[proofplan] On $\Omega':=\Omega\setminus g^{-1}(0)$ the equation $\sum_{j=1}^m g_jh_j=f$ has a smooth pointwise minimal solution $\sigma_j=\overline{g_j}\,f/|g|^2$, which is not holomorphic. We correct it to a holomorphic solution $h=\sigma-u$, where the correction $u$ is a section of the relation bundle $S:=\ker g$ solving $\bar\partial u=\bar\partial\sigma$. The weighted $L^2$ existence theorem for $\bar\partial$ (Bochner–Kodaira–Nakano/Hörmander), applied with the plurisubharmonic weight $\varphi+\beta\log|g|^2$ for $\beta:=\alpha q$, together with Skoda's pointwise curvature inequality, bounds the correction by the factor $q/(\beta-q)=1/(\alpha-1)$; here the rank bound $q=\min\{n,m-1\}$ controls the curvature and the hypothesis $\alpha>1$ keeps $\beta>q$. Since $\sigma\perp S$ pointwise, the Pythagorean identity $|h|^2=|\sigma|^2+|u|^2$ converts this into the sharp constant $1+\tfrac{1}{\alpha-1}=\tfrac{\alpha}{\alpha-1}$. The singularities along $g^{-1}(0)$ and of $\varphi$ are removed by regularising $\log|g|^2$ to $\log(|g|^2+\varepsilon^2)$, exhausting $\Omega$ by smooth pseudoconvex subdomains, and passing to a weak-$L^2$ limit via Banach–Alaoglu, after which $h$ extends holomorphically across the measure-zero analytic set $g^{-1}(0)$. [/proofplan]

[step:Dispose of the degenerate cases and fix the notation $Z$, $\Omega'$, $\beta$ and $I$] If $g\equiv 0$ on a connected component of $\Omega$, then $|g|\equiv 0$ there and the hypothesis forces $f\equiv 0$ on that component (otherwise the integrand $|f|^2e^{-\varphi}|g|^{-2(\alpha q+1)}$ equals $+\infty$ on a set of positive $\mathcal L^{2n}$-measure); there $h:=0$ satisfies both conclusions. We may therefore assume that on every connected component of $\Omega$ some $g_j$ is not identically zero, so that \begin{align*} Z:=\{x\in\Omega:g(x)=0\}=\bigcap_{j=1}^m g_j^{-1}(0) \end{align*} is a proper analytic subset of $\Omega$; hence $\mathcal L^{2n}(Z)=0$. Set $\Omega':=\Omega\setminus Z$, an [open set](/page/Open%20Set) of full measure on which $|g|>0$. Write $q:=\min\{n,m-1\}$ and $\beta:=\alpha q$; since $\alpha>1$ we have $\beta\ge q$, with $\beta>q$ as soon as $q\ge 1$. Denote by \begin{align*} I:=\int_\Omega \frac{|f|^2 e^{-\varphi}}{|g|^{2(\beta+1)}}\,d\mathcal L^{2n}<\infty \end{align*} the finite quantity of the hypothesis (note $2(\beta+1)=2(\alpha q+1)$). Because $\mathcal L^{2n}(Z)=0$, each integral over $\Omega$ below equals the corresponding integral over $\Omega'$. [/step]

[step:Reduce the division problem to a $\bar\partial$-equation valued in the relation bundle $\ker g$]Define the pointwise minimal solution \begin{align*} \sigma:\Omega'&\to\mathbb C^m, & \sigma(x)&=\frac{f(x)}{|g(x)|^2}\big(\overline{g_1(x)},\dots,\overline{g_m(x)}\big). \end{align*} Then $\sigma$ is smooth on $\Omega'$ (the denominator is nonzero there) and \begin{align*} g\cdot\sigma:=\sum_{j=1}^m g_j\sigma_j=\frac{f}{|g|^2}\sum_{j=1}^m|g_j|^2=f,\qquad |\sigma|^2=\frac{|f|^2}{|g|^2}. \end{align*} For $x\in\Omega'$ let \begin{align*} S_x:=\ker\big(g(x)\big)=\Big\{\xi\in\mathbb C^m:\sum_{j=1}^m g_j(x)\xi_j=0\Big\}, \end{align*} an $(m-1)$-dimensional subspace, and let $S:=\bigsqcup_{x\in\Omega'}S_x$ be the relation bundle. Its orthogonal complement is $S_x^\perp=\mathbb C\cdot\overline{g(x)}$, and $\sigma(x)=(f/|g|^2)\,\overline{g(x)}\in S_x^\perp$; thus $\sigma\perp S$ pointwise. Since each $g_j$ is holomorphic, $\bar\partial g_j=0$, so \begin{align*} g\cdot\bar\partial\sigma=\bar\partial(g\cdot\sigma)=\bar\partial f=0, \end{align*} i.e. the $\mathbb C^m$-valued $(0,1)$-form $\bar\partial\sigma$ takes values in $S$. The division problem reduces to finding a section $u$ of $S$ (a smooth map $u:\Omega'\to\mathbb C^m$ with $u(x)\in S_x$) solving \begin{align*} \bar\partial u=\bar\partial\sigma\quad\text{on }\Omega'. \tag{$\star$} \end{align*} Indeed, $h:=\sigma-u$ then satisfies $\bar\partial h=\bar\partial\sigma-\bar\partial u=0$ (so $h$ is holomorphic on $\Omega'$) and $g\cdot h=g\cdot\sigma-g\cdot u=f-0=f$. Moreover, as $u(x)\in S_x$ and $\sigma(x)\in S_x^\perp$, \begin{align*} |h|^2=|\sigma-u|^2=|\sigma|^2+|u|^2\quad\text{pointwise on }\Omega'. \tag{$\diamond$} \end{align*}[/step]

[guided]Why this particular $\sigma$? Among all $\xi\in\mathbb C^m$ solving $g(x)\cdot\xi=f(x)$ at a fixed $x\in\Omega'$, the one of smallest Euclidean norm is the [orthogonal projection](/theorems/437) of $0$ onto the affine solution space, i.e. the unique solution lying in $S_x^\perp=\mathbb C\cdot\overline{g(x)}$. Writing $\xi=\lambda\overline{g(x)}$ and imposing $g\cdot\xi=\lambda|g|^2=f$ gives $\lambda=f/|g|^2$, so $\sigma_j=\overline{g_j}f/|g|^2$. This is smooth on $\Omega'$ but in general not holomorphic, hence not yet admissible. To repair holomorphy without spoiling the identity $g\cdot h=f$, we seek $h=\sigma-u$ with $g\cdot u=0$, i.e. $u$ a section of $S=\ker g$; then automatically $g\cdot h=g\cdot\sigma=f$. Holomorphy of $h$ means $\bar\partial h=0$, i.e. $\bar\partial u=\bar\partial\sigma$, which is $(\star)$. Is the datum $\bar\partial\sigma$ admissible, i.e. $S$-valued, so that we may seek $u$ in $S$? Differentiating the holomorphic identity $g\cdot\sigma=f$ and using $\bar\partial g_j=0$ gives \begin{align*} 0=\bar\partial f=\bar\partial(g\cdot\sigma)=g\cdot\bar\partial\sigma, \end{align*} so $\bar\partial\sigma$ is indeed $S$-valued. Finally, the feature that makes the final constant sharp: since $u(x)\in S_x$ while $\sigma(x)\in S_x^\perp$, the two are orthogonal, so there is no cross term and $|h|^2=|\sigma-u|^2=|\sigma|^2+|u|^2$ — the norm of the answer splits into the fixed minimal-solution part $|\sigma|^2$ and the correctable part $|u|^2$, which is $(\diamond)$.[/guided]

[step:Bound the correction by the weighted $L^2$ estimate, where $\alpha>1$ produces the factor $1/(\alpha-1)$]If $q=0$ then $m=1$ and $S=\{0\}$; here $\sigma=f/g_1$ is already holomorphic on $\Omega'$, so $\bar\partial\sigma=0$ and we take $u\equiv 0$, for which the estimate below holds with right-hand side $0$. Assume henceforth $q\ge 1$, so that $\beta=\alpha q>q$. Equip $\Omega\subset\mathbb C^n$ with the standard Kähler form \begin{align*} \omega:=i\sum_{k=1}^n dz_k\wedge d\bar z_k, \end{align*} and let $\Lambda$ denote the pointwise adjoint of the Lefschetz operator $\tau\mapsto\omega\wedge\tau$ with respect to the induced Hermitian