[proofplan] We put the generic CR submanifold into local graph coordinates and transfer the [Hausdorff measure](/page/Hausdorff%20Measure) to a smooth positive density in those coordinates. The analytic input is the standard Baouendi-Treves local approximation theorem for $L^2$ distributional CR functions on a generic graph: its Gaussian kernels define entire functions whose restrictions converge in $L^2$ on compact subsets. After applying that local theorem with a cutoff supported in the coordinate patch, we approximate the resulting entire functions uniformly on a compact ambient neighbourhood by holomorphic polynomials and use the triangle inequality. [/proofplan]

[step:Put $M$ into generic graph coordinates near $p$] Let $d:=\operatorname{codim}_{\mathbb R}M$ and $m:=N-d$. Since $M$ is a $C^\infty$ embedded generic CR submanifold, the local graph normal form for generic submanifolds gives, after translating $p$ to $0$ and applying a complex-linear change of coordinates in $\mathbb C^N$, an open neighbourhood $W\subset\mathbb C^m\times\mathbb R^d$ of $0$, an open neighbourhood $U_0\subset M$ of $p$, and a $C^\infty$ diffeomorphism $\Phi:W\to U_0$ of the form \begin{align*} \Phi(z,s)=(z,s+i\varphi(z,s)). \end{align*} Here $\varphi:W\to\mathbb R^d$ is a $C^\infty$ map satisfying $\varphi(0)=0$ and $D\varphi_0=0$, where $D\varphi_0:\mathbb R^{2m+d}\to\mathbb R^d$ is the total derivative at $0$. Define the coordinate embedding $Z:W\to\mathbb C^N$ by \begin{align*} Z(q)=\Phi(q). \end{align*} A point of $W$ will be denoted by $q$, and an independent integration variable in $W$ will be denoted by $q'$. Choose an open neighbourhood $U\subset U_0$ of $p$ such that the closure of $U$ in $M$ is compact and contained in $U_0$. It is enough to prove the asserted approximation for every $V\Subset U$, because the theorem only requires one sufficiently small neighbourhood of $p$. The measure $\mu_M=\mathcal H^{2m+d}|_M$ pulls back under $\Phi$ to a smooth positive density. Thus there is a $C^\infty$ function $J:W\to(0,\infty)$ such that every Borel function $a:U_0\to\mathbb C$ for which either side is defined satisfies \begin{align*} \int_{U_0} a(\zeta)\,d\mu_M(\zeta)=\int_W a(\Phi(q))J(q)\,d\mathcal L^{2m+d}(q). \end{align*} [/step]

[step:Localize the CR function without changing it near $V$] Fix an [open set](/page/Open%20Set) $V\Subset U$. Choose open sets $V_1,V_2\subset U_0$ such that $V\Subset V_1\Subset V_2\Subset U$. Choose a cutoff function $\eta:M\to[0,1]$ with $\eta\in C_c^\infty(V_2;\mathbb R)$ and $\eta=1$ on a neighbourhood of $\overline{V_1}$. Let $f\in L^2_{\mathrm{loc}}(U,\mu_M;\mathbb C)$ be CR in the distribution sense on $U$. Define $g:U_0\to\mathbb C$ by \begin{align*} g(\zeta)=\eta(\zeta)f(\zeta) \end{align*} for $\zeta\in U$, and set $g(\zeta)=0$ for $\zeta\in U_0\setminus\operatorname{supp}\eta$. Since $\operatorname{supp}\eta$ is compact in $U$ and $f\in L^2_{\mathrm{loc}}(U,\mu_M;\mathbb C)$, the function $g$ belongs to $L^2(U_0,\mu_M;\mathbb C)$. Define $h:W\to\mathbb C$ by \begin{align*} h(q)=g(\Phi(q)). \end{align*} Then $h\in L^2(W,J\,d\mathcal L^{2m+d};\mathbb C)$, the support of $h$ is a compact subset of $W$, and $h(q)=f(\Phi(q))$ for every $q\in\Phi^{-1}(V_1)$. [/step]

[step:Apply the Baouendi-Treves local theorem for $L^2$ CR functions]We use the standard Baouendi-Treves local approximation theorem in its $L^2$ distributional form. In the notation needed here, it says the following. Let $Z:W\to\mathbb C^N$ be a $C^\infty$ generic graph of the form $Z(z,s)=(z,s+i\varphi(z,s))$ with $\varphi(0)=0$ and $D\varphi_0=0$. If $K_0\Subset K_1\Subset W$ are compