{\partial }}_x{T}_U-{\mathrm{\partial }}_y{T}_V=0$ and ${\mathrm{\partial }}_x{T}_V+{\mathrm{\partial }}_y{T}_U=0$ follow from step 3.1. By [F9], these distributions are the regular distributions of $U_x-V_y$ and $V_x+U_y$; [F10] makes both continuous functions zero almost everywhere. They vanish everywhere: if either were nonzero at a point, continuity would keep its modulus positive on a ball of positive measure by [F12]. Thus $U,V$ satisfy the Cauchy–Riemann equations at every point. By [F13], $H:=U+iV$ is holomorphic on $Y$ and represents $h$.

Fix $z_0\in W$. Choose a convex ball $B_z$ with $z_0\in B_z$ and $\overline{B_z}\subset W$, and a convex ball $B_w$ with $\overline{B_w}\subset Y$ such that $\mathrm{\Phi }(\overline{B_z})\subset B_w$; this is possible by continuity of $\mathrm{\Phi }$ and openness of $Y$. By [F16], the closed balls are compact and the derivatives of the smooth $H$ through order $k+2$ are bounded on $\overline{B_w}$. The chart bounds in [F1] bound the derivatives of $\mathrm{\Phi }$ through order $k+1$ on $B_z$, with the top-order $\alpha$-seminorm finite. Repeated use of the chain rule [F15] and the coordinate product rule expresses each derivative of $H\circ \mathrm{\Phi }$ through order $k+1$ as a finite sum of products of derivatives of $H$ composed with $\mathrm{\Phi }$ and derivatives of $\mathrm{\Phi }$. The mean-value bound [F15] makes each composed derivative of $H$ Lipschitz on $B_z$; it also makes derivatives of $\mathrm{\Phi }$ through order $k$ Lipschitz there. These fields are bounded, while derivatives of $\mathrm{\Phi }$ of order $k+1$ are $\alpha$-Hölder by [F1]. The inequality $[FG{]}_{0,\alpha }\le \mathrm{\parallel }F{\mathrm{\parallel }}_{\mathrm{\infty }}[G{]}_{0,\alpha }+\mathrm{\parallel }G{\mathrm{\parallel }}_{\mathrm{\infty }}[F{]}_{0,\alpha }$, from [F15], shows each finite product and sum has the same local Hölder bound. Thus $H\circ \mathrm{\Phi }$ has the required norm on each such ball. For any $O⋐W$, compactness of $\bar{O}$ gives a finite subcover by these balls; [F20] gives a Lebesgue number for that cover. Pairs in $O$ closer than this number lie in one ball and use its Hölder bound; pairs farther apart are controlled by the bounded derivative suprema and the positive lower distance. Hence $H\circ \mathrm{\Phi }\in C^{k+1,\alpha }(O)$ for every $O⋐W$, so it belongs to $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k+1,\alpha }(W)$.

Since $h=f\circ {\mathrm{\Phi }}^{-1}$ as an a.e. class and $H=h$ almost everywhere on $Y$, composition by the C${}^1$ diffeomorphism $\mathrm{\Phi }$ preserves this equality by [F4]. Hence $f=H\circ \mathrm{\Phi }$ almost everywhere on $W$. Steps 4.1 and 5.1 give the asserted holomorphic factor and the local $C^{k+1,\alpha }$ representative. The argument used only the nondegeneracy of $\mathrm{\Phi }$ and never divided by $f_z$ or assumed $f$ injective.