[proofplan] We prove uniform approximation on an arbitrary compact set $K \subset \Omega_0$. First we choose a cutoff $\chi$ that equals $1$ near the $\mathcal{O}(\Omega_1)$-hull of $K$, so that $\bar\partial(\chi f)$ is supported away from that hull. Holomorphic separation of this support from $K$ produces a plurisubharmonic weight that is very negative near $K$ and non-negative on the support of the $\bar\partial$-error. Hörmander's weighted $L^2$ theorem for $\bar\partial$ then solves away the error with exponentially small $L^2$ norm near $K$, and the submean estimate converts this local $L^2$ smallness into uniform approximation on $K$. [/proofplan]

[step:Choose a cutoff whose $\bar\partial$-error is supported away from the hull]Fix $f \in \mathcal{O}(\Omega_0)$, a compact set $K \subset \Omega_0$, and $\varepsilon > 0$. If $K=\varnothing$, there is nothing to prove, so assume $K \neq \varnothing$. Define the $\mathcal{O}(\Omega_1)$-hull of $K$ by \begin{align*} H := \widehat{K}_{\Omega_1} = \left\{ z \in \Omega_1 : |g(z)| \leq \sup_{\zeta \in K} |g(\zeta)| \text{ for every } g \in \mathcal{O}(\Omega_1) \right\}. \end{align*} By hypothesis, $H$ is compact and $H \subset \Omega_0$. Choose an [open set](/page/Open%20Set) $U \subset \mathbb{C}^n$ such that \begin{align*} H \subset U \Subset \Omega_0. \end{align*} Choose $\chi \in C_c^\infty(\Omega_0)$ such that $\chi=1$ on a neighbourhood of $\overline{U}$. Define the smooth compactly supported function \begin{align*} h:\Omega_1 &\to \mathbb{C} \\ z &\mapsto \begin{cases} \chi(z)f(z), & z \in \Omega_0,\\ 0, & z \in \Omega_1 \setminus \operatorname{supp}\chi. \end{cases} \end{align*} This definition is smooth because $\operatorname{supp}\chi \Subset \Omega_0$. Define the smooth compactly supported $(0,1)$-form \begin{align*} \alpha := \bar\partial h = \sum_{j=1}^n \alpha_j\,d\bar z_j, \qquad \alpha_j:\Omega_1 \to \mathbb{C}, \qquad \alpha_j=\frac{\partial h}{\partial \bar z_j}. \end{align*} Since $f$ is holomorphic on $\Omega_0$ and $\chi=1$ on a neighbourhood of $\overline{U}$, we have $\alpha=0$ on $U$. Therefore the compact set \begin{align*} S:=\operatorname{supp}\alpha \end{align*} satisfies \begin{align*} S \subset \Omega_1 \setminus H. \end{align*} Moreover $\bar\partial\alpha=\bar\partial^2 h=0$. If $S=\varnothing$, then $\alpha=0$ on $\Omega_1$, so $\bar\partial h=0$ on $\Omega_1$. Since $h$ is smooth, $h \in \mathcal{O}(\Omega_1)$. Because $h=f$ on $U$ and $K \subset U$, we have \begin{align*} \sup_{z \in K}|h(z)-f(z)|=0<\varepsilon, \end{align*} and the desired approximation follows. Hence, for the rest of the proof, assume $S\neq\varnothing$.[/step]

[guided]The approximation will come from correcting the non-[holomorphic function](/page/Holomorphic%20Function) $\chi f$. We choose $\chi$ so that $\chi f$ agrees with $f$ near $K$, but has compact support inside $\Omega_0$ and can therefore be treated as a smooth function on the larger domain $\Omega_1$. The hull \begin{align*} H := \widehat{K}_{\Omega_1} = \left\{ z \in \Omega_1 : |g(z)| \leq \sup_{\zeta \in K} |g(\zeta)| \text{ for every } g \in \mathcal{O}(\Omega_1) \right\} \end{align*} is compactly contained in $\Omega_0$ by hypothesis. Hence we