[proofplan] The proof is a standard reduction from the global estimate to local boundary estimates. Interior pieces are controlled by elliptic regularity for the complex Laplacian, while boundary pieces are flattened, decomposed microlocally into positive, negative, and elliptic tangential frequency cones, and estimated by the Morrey-Kohn-Hörmander identity. Strong pseudoconvexity gives a uniform positive Levi lower bound on the positive microlocal cone; the negative cone is reduced to the positive cone by the adjoint symmetry in degree $q$, and the elliptic cone is handled by tangential ellipticity. A finite [partition of unity](/page/Partition%20of%20Unity) then patches the local estimates and absorbs commutator errors into the $L^2$ term. [/proofplan]

[step:State the quadratic form and the target norm] Let $\Lambda^{0,q}$ denote the complex vector bundle of $(0,q)$-covectors over $\overline{\Omega}$. Choose a smooth defining function $\rho:\mathbb{C}^n\to\mathbb{R}$ for $\Omega$, so that $\Omega=\{\rho<0\}$ near $\partial\Omega$ and $d\rho\ne0$ on $\partial\Omega$. Let \begin{align*} \bar{\partial}_q:C^\infty(\overline{\Omega};\Lambda^{0,q})&\to C^\infty(\overline{\Omega};\Lambda^{0,q+1}) \end{align*} denote the Cauchy-Riemann differential on $(0,q)$-forms, viewed as an unbounded operator on $L^2(\Omega;\Lambda^{0,q})$ with respect to $\mathcal{L}^{2n}$. Let \begin{align*} \bar{\partial}_q^*:\operatorname{Dom}(\bar{\partial}_q^*)\subset L^2(\Omega;\Lambda^{0,q+1})&\to L^2(\Omega;\Lambda^{0,q}) \end{align*} denote its Hilbert-space adjoint, and write $\bar{\partial}^*$ for the adjoint in the degree determined by the form under discussion. The [$\bar{\partial}$-Neumann boundary condition](/page/Dbar-Neumann%20Boundary%20Condition) for a smooth $(0,q)$-form $u$ is $u\lrcorner\bar{\partial}\rho=0$ on $\partial\Omega$, equivalently $u\in\operatorname{Dom}(\bar{\partial}^*)$. Define the smooth form space with Neumann boundary condition \begin{align*} \mathcal{D}_q := C^\infty(\overline{\Omega};\Lambda^{0,q}) \cap \operatorname{Dom}(\bar{\partial}^*). \end{align*} For $u \in \mathcal{D}_q$, define the $\bar{\partial}$-Neumann energy $Q_q: \mathcal{D}_q \to [0,\infty)$ by \begin{align*} Q_q(u) := \|\bar{\partial}u\|_{L^2(\Omega)}^2 + \|\bar{\partial}^*u\|_{L^2(\Omega)}^2 + \|u\|_{L^2(\Omega)}^2. \end{align*} Here $L^2(\Omega)$ is taken with respect to $2n$-dimensional Lebesgue measure $\mathcal{L}^{2n}$ after identifying $\mathbb{C}^n$ with $\mathbb{R}^{2n}$. The desired estimate is equivalent to finding $C_q>0$ such that \begin{align*} \|u\|_{H^{1/2}(\Omega)}^2 \le C_q Q_q(u) \end{align*} for every $u \in \mathcal{D}_q$. [/step]

[step:Control the interior part by elliptic regularity] Choose an [open set](/page/Open%20Set) $V_0 \subset \Omega$ with $\overline{V_0} \subset \Omega$ and a cutoff function $\zeta_0: \Omega \to [0,1]$ satisfying $\zeta_0 \in C_c^\infty(V_0)$. Since $\zeta_0 u$ has compact support in the interior of $\Omega$, the boundary condition in $\operatorname{Dom}(\bar{\partial}^*)$ is irrelevant on $\operatorname{supp}\zeta_0$. The interior elliptic estimate for the complex Laplacian $\Box_q=\bar{\partial}\bar{\partial}^*+\bar{\partial}^*\bar{\partial}$ first gives a constant $C_0'>0$, depending only on $\zeta_0$ and $\Omega$, such that \begin{align*} \|\zeta_0 u\|_{H^1(\Omega)}^2 \le C_0'\left(\|\bar{\partial}(\zeta_0u)\|_{L^2(\Omega)}^2+\|\bar{\partial}^*(\zeta_0u)\|_{L^2(\Omega)}^2+\|\zeta_0u\|_{L^2(\Omega)}^2\right). \end{align*} The cutoff commutators are zero-order operators: $\bar{\partial}(\zeta_0u)=\zeta_0\bar{\partial}u+(\bar{\partial}\zeta_0)\wedge u$, and $\bar{\partial}^*(\zeta_0u)=\zeta_0\bar{\partial}^*u+R_0u$ for a smooth bundle map $R_0$ with bounded coefficients depending only on $\zeta_0$. Hence, after increasing the constant to $C_0>0$, \begin{align*} \|\zeta_0 u\|_{H^1(\Omega)}^2 \le C_0\left(\|\bar{\partial}u\|_{L^2(\Omega)}^2+\|\bar{\partial}^*u\|_{L^2(\Omega)}^2+\|u\|_{L^2(\Omega)}^2\right)=C_0Q_q(u). \end{align*} Since the continuous embedding $H^1(\Omega)\hookrightarrow H^{1/2}(\Omega)$ has norm $E_0>0$, it follows that \begin{align*} \|\zeta_0 u\|_{H^{1/2}(\Omega)}^2 \le E_0^2C_0Q_q(u). \end{align*} [/step]

[step:Flatten the boundary and invoke the precise local Kohn estimate]Let $p\in \partial\Omega$. Use the fixed defining function $\rho:\mathbb{C}^n\to\mathbb{R}$ from the first step. Since $\partial\Omega$ is $C^\infty$ and $d\rho_p\ne0$, choose a boundary chart $\Phi_p:W_p\to U_p\subset\mathbb{R}^{2n}$ with $\Phi_p(p)=0$ such that \begin{align*} \Phi_p(W_p\cap\Omega)=U_p\cap\{x_{2n}<0\}, \qquad \Phi_p(W_p\cap\partial\Omega)=U_p\cap\{x_{2n}=0\}. \end{align*} The flattening coordinate $x_{2n}$ is used only to separate normal and tangential variables; the [Levi form](/page/Levi%20Form) is always the Hermitian form $\mathcal{L}_\rho$ determined by the defining function $\rho$ and the ambient complex structure on $\mathbb{C}^n$. Let $\zeta_p:\Omega\to[0,1]$ be a function in $C_c^\infty(W_p)$, and let $v_p$ denote the coefficient vector of the pulled-back $(0,q)$-form $(\Phi_p^{-1})^*(\zeta_pu)$ in a smooth unitary coframe adapted to $\bar{\partial}\rho$ along the boundary. We use the [Local Kohn Subelliptic Estimate for Strongly Pseudoconvex Boundaries](/page/Local%20Kohn%20Subelliptic%20Estimate%20for%20Strongly%20Pseudoconvex%20Boundaries). In the notation of the present chart and defining function, this result states the following. If the boundary coefficients are $C^\infty$, the coefficient vector satisfies the standard tangential $\bar{\partial}$-Neumann boundary condition, and there is $\lambda_p>0$ such that \begin{align*} \mathcal{L}_\rho(z;\xi)\ge \lambda_p|\xi|^2 \end{align*} for every $z\in W_p'\cap\partial\Omega$ and every $\xi\in T_z^{1,0}(\partial\Omega)$, then for every degree $1\le q\le n$ there are a neighbourhood $W_p'\subset W_p$ of $p$ and a constant $C_p>0$ such that every $u\in\mathcal{D}_q$ with $\operatorname{supp}u\subset W_p'$ satisfies \begin{align*} \|u\|_{H^{1/2}(\Omega)}^2 \le C_pQ_q(u). \end{align*} The $H^{1/2}$ norm here is the restriction norm induced from $H^{1/2}(\mathbb{R}^{2n})$ after extension by zero in the chart; changing charts only changes the estimate by a finite norm-equivalence constant because the transition maps are $C^\infty$ diffeomorphisms on a finite-dimensional boundary atlas. The hypotheses of the cited local estimate are satisfied. The coefficients in the flattened boundary chart are $C^\infty$ because $\partial\Omega$ is smoothly bounded and the adapted unitary coframe is chosen smoothly after shrinking $W_p$ if necessary. The condition $u\in\operatorname{Dom}(\bar{\partial}^*)$ means $u\lrcorner\bar{\partial}\rho=0$ on $\partial\Omega$; in the adapted coframe this is exactly the vanishing of the normal coefficient components, which is the standard tangential Neumann boundary condition used in the local theorem. Strong pseudoconvexity means $\mathcal{L}_\rho(z;\xi)>0$ for every $z\in\partial\Omega$ and every nonzero $\xi\in T_z^{1,0}(\partial\Omega)$. By continuity of $\mathcal{L}_\rho$ and compactness of the unit complex tangent sphere over a sufficiently small boundary neighbourhood, shrink $W_p$ to $W_p'$ and choose $\lambda_p>0$ so that the displayed lower bound holds on $W_p'\cap\partial\Omega$. For clarity, we record what is contained inside the cited theorem rather than using it as an additional unproved step. Let $T^*(\partial\Omega)$ denote the tangential cotangent bundle, and let $\tau\in T_z^*(\partial\Omega)$ denote a tangential covector at $z\in\partial\Omega$. In $T^*(\partial\Omega)$, the proof of the local theorem decomposes high tangential frequencies into positive, negative, and elliptic cones by order-zero tangential pseudodifferential operators \begin{align*} \Psi_p^+,\Psi_p^-,\Psi_p^0:C_c^\infty(W_p'\cap\partial\Omega;\Lambda^{0,q})&\to C^\infty(W_p'\cap\partial\Omega;\Lambda^{0,q}), \end{align*} whose scalar principal symbols $\psi_p^+(z,\tau)$, $\psi_p^-(z,\tau)$, and $\psi_p^0(z,\tau)$ form a smooth [partition of unity](/page/Partition%20of%20Unity) for $|\tau|\ge1$. The positive cone estimate is obtained from the [Morrey-Kohn-Hörmander Identity](/page/Morrey-Kohn-H%C3%B6rmander%20Identity) and the Levi lower bound. For $1\le q<n$, the negative cone is treated inside the local theorem by the conjugate Hodge-star coefficient map $\mathcal{J}_q:\Lambda^{0,q}\to\Lambda^{0,n-q}$, which sends each coefficient indexed by a multi-index $J$ to the signed conjugate coefficient indexed by the complementary multi-index and preserves the adapted boundary condition. For the endpoint $q=n$, the local theorem uses the top-degree Morrey-Kohn-Hörmander identity directly: there are no $(0,n+1)$ components, and the boundary contribution is controlled through $\bar{\partial}^*$ and the positive Levi trace term rather than by reduction to degree $0$. The elliptic cone is handled by the [Tangential Elliptic Estimate](/page/Tangential%20Elliptic%20Estimate), because the tangential principal symbol is elliptic on $\operatorname{supp}\Psi_p^0$. Commutators between the cutoffs, the chart coefficients, and $\bar{\partial}$ or $\bar{\partial}^*$ are tangential pseudodifferential operators of order $0$, so their $L^2$ contributions are bounded by a constant multiple of $\|u\|_{L^2(\Omega)}^2$ and are included in $Q_q(u)$.[/step]

[guided]Fix $p\in\partial\Omega$. The local problem must be put into a coordinate system where the normal and tangential directions are separated, but the Levi form must still be computed from a genuine defining function for the original complex domain. Use the smooth defining function $\rho:\mathbb{C}^n\to\mathbb{R}$ fixed in the first step. Since $\partial\Omega$ is $C^\infty$ and $d\rho_p\ne0$, choose a boundary chart $\Phi_p:W_p\to U_p\subset\mathbb{R}^{2n}$ with $\Phi_p(p)=0$ such that \begin{align*} \Phi_p(W_p\cap\Omega)=U_p\cap\{x_{2n}<0\}, \qquad \Phi_p(W_p\cap\partial\Omega)=U_p\cap\{x_{2n}=0\}. \end{align*} The coordinate $x_{2n}$ is only a flattened normal coordinate. It is not being used as the defining function for the Levi form. The [Levi form](/page/Levi%20Form) used in the estimate is $\mathcal{L}_\rho$, the Hermitian form associated with $\rho$ and the ambient complex structure of $\mathbb{C}^n$. If $\zeta_p:\Omega\to[0,1]$ is a cutoff in $C_c^\infty(W_p)$, define $v_p$ to be the coefficient vector of the pulled-back form $(\Phi_p^{-1})^*(\zeta_pu)$ in a smooth unitary coframe adapted to $\bar{\partial}\rho$ along the boundary. We now apply the [Local Kohn Subelliptic Estimate for Strongly Pseudoconvex Boundaries](/page/Local%20Kohn%20Subelliptic%20Estimate%20for%20Strongly%20Pseudoconvex%20Boundaries). The theorem requires smooth boundary coefficients in a flattened chart, the standard $\bar{\partial}$-Neumann tangential boundary condition, and a positive lower bound for the Levi form. The first hypothesis holds because the boundary chart and adapted coframe are $C^\infty$ after shrinking $W_p$ if necessary. The second hypothesis is the invariant boundary condition written in local coefficients: for a smooth $(0,q)$-form, $u\in\operatorname{Dom}(\bar{\partial}^*)$ is equivalent to $u\lrcorner\bar{\partial}\rho=0$ on $\partial\Omega$, and in the adapted coframe this says exactly that the normal coefficient components vanish. For the Levi condition, strong pseudoconvexity gives $\mathcal{L}_\rho(z;\xi)>0$ for each nonzero $\xi\in T_z^{1,0}(\partial\Omega)$. Continuity of $\mathcal{L}_\rho$ and compactness of the unit complex tangent sphere over a small boundary neighbourhood allow us to shrink $W_p$ to $W_p'$ and choose $\lambda_p>0$ so that \begin{align*} \mathcal{L}_\rho(z;\xi)\ge \lambda_p|\xi|^2 \end{align*} for every $z\in W_p'\cap\partial\Omega$ and every $\xi\in T_z^{1,0}(\partial\Omega)$. The cited local theorem then gives a constant $C_p>0$ such that every $u\in\mathcal{D}_q$ supported in $W_p'$ satisfies \begin{align*} \|u\|_{H^{1/2}(\Omega)}^2 \le C_pQ_q(u). \end{align*} Here the local $H^{1/2}$ norm is the restriction norm inherited from $H^{1/2}(\mathbb{R}^{2n})$ after extension by zero in the flattened chart, and smooth changes of chart alter this norm only by a fixed equivalence constant. This is why the local estimate can be inserted into the global $H^{1/2}(\Omega)$ estimate after multiplying by cutoffs. The mechanism inside the local theorem is the following. This paragraph records the ingredients of the cited result, so the microlocal terms are not additional hypotheses in the global patching argument. Let $T^*(\partial\Omega)$ denote the tangential cotangent bundle, and let $\tau\in T_z^*(\partial\Omega)$ denote a tangential covector at $z\in\partial\Omega$. The local proof chooses order-zero tangential pseudodifferential operators \begin{align*} \Psi_p^+,\Psi_p^-,\Psi_p^0:C_c^\infty(W_p'\cap\partial\Omega;\Lambda^{0,q})&\to C^\infty(W_p'\cap\partial\Omega;\Lambda^{0,q}), \end{align*} with scalar principal symbols $\psi_p^+(z,\tau)$, $\psi_p^-(z,\tau)$, and $\psi_p^0(z,\tau)$ forming a smooth [partition of unity](/page/Partition%20of%20Unity) on $|\tau|\ge 1$. These cutoffs separate positive, negative, and elliptic tangential frequency cones. The positive cone is controlled by the [Morrey-Kohn-Hörmander Identity](/page/Morrey-Kohn-H%C3%B6rmander%20Identity), whose boundary term contains the Levi form and therefore gains one half tangential derivative from the displayed lower bound. For $1\le q<n$, the negative cone is handled by the conjugate Hodge-star coefficient map $\mathcal{J}_q:\Lambda^{0,q}\to\Lambda^{0,n-q}$, which sends each coefficient indexed by a multi-index $J$ to the signed conjugate coefficient indexed by the complementary multi-index and preserves the adapted Neumann boundary condition. The endpoint $q=n$ is part of the cited local theorem but is not obtained by reducing to degree $0$; instead, the top-degree Morrey-Kohn-Hörmander identity controls the negative-frequency contribution directly through $\bar{\partial}^*$ and the positive Levi trace term. On the support of $\Psi_p^0$, the tangential principal symbol is elliptic, and the [Tangential Elliptic Estimate](/page/Tangential%20Elliptic%20Estimate) gives the same $H^{1/2}$ bound. Finally, commutators such as $[\bar{\partial},\Psi_p^\pm]$ and $[\bar{\partial}^*,\Psi_p^\pm]$ have order $0$, so their contributions are bounded by a constant multiple of $\|u\|_{L^2(\Omega)}^2$ and are absorbed into $Q_q(u)$.[/guided]