inner product on forms (the Lefschetz contraction). Define the plurisubharmonic weight \begin{align*} \Phi:\Omega\to[-\infty,\infty),\qquad \Phi:=\varphi+\beta\log|g|^2, \end{align*} so that $e^{-\Phi}=e^{-\varphi}|g|^{-2\beta}$, and the Hermitian curvature endomorphism acting on $S$-valued $(0,1)$-forms \begin{align*} B:=[\,\Theta,\ \Lambda\,],\qquad \Theta:=i\partial\bar\partial\Phi\otimes\mathrm{Id}_S+\Theta_S, \end{align*} where $\Theta_S$ is the Chern curvature of the subbundle $S\hookrightarrow\Omega\times\mathbb C^m$. Since the ambient bundle $\Omega\times\mathbb C^m$ carries the flat product connection, its curvature vanishes, and the Gauss–Codazzi relation for a holomorphic subbundle, $\Theta_S=\Theta_{\Omega\times\mathbb C^m}\big|_S-\beta_S^*\wedge\beta_S$, reduces to $\Theta_S=-\beta_S^*\wedge\beta_S$, where $\beta_S$ is the second fundamental form of $S$ in $\Omega\times\mathbb C^m$; the second fundamental form thus lowers the curvature, and this sign is what the estimate below relies on. The derivation of this relation and of the resulting curvature bound is precisely the content of Skoda's pointwise inequality stated next. We invoke two results. First, the **weighted $L^2$ existence theorem for $\bar\partial$ with curvature lower bound** (Bochner–Kodaira–Nakano/Hörmander), which we state in full as the result being applied: if $\Omega$ is pseudoconvex, $\Phi$ is plurisubharmonic, and $B$ is positive definite on $S$-valued $(0,1)$-forms, then for every $\bar\partial$-closed $S$-valued $(0,1)$-form $w$ with $\int_\Omega\langle B^{-1}w,w\rangle\,e^{-\Phi}\,d\mathcal L^{2n}<\infty$ there is a section $u$ of $S$ with $\bar\partial u=w$ and \begin{align*} \int_\Omega |u|^2 e^{-\Phi}\,d\mathcal L^{2n}\le\int_\Omega\langle B^{-1}w,w\rangle\, e^{-\Phi}\,d\mathcal L^{2n}. \end{align*} We verify its hypotheses: $\Omega$ is pseudoconvex (given); $\Phi=\varphi+\beta\log|g|^2$ is plurisubharmonic, since $\varphi$ is plurisubharmonic (given) and $\log|g|^2$ is the composition of the plurisubharmonic function $\zeta\mapsto\log|\zeta|^2$ on $\mathbb C^m$ with the holomorphic map $g$, hence plurisubharmonic ([Log-Modulus of a Holomorphic Function is PSH](/theorems/3405), [Stability Properties of PSH Functions](/theorems/3404)); the datum $w:=\bar\partial\sigma$ is $\bar\partial$-closed and $S$-valued by Step 2; and positivity of $B$ is supplied by the second result. The crude unweighted bound $\|u\|_{L^2}\le C_\Omega\|w\|_{L^2}$ furnished by [$\bar\partial$-Solvability on Pseudoconvex Domains](/theorems/3493) does not suffice, as its constant $C_\Omega$ is not explicit and cannot produce the sharp factor $\alpha/(\alpha-1)$; the curvature form of the estimate is essential. Second, **Skoda's pointwise curvature inequality for a surjective bundle morphism**, which we likewise state in full as the result being applied: when $\beta>q$, the endomorphism $B$ is positive definite on $S$-valued $(0,1)$-forms and \begin{align*} \langle B^{-1}\,\bar\partial\sigma,\bar\partial\sigma\rangle\le\frac{q}{\beta-q}\,\frac{|f|^2}{|g|^2}\qquad\text{pointwise on }\Omega'. \end{align*} The number $q=\min\{n,m-1\}$ is the maximal rank of $i\partial\bar\partial\log|g|^2$ (it cannot exceed the dimension $n$ of $\Omega$, nor the rank $m-1$ of the relation bundle), and $\beta-q=q(\alpha-1)>0$ is exactly the gap created by $\alpha>1$. Combining the two ingredients with $w=\bar\partial\sigma$ and $e^{-\Phi}=e^{-\varphi}|g|^{-2\beta}$, for any smooth plurisubharmonic weight the existence theorem furnishes a solution $u$ of $\bar\partial u=\bar\partial\sigma$ with \begin{align*} \int_{\Omega}\frac{|u|^2e^{-\varphi}}{|g|^{2\beta}}\,d\mathcal L^{2n} \le\frac{q}{\beta-q}\int_{\Omega}\frac{|f|^2e^{-\varphi}}{|g|^{2(\beta+1)}}\,d\mathcal L^{2n} =\frac{q}{\beta-q}\,I, \end{align*} and with $\beta=\alpha q$, $q\ge 1$ we get $\dfrac{q}{\beta-q}=\dfrac{q}{\alpha q-q}=\dfrac{1}{\alpha-1}$. This is where the hypothesis $\alpha>1$ is consumed. Because $\Phi$ is singular along $Z$ (where $\log|g|^2=-\infty$) and $\varphi$ need not be smooth, the existence theorem is applied not to $\Phi$ directly but to the smooth plurisubharmonic approximations of Step 4, which recover this bound in the limit.[/step]

[guided]The strategy is to solve $(\star)$ with a quantitative $L^2$ bound, and the only flexible quantity is the weight. We choose $\Phi=\varphi+\beta\log|g|^2$ with $\beta=\alpha q$: the factor $e^{-\Phi}=e^{-\varphi}|g|^{-2\beta}$ is exactly the density appearing in the target estimate, and its $\log|g|^2$ part injects curvature concentrated near $Z$ that controls the singularity of $\sigma$. The first ingredient is the curvature form of the Bochner–Kodaira–Nakano/Hörmander $L^2$ existence theorem: for $\Omega$ pseudoconvex, $\Phi$ plurisubharmonic, and $B=[\,\Theta,\Lambda\,]$ positive definite on $S$-valued $(0,1)$-forms (here $\Theta=i\partial\bar\partial\Phi\otimes\mathrm{Id}_S+\Theta_S$ combines the Hessian of the weight with the curvature $\Theta_S$ of $S\subset\Omega\times\mathbb C^m$, and $\Lambda$ is the contraction adjoint to $\omega\wedge\cdot$), every $\bar\partial$-closed $S$-valued datum $w$ of finite inverse-curvature energy has a primitive $u$ with \begin{align*} \int_\Omega|u|^2e^{-\Phi}\,d\mathcal L^{2n}\le\int_\Omega\langle B^{-1}w,w\rangle\,e^{-\Phi}\,d\mathcal L^{2n}. \end{align*} We check the hypotheses: pseudoconvexity is given; $\Phi$ is plurisubharmonic because $\varphi$ is and $\log|g|^2=(\log|\cdot|^2)\circ g$ is plurisubharmonic ([Log-Modulus of a Holomorphic Function is PSH](/theorems/3405), [Stability Properties of PSH Functions](/theorems/3404)); positivity of $B$ is the content of the second ingredient. Why not the unweighted bound $\|u\|_{L^2}\le C_\Omega\|w\|_{L^2}$ of [$\bar\partial$-Solvability on Pseudoconvex Domains](/theorems/3493)? Because $C_\Omega$ is inexplicit and cannot give the sharp $\alpha/(\alpha-1)$; only the curvature term controls the constant. The second ingredient is Skoda's pointwise inequality: when $\beta>q$, $B$ is positive definite and \begin{align*} \langle B^{-1}\bar\partial\sigma,\bar\partial\sigma\rangle\le\frac{q}{\beta-q}\,\frac{|f|^2}{|g|^2}\qquad\text{on }\Omega'. \end{align*} Here $q=\min\{n,m-1\}$ bounds the rank of the obstruction $i\partial\bar\partial\log|g|^2$, and $\beta-q=q(\alpha-1)$ is positive precisely because $\alpha>1$. What fails at $\alpha=1$? Then $\beta=q$, $B$ degenerates, $B^{-1}$ blows up, and no finite bound survives — the constant $\alpha/(\alpha-1)\to\infty$. Multiplying through and integrating, with $e^{-\Phi}=e^{-\varphi}|g|^{-2\beta}$ and $\langle B^{-1}\bar\partial\sigma,\bar\partial\sigma\rangle e^{-\Phi}\le\frac{q}{\beta-q}|f|^2|g|^{-2(\beta+1)}e^{-\varphi}$, \begin{align*} \int_{\Omega}\frac{|u|^2e^{-\varphi}}{|g|^{2\beta}}\,d\mathcal L^{2n} \le\frac{q}{\beta-q}\int_{\Omega}\frac{|f|^2e^{-\varphi}}{|g|^{2(\beta+1)}}\,d\mathcal L^{2n} =\frac{q}{\beta-q}\,I=\frac{1}{\alpha-1}\,I, \end{align*} using $\beta=\alpha q$. Since $\Phi$ is singular along $Z$ and $\varphi$ need not be smooth, this bound is realised through the regularisation of the next step rather than by a single application of the theorem.[/guided]

[step:Regularise the singular weight $\log|g|^2$ and pass to a weak-$L^2$ limit] Since $\Phi=\varphi+\beta\log|g|^2=-\infty$ on $Z$ and $\varphi$ may be singular, we approximate. As $\Omega$ is pseudoconvex it admits a smooth plurisubharmonic exhaustion; choose smooth bounded pseudoconvex subdomains $\Omega_k\Subset\Omega_{k+1}$ with $\bigcup_k\Omega_k=\Omega$, smooth plurisubharmonic $\varphi_\nu\downarrow\varphi$ on each $\Omega_k$, and for $\varepsilon>0$ the smooth plurisubharmonic functions $\log(|g|^2+\varepsilon^2)\downarrow\log|g|^2$ as $\varepsilon\downarrow 0$. The weights \begin{align*} \Phi_{\nu,\varepsilon}:=\varphi_\nu+\beta\log(|g|^2+\varepsilon^2) \end{align*} are smooth and plurisubharmonic. Skoda's pointwise inequality applies verbatim to the regularised weight: replacing $|g|^2$ by $|g|^2+\varepsilon^2$ leaves the rank bound $q=\min\{n,m-1\}$ and the gap $\beta-q$ unchanged and does not alter the datum $\bar\partial\sigma$, so the regularised curvature endomorphism $B_\varepsilon:=[\,i\partial\bar\partial\Phi_{\nu,\varepsilon}\otimes\mathrm{Id}_S+\Theta_S,\ \Lambda\,]$ is positive definite on $S$-valued $(0,1)$-forms and satisfies \begin{align*} \langle B_\varepsilon^{-1}\,\bar\partial\sigma,\bar\partial\sigma\rangle\le\frac{q}{\beta-q}\,\frac{|f|^2}{|g|^2+\varepsilon^2}\qquad\text{on }\Omega_k\cap\Omega'. \end{align*} Multiplying by $e^{-\Phi_{\nu,\varepsilon}}=e^{-\varphi_\nu}(|g|^2+\varepsilon^2)^{-\beta}$ yields exactly the regularised integrand of the bound below. Hence Step 3, applied on $\Omega_k$ with the smooth weight $\Phi_{\nu,\varepsilon}$, furnishes sections $u_{k,\nu,\varepsilon}$ of $S$ solving $\bar\partial u_{k,\nu,\varepsilon}=\bar\partial\sigma$ on $\Omega_k\cap\Omega'$ with \begin{align*} \int_{\Omega_k}\frac{|u_{k,\nu,\varepsilon}|^2e^{-\varphi_\nu}}{(|g|^2+\varepsilon^2)^{\beta}}\,d\mathcal L^{2n} \le\frac{q}{\beta-q}\int_{\Omega_k}\frac{|f|^2e^{-\varphi_\nu}}{(|g|^2+\varepsilon^2)^{\beta+1}}\,d\mathcal L^{2n} \le\frac{q}{\beta-q}\,I, \end{align*} the last inequality because $|g|^2+\varepsilon^2\ge|g|^2$ and $\varphi_\nu\ge\varphi$ make the integrand dominated by that of $I$, which is finite by hypothesis; the [Monotone Convergence Theorem](/theorems/509) governs the passages $\varepsilon\downarrow 0$, $\nu\uparrow\infty$, $k\uparrow\infty$. The family $\{u_{k,\nu,\varepsilon}\}$ is thus bounded in the [Hilbert space](/page/Hilbert%20Space) $L^2_{\mathrm{loc}}\big(\Omega',|g|^{-2\beta}e^{-\varphi}\,d\mathcal L^{2n};\mathbb C^m\big)$. By [Sequential Banach–Alaoglu](/theorems/496) (a bounded sequence in a [Hilbert space](/page/Hilbert%20Space), being reflexive, has a weakly convergent subsequence) together with a diagonal argument over an exhaustion of $\Omega'$, a subsequence converges weakly to a section $u$ of $S$ on $\Omega'$. [Weak convergence](/page/Weak%20Convergence) commutes with the (constant-coefficient, distributional) operator $\bar\partial$, so $\bar\partial u=\bar\partial\sigma$ holds on $\Omega'$ in the sense of distributions; and by weak [Lower Semicontinuity of the Norm](/theorems/215), \begin{align*} \int_{\Omega'}\frac{|u|^2e^{-\varphi}}{|g|^{2\beta}}\,d\mathcal L^{2n}\le\frac{q}{\beta-q}\,I=\frac{1}{\alpha-1}\,I. \end{align*} [/step]

[step:Assemble the holomorphic solution, extend across $Z$, and derive the constant $\alpha/(\alpha-1)$] Set $h:=\sigma-u:\Omega'\to\mathbb C^m$. Then $\bar\partial h=\bar\partial\sigma-\bar\partial u=0$ on $\Omega'$, so by elliptic regularity for the Cauchy–Riemann system ([Higher Interior Elliptic Regularity](/theorems/96)) $h$ is holomorphic on $\Omega'$, and $g\cdot h=g\cdot\sigma-g\cdot u=f-0=f$ there. By the pointwise orthogonality $(\diamond)$, \begin{align*} \int_{\Omega'}\frac{\sum_{j=1}^m|h_j|^2e^{-\varphi}}{|g|^{2\beta}}\,d\mathcal L^{2n} =\int_{\Omega'}\frac{|\sigma|^2e^{-\varphi}}{|g|^{2\beta}}\,d\mathcal L^{2n} +\int_{\Omega'}\frac{|u|^2e^{-\varphi}}{|g|^{2\beta}}\,d\mathcal L^{2n}. \end{align*} Since $|\sigma|^2=|f|^2/|g|^2$, the first term equals $\int_{\Omega'}|f|^2|g|^{-2(\beta+1)}e^{-\varphi}\,d\mathcal L^{2n}=I$; the second is $\le\frac{1}{\alpha-1}I$ by Step 4. Hence \begin{align*} \int_{\Omega'}\frac{\sum_{j=1}^m|h_j|^2e^{-\varphi}}{|g|^{2\beta}}\,d\mathcal L^{2n}\le I+\frac{1}{\alpha-1}I=\frac{\alpha}{\alpha-1}\,I. \end{align*} It remains to extend $h$ across $Z$. On any compact $K\subset\Omega$ we have $|g|^{-2\beta}\ge(\sup_K|g|^2)^{-\beta}>0$ and $e^{-\varphi}\ge e^{-\sup_K\varphi}>0$ (a plurisubharmonic $\varphi$ is locally bounded above), so the finiteness just proved gives $\int_{K\setminus Z}|h|^2\,d\mathcal L^{2n}<\infty$; thus $h\in L^2_{\mathrm{loc}}(\Omega)^m$ is holomorphic on $\Omega\setminus Z$ with $Z$ a proper analytic subset. By [Riemann's Removable Singularity Theorem](/theorems/3356) in its $L^2$ form across analytic sets (cf. [Removability of Codimension-Two Singularities](/theorems/3383)), $h$ extends to $h\in\mathcal O(\Omega)^m$. The identity $g\cdot h=f$ then holds on all of $\Omega$ by the [Identity Principle](/theorems/3357), and $\mathcal L^{2n}(Z)=0$ lets us replace $\Omega'$ by $\Omega$ in the bound. Recalling $\beta=\alpha q$, we have produced $h_1,\dots,h_m\in\mathcal O(\Omega)$ with $f=\sum_{j=1}^m g_jh_j$ and \begin{align*} \int_\Omega\frac{\sum_{j=1}^m|h_j|^2e^{-\varphi}}{|g|^{2\alpha q}}\,d\mathcal L^{2n}\le\frac{\alpha}{\alpha-1}\int_\Omega\frac{|f|^2e^{-\varphi}}{|g|^{2(\alpha q+1)}}\,d\mathcal L^{2n}, \end{align*} which is exactly the assertion of the theorem. $\blacksquare$ [/step]