may choose an [open set](/page/Open%20Set) $U$ with \begin{align*} H \subset U \Subset \Omega_0. \end{align*} We then choose $\chi \in C_c^\infty(\Omega_0)$ with $\chi=1$ on a neighbourhood of $\overline{U}$. Now define \begin{align*} h:\Omega_1 &\to \mathbb{C} \\ z &\mapsto \begin{cases} \chi(z)f(z), & z \in \Omega_0,\\ 0, & z \in \Omega_1 \setminus \operatorname{supp}\chi. \end{cases} \end{align*} Because $\operatorname{supp}\chi$ is compactly contained in $\Omega_0$, the zero extension is smooth across $\partial\Omega_0$. Define \begin{align*} \alpha := \bar\partial h = \sum_{j=1}^n \alpha_j\,d\bar z_j, \qquad \alpha_j:\Omega_1 \to \mathbb{C}, \qquad \alpha_j=\frac{\partial h}{\partial \bar z_j}. \end{align*} On the set where $\chi=1$, we have $h=f$, and $f$ is holomorphic. Therefore $\bar\partial h=0$ on a neighbourhood of $\overline U$. Thus the support \begin{align*} S:=\operatorname{supp}\alpha \end{align*} is compact and lies outside $H$. This separation from the hull is the point of placing the cutoff around $H$, not merely around $K$. There is one degenerate case to remove before constructing separating functions. If $S=\varnothing$, then $\alpha=0$ on $\Omega_1$, so $\bar\partial h=0$ on $\Omega_1$. Since $h$ is smooth, this means $h$ is holomorphic on $\Omega_1$. Also $h=f$ on $U$, and $K\subset U$, so \begin{align*} \sup_{z \in K}|h(z)-f(z)|=0<\varepsilon. \end{align*} Thus the approximation is exact in this case. We therefore continue only in the case $S\neq\varnothing$.[/guided]

[step:Separate the support of the error from $K$ by holomorphic functions]For each point $p \in S$, the inclusion $p \notin H$ gives a function $a_p \in \mathcal{O}(\Omega_1)$ such that \begin{align*} |a_p(p)| > \sup_{\zeta \in K}|a_p(\zeta)|. \end{align*} Since $|a_p(p)|>0$, define the normalized [holomorphic function](/page/Holomorphic%20Function) \begin{align*} b_p:\Omega_1 &\to \mathbb{C} \\ z &\mapsto \frac{a_p(z)}{a_p(p)}. \end{align*} Then $|b_p(p)|=1$ and \begin{align*} r_p:=\sup_{\zeta \in K}|b_p(\zeta)|<1. \end{align*} Choose $q_p \in \mathbb{N}$ such that $r_p^{q_p}<e^{-3}$. Choose a real number $c_p>1$ satisfying \begin{align*} c_p r_p^{q_p}<e^{-3}, \end{align*} which is possible because $r_p^{q_p}<e^{-3}$. Define \begin{align*} g_p:\Omega_1 &\to \mathbb{C} \\ z &\mapsto c_p b_p(z)^{q_p}. \end{align*} Then $g_p \in \mathcal{O}(\Omega_1)$ and \begin{align*} \sup_{\zeta \in K}|g_p(\zeta)| < e^{-3}, \qquad |g_p(p)| = c_p > 1. \end{align*} By continuity, there is an open neighbourhood $V_p \subset \Omega_1$ of $p$ such that \begin{align*} |g_p(z)|>1 \qquad \text{for every } z \in V_p. \end{align*} Since $S$ is compact, choose points $p_1,\dots,p_m \in S$ such that \begin{align*} S \subset \bigcup_{\ell=1}^m V_{p_\ell}. \end{align*} Define \begin{align*} \psi:\Omega_1 &\to [-\infty,\infty)\\ z &\mapsto \max_{1 \leq \ell \leq m}\log |g_{p_\ell}(z)|. \end{align*} A finite maximum of plurisubharmonic functions is plurisubharmonic, so $\psi$ is plurisubharmonic on $\Omega_1$. The construction gives \begin{align*} \psi(z) \geq 0 \qquad \text{for every } z \in S. \end{align*} Also, since $\sup_K |g_{p_\ell}|<e^{-3}$ for every $\ell$, continuity and compactness of $K$ give an [open set](/page/Open%20Set) $W \subset U$ with $K \subset W \Subset U$ such that \begin{align*} \psi(z) \leq -2 \qquad \text{for every } z \in W. \end{align*}[/step]