[step:Patch the local estimates over the compact boundary] Since $\Omega$ is bounded and $\partial\Omega$ is closed in $\mathbb{C}^n \cong \mathbb{R}^{2n}$, the boundary $\partial\Omega$ is compact. Choose finitely many boundary points $p_1,\dots,p_N\in\partial\Omega$ such that the corresponding neighbourhoods $W_{p_j}'$ cover $\partial\Omega$. Choose functions $\zeta_j:\Omega\to[0,1]$ in $C_c^\infty(W_{p_j}')$ for $1\le j\le N$ and keep the interior cutoff $\zeta_0$ so that \begin{align*} \sum_{j=0}^N \zeta_j^2=1 \end{align*} on a neighbourhood of $\overline{\Omega}$. We use the restriction definition of $H^{1/2}(\Omega)$: a distribution $f$ belongs to $H^{1/2}(\Omega)$ when it is the restriction to $\Omega$ of some $F\in H^{1/2}(\mathbb{R}^{2n})$, with norm the infimum of $\|F\|_{H^{1/2}(\mathbb{R}^{2n})}$ over all such extensions. By the [Multiplier Theorem for Sobolev Spaces](/page/Multiplier%20Theorem%20for%20Sobolev%20Spaces), multiplication by each $\zeta_j\in C_c^\infty$ is a bounded operator $H^{1/2}(\Omega)\to H^{1/2}(\Omega)$; let $M_j>0$ denote its operator norm. We also use the [Localization Inequality for Fractional Sobolev Norms](/page/Localization%20Inequality%20for%20Fractional%20Sobolev%20Norms), applied to the finite smooth partition satisfying $\sum_{j=0}^N\zeta_j^2=1$ on a neighbourhood of $\overline{\Omega}$. It gives \begin{align*} \|u\|_{H^{1/2}(\Omega)}^2 \le C_{\mathrm{part}}\sum_{j=0}^N \|\zeta_j u\|_{H^{1/2}(\Omega)}^2 \end{align*} for a constant $C_{\mathrm{part}}>0$ depending only on the chosen partition and the norm-equivalence constants for the finite family of charts. For each $1\le j\le N$, define the cutoff-energy constant $A_j>0$ as follows. Since $\zeta_j\in C_c^\infty(W_{p_j}')$, multiplication by $\zeta_j$ preserves $\operatorname{Dom}(\bar{\partial}^*)$ and \begin{align*} \bar{\partial}(\zeta_j u)=\zeta_j\bar{\partial}u+(\bar{\partial}\zeta_j)\wedge u. \end{align*} The adjoint commutator is a zero-order operator depending only on $\zeta_j$, so \begin{align*} \bar{\partial}^*(\zeta_j u)=\zeta_j\bar{\partial}^*u+R_j u, \end{align*} where $R_j:C^\infty(\overline{\Omega};\Lambda^{0,q})\to C^\infty(\overline{\Omega};\Lambda^{0,q-1})$ is a smooth bundle map with bounded coefficients. Let \begin{align*} A_j:=3\max\left\{\|\zeta_j\|_{L^\infty(\Omega)}^2,\|\bar{\partial}\zeta_j\|_{L^\infty(\Omega)}^2,\|R_j\|_{L^\infty(\Omega)}^2,1\right\}. \end{align*} Using $(a+b)^2\le 2a^2+2b^2$ and boundedness of the coefficient multipliers gives \begin{align*} Q_q(\zeta_j u)\le A_j Q_q(u). \end{align*} Applying the interior estimate to $\zeta_0u$ and the boundary estimate to each $\zeta_ju$ gives \begin{align*} \|u\|_{H^{1/2}(\Omega)}^2 \le C_{\mathrm{part}}\left(E_0^2C_0Q_q(u)+\sum_{j=1}^N C_{p_j}A_jQ_q(u)\right). \end{align*} Set \begin{align*} K_q:=C_{\mathrm{part}}\left(E_0^2C_0+\sum_{j=1}^N C_{p_j}A_j\right), \qquad C_q:=K_q^{1/2}. \end{align*} Then \begin{align*} \|u\|_{H^{1/2}(\Omega)}^2 \le K_qQ_q(u)=C_q^2Q_q(u). \end{align*} [/step]

[step:Take square roots and identify the subelliptic estimate] Since both sides of the preceding inequality are nonnegative and $C_q=K_q^{1/2}$, taking square roots gives \begin{align*} \|u\|_{H^{1/2}(\Omega)} \le C_q\left(\|\bar{\partial}u\|_{L^2(\Omega)}^2+\|\bar{\partial}^*u\|_{L^2(\Omega)}^2+\|u\|_{L^2(\Omega)}^2\right)^{1/2}. \end{align*} Using the elementary inequality $(a^2+b^2+c^2)^{1/2}\le a+b+c$ for $a,b,c\ge0$, we obtain \begin{align*} \|u\|_{H^{1/2}(\Omega)} \le C_q\left(\|\bar{\partial}u\|_{L^2(\Omega)}+\|\bar{\partial}^*u\|_{L^2(\Omega)}+\|u\|_{L^2(\Omega)}\right). \end{align*} This is exactly the subelliptic estimate of order $1/2$ for the $\bar{\partial}$-Neumann problem in degree $q$. [/step]