[guided]We now convert the hull hypothesis into a plurisubharmonic weight. Because $S$ is disjoint from the hull $H$, every point of $S$ can be separated from $K$ by a [holomorphic function](/page/Holomorphic%20Function) on $\Omega_1$. Fix $p \in S$. Since $p \notin H$, the definition of $H$ gives some $a_p \in \mathcal{O}(\Omega_1)$ with \begin{align*} |a_p(p)| > \sup_{\zeta \in K}|a_p(\zeta)|. \end{align*} A scalar multiple alone would preserve the ratio between the two sides, so we first amplify the separation by taking a power. Since $|a_p(p)|>0$, define \begin{align*} b_p:\Omega_1 &\to \mathbb{C} \\ z &\mapsto \frac{a_p(z)}{a_p(p)}. \end{align*} Then $|b_p(p)|=1$ and \begin{align*} r_p:=\sup_{\zeta \in K}|b_p(\zeta)|<1. \end{align*} Choose $q_p \in \mathbb{N}$ such that $r_p^{q_p}<e^{-3}$. Because this inequality is strict, choose $c_p>1$ with \begin{align*} c_p r_p^{q_p}<e^{-3}. \end{align*} Now define \begin{align*} g_p:\Omega_1 &\to \mathbb{C} \\ z &\mapsto c_p b_p(z)^{q_p}. \end{align*} Then $g_p$ is holomorphic on $\Omega_1$ and satisfies \begin{align*} \sup_{\zeta \in K}|g_p(\zeta)| < e^{-3}, \qquad |g_p(p)| = c_p > 1. \end{align*} Continuity of $g_p$ gives an open neighbourhood $V_p$ of $p$ on which $|g_p|>1$. The compactness of $S$ lets us pass from pointwise separation to finitely many holomorphic functions. Choose $p_1,\dots,p_m \in S$ such that \begin{align*} S \subset \bigcup_{\ell=1}^m V_{p_\ell}. \end{align*} Define \begin{align*} \psi:\Omega_1 &\to [-\infty,\infty)\\ z &\mapsto \max_{1 \leq \ell \leq m}\log |g_{p_\ell}(z)|. \end{align*} Each function $\log|g_{p_\ell}|$ is plurisubharmonic, and a finite maximum of plurisubharmonic functions is plurisubharmonic. Thus $\psi$ is plurisubharmonic. The inequalities built into the construction have opposite signs on the two regions we care about. On $S$, at least one of the functions satisfies $|g_{p_\ell}(z)|>1$, so \begin{align*} \psi(z) \geq 0 \qquad \text{for every } z \in S. \end{align*} On $K$, all the functions satisfy $|g_{p_\ell}|<e^{-3}$. By continuity and compactness of $K$, this strict inequality persists on some open neighbourhood $W$ of $K$. Shrinking $W$ if necessary, we may assume $W \Subset U$, and then \begin{align*} \psi(z) \leq -2 \qquad \text{for every } z \in W. \end{align*} This sign gap is the mechanism that will force the $\bar\partial$ solution to be small near $K$.[/guided]

[step:Solve the $\bar\partial$ equation with exponentially weighted control]Let $\mathcal{L}^{2n}$ denote Lebesgue measure on $\mathbb{C}^n\cong\mathbb{R}^{2n}$. For $N \in \mathbb{N}$, define the plurisubharmonic weight \begin{align*} \Phi_N:\Omega_1 &\to [-\infty,\infty)\\ z &\mapsto N\psi(z)+|z|^2. \end{align*} The term $|z|^2$ has Levi form equal to the standard Hermitian metric on $\mathbb{C}^n$, so the Levi form of $\Phi_N$ is bounded below by that metric in the sense of currents. Since $\psi$ is a finite maximum of functions of the form $\log|g|$, the weight $\Phi_N$ is an extended-valued plurisubharmonic function and may take the value $-\infty$. By the singular-weight form of the Hörmander $L^2$ existence theorem for $\bar\partial$ on pseudoconvex domains, applied on $\Omega_1$ to the $\bar\partial$-closed compactly supported $(0,1)$-form $\alpha$ and the extended-valued plurisubharmonic weight $\Phi_N$, there exists a function \begin{align*} u_N \in L^2_{\mathrm{loc}}(\Omega_1) \end{align*} such that \begin{align*} \bar\partial u_N=\alpha \end{align*} in the sense of distributions and \begin{align*} \int_{\Omega_1}|u_N(z)|^2 e^{-\Phi_N(z)}\,d\mathcal{L}^{2n}(z) \leq \int_{\Omega_1}|\alpha(z)|^2 e^{-\Phi_N(z)}\,d\mathcal{L}^{2n}(z). \end{align*} Here \begin{align*} |\alpha(z)|^2:=\sum_{j=1}^n|\alpha_j(z)|^2. \end{align*} (external citation needed: singular-weight Hörmander $L^2$ existence theorem for $\bar\partial$ on pseudoconvex domains) Since $\operatorname{supp}\alpha=S$ and $\psi \geq 0$ on $S$, we have \begin{align*} \int_{\Omega_1}|\alpha(z)|^2 e^{-\Phi_N(z)}\,d\mathcal{L}^{2n}(z) = \int_S|\alpha(z)|^2 e^{-N\psi(z)-|z|^2}\,d\mathcal{L}^{2n}(z) \leq A, \end{align*} where the finite constant $A$ is defined by \begin{align*} A:=\int_S|\alpha(z)|^2\,d\mathcal{L}^{2n}(z). \end{align*} Let \begin{align*} R:=\sup_{z \in W}|z|. \end{align*} Since $\psi \leq -2$ on $W$, we have for $z \in W$ \begin{align*} e^{-\Phi_N(z)} = e^{-N\psi(z)-|z|^2} \geq e^{2N-R^2}. \end{align*} Therefore \begin{align*} e^{2N-R^2} \int_W |u_N(z)|^2\,d\mathcal{L}^{2n}(z) \leq \int_W |u_N(z)|^2 e^{-\Phi_N(z)}\,d\mathcal{L}^{2n}(z) \leq A, \end{align*} and hence \begin{align*} \int_W |u_N(z)|^2\,d\mathcal{L}^{2n}(z) \leq A e^{-2N+R^2}. \end{align*}[/step]

[guided]We now use the weight to solve the equation $\bar\partial u_N=\alpha$ in such a way that $u_N$ is small near $K$. Let $\mathcal{L}^{2n}$ denote Lebesgue measure on $\mathbb{C}^n\cong\mathbb{R}^{2n}$. For each $N \in \mathbb{N}$, define \begin{align*} \Phi_N:\Omega_1 &\to [-\infty,\infty)\\ z &\mapsto N\psi(z)+|z|^2. \end{align*} The function $\psi$ is plurisubharmonic, and $|z|^2$ is strictly plurisubharmonic with Levi form equal to the standard Hermitian metric. Because $\psi$ is a finite maximum of functions $\log|g_{p_\ell}|$, it may take the value $-\infty$ at common zeros of the $g_{p_\ell}$. Thus $\Phi_N$ is an extended-valued plurisubharmonic Hörmander weight, and the version of the $L^2$ theorem used below is the singular-weight version. We apply the singular-weight Hörmander $L^2$ existence theorem for $\bar\partial$ on the pseudoconvex domain $\Omega_1$. The hypotheses are satisfied: $\Omega_1$ is pseudoconvex by assumption, $\Phi_N$ is an extended-valued plurisubharmonic weight whose Levi form is bounded below by the standard Hermitian metric in the sense of currents, $\alpha$ is compactly supported, and $\bar\partial\alpha=0$. Hence there exists \begin{align*} u_N \in L^2_{\mathrm{loc}}(\Omega_1) \end{align*} such that \begin{align*} \bar\partial u_N=\alpha \end{align*} in the sense of distributions and \begin{align*} \int_{\Omega_1}|u_N(z)|^2 e^{-\Phi_N(z)}\,d\mathcal{L}^{2n}(z) \leq \int_{\Omega_1}|\alpha(z)|^2 e^{-\Phi_N(z)}\,d\mathcal{L}^{2n}(z), \end{align*} where \begin{align*} |\alpha(z)|^2:=\sum_{j=1}^n|\alpha_j(z)|^2. \end{align*} (external citation needed: singular-weight Hörmander $L^2$ existence theorem for $\bar\partial$ on pseudoconvex domains) The right-hand side is uniformly bounded in $N$. Indeed, $\alpha$ is supported on $S$, and on $S$ we constructed $\psi \geq 0$. Therefore \begin{align*} \int_{\Omega_1}|\alpha(z)|^2 e^{-\Phi_N(z)}\,d\mathcal{L}^{2n}(z) = \int_S|\alpha(z)|^2 e^{-N\psi(z)-|z|^2}\,d\mathcal{L}^{2n}(z) \leq \int_S|\alpha(z)|^2\,d\mathcal{L}^{2n}(z). \end{align*} Define \begin{align*} A:=\int_S|\alpha(z)|^2\,d\mathcal{L}^{2n}(z). \end{align*} This constant is finite because $\alpha$ is smooth and $S$ is compact. Near $K$, the same weight has the opposite effect. Let \begin{align*} R:=\sup_{z \in W}|z|. \end{align*} Since $W$ is relatively compact, $R<\infty$. Since $\psi \leq -2$ on $W$, for $z \in W$ we get \begin{align*} e^{-\Phi_N(z)} = e^{-N\psi(z)-|z|^2} \geq e^{2N-R^2}. \end{align*} Thus the weighted estimate forces the unweighted $L^2$ norm of $u_N$ on $W$ to decay exponentially: \begin{align*} e^{2N-R^2} \int_W |u_N(z)|^2\,d\mathcal{L}^{2n}(z) \leq \int_W |u_N(z)|^2 e^{-\Phi_N(z)}\,d\mathcal{L}^{2n}(z) \leq A, \end{align*} so \begin{align*} \int_W |u_N(z)|^2\,d\mathcal{L}^{2n}(z) \leq A e^{-2N+R^2}. \end{align*}[/guided]

[step:Convert the local $L^2$ estimate into uniform smallness on $K$]Define the distribution $G_N$ on $\Omega_1$ by \begin{align*} G_N:=h-u_N. \end{align*} Since $\bar\partial h=\alpha$ and $\bar\partial u_N=\alpha$, we have \begin{align*} \bar\partial G_N=0 \end{align*} in the sense of distributions. By the Weyl lemma for the $\bar\partial$ operator, $G_N$ is represented by a [holomorphic function](/page/Holomorphic%20Function). We choose this representative and denote it by $F_N \in \mathcal{O}(\Omega_1)$. (external citation needed: Weyl lemma for the $\bar\partial$ operator) Because $\chi=1$ on $W$, we have $h=f$ on $W$. The equality $G_N=h-u_N$ therefore gives \begin{align*} F_N=f-u_N \end{align*} almost everywhere on $W$. Redefine $u_N$ on $W$ by the [holomorphic function](/page/Holomorphic%20Function) $f-F_N$, leaving its $L^2$ equivalence class unchanged. With this representative, for every $z \in W$ we have \begin{align*} F_N(z)-f(z)=-u_N(z). \end{align*} Let \begin{align*} \rho:=\frac{1}{2}\operatorname{dist}(K,\mathbb{C}^n\setminus W). \end{align*} Since $K \subset W$ and $K$ is compact, $\rho>0$. For every $z \in K$, the Euclidean ball $B(z,\rho)$ satisfies $B(z,\rho)\subset W$. Since $u_N=f-F_N$ is holomorphic on $W$, the submean inequality applied to the plurisubharmonic function $|u_N|^2$ gives \begin{align*} |u_N(z)|^2 \leq \frac{1}{\mathcal{L}^{2n}(B(z,\rho))} \int_{B(z,\rho)} |u_N(\zeta)|^2\,d\mathcal{L}^{2n}(\zeta). \end{align*} Let \begin{align*} \beta_{2n}:=\mathcal{L}^{2n}(B(0,1)). \end{align*} Then $\mathcal{L}^{2n}(B(z,\rho))=\beta_{2n}\rho^{2n}$, and because $B(z,\rho)\subset W$, \begin{align*} |u_N(z)|^2 \leq \frac{1}{\beta_{2n}\rho^{2n}} \int_W |u_N(\zeta)|^2\,d\mathcal{L}^{2n}(\zeta) \leq \frac{A}{\beta_{2n}\rho^{2n}}e^{-2N+R^2}. \end{align*} Choose $N$ large enough that \begin{align*} \left(\frac{A}{\beta_{2n}\rho^{2n}}\right)^{1/2}e^{-N+R^2/2}<\varepsilon. \end{align*} Then for every $z \in K$, \begin{align*} |F_N(z)-f(z)|=|u_N(z)|<\varepsilon. \end{align*} Thus \begin{align*} \sup_{z \in K}|F_N(z)-f(z)|<\varepsilon. \end{align*}[/step]

[guided]The solution $u_N$ removes exactly the $\bar\partial$-error of $h$, but $u_N$ is initially only an $L^2_{\mathrm{loc}}$ equivalence class. Define the distribution \begin{align*} G_N:=h-u_N \end{align*} on $\Omega_1$. Then, in the distributional sense, \begin{align*} \bar\partial G_N = \bar\partial h-\bar\partial u_N = \alpha-\alpha = 0. \end{align*} By the Weyl lemma for the $\bar\partial$ operator, $G_N$ is represented by a [holomorphic function](/page/Holomorphic%20Function) on $\Omega_1$. We choose this representative and call it $F_N \in \mathcal{O}(\Omega_1)$. (external citation needed: Weyl lemma for the $\bar\partial$ operator) On the neighbourhood $W$ of $K$, the cutoff is identically $1$, so $h=f$. Hence the distributional identity $G_N=h-u_N$ gives \begin{align*} F_N=f-u_N \end{align*} almost everywhere on $W$. To make pointwise estimates legitimate, we now choose the representative of $u_N$ on $W$ to be the [holomorphic function](/page/Holomorphic%20Function) $f-F_N$. This change is only on a null set and preserves the previous $L^2$ estimates. With this representative, \begin{align*} F_N(z)-f(z) = -u_N(z) \qquad \text{for every } z \in W. \end{align*} Thus it remains only to turn the $L^2$ smallness of $u_N$ on $W$ into pointwise smallness on $K$. Define \begin{align*} \rho:=\frac{1}{2}\operatorname{dist}(K,\mathbb{C}^n\setminus W). \end{align*} Because $K$ is compact and $K \subset W$ with $W$ open, $\rho>0$. For every $z \in K$, the ball $B(z,\rho)$ is contained in $W$. Since $u_N=f-F_N$ on $W$ and both $f$ and $F_N$ are holomorphic there, $u_N$ is holomorphic on $W$. Hence $|u_N|^2$ is plurisubharmonic, and the submean inequality gives \begin{align*} |u_N(z)|^2 \leq \frac{1}{\mathcal{L}^{2n}(B(z,\rho))} \int_{B(z,\rho)} |u_N(\zeta)|^2\,d\mathcal{L}^{2n}(\zeta). \end{align*} Let \begin{align*} \beta_{2n}:=\mathcal{L}^{2n}(B(0,1)). \end{align*} Then \begin{align*} \mathcal{L}^{2n}(B(z,\rho))=\beta_{2n}\rho^{2n}. \end{align*} Since $B(z,\rho)\subset W$, the previous $L^2$ estimate yields \begin{align*} |u_N(z)|^2 \leq \frac{1}{\beta_{2n}\rho^{2n}} \int_W |u_N(\zeta)|^2\,d\mathcal{L}^{2n}(\zeta) \leq \frac{A}{\beta_{2n}\rho^{2n}}e^{-2N+R^2}. \end{align*} The right-hand side tends to $0$ as $N \to \infty$. Choose $N$ so large that \begin{align*} \left(\frac{A}{\beta_{2n}\rho^{2n}}\right)^{1/2}e^{-N+R^2/2}<\varepsilon. \end{align*} Then for every $z \in K$, \begin{align*} |F_N(z)-f(z)| = |u_N(z)| < \varepsilon. \end{align*} This is precisely the required uniform approximation on $K$ by a function holomorphic on $\Omega_1$.[/guided]

[step:Conclude density of restrictions from $\Omega_1$ to $\Omega_0$] We have shown that for an arbitrary $f \in \mathcal{O}(\Omega_0)$, compact set $K \subset \Omega_0$, and $\varepsilon>0$, there exists $F_N \in \mathcal{O}(\Omega_1)$ such that \begin{align*} \sup_{z \in K}|F_N(z)-f(z)|<\varepsilon. \end{align*} Therefore the restrictions of functions in $\mathcal{O}(\Omega_1)$ are dense in $\mathcal{O}(\Omega_0)$ for the compact-open topology. Hence $(\Omega_0,\Omega_1)$ is a Runge pair. [/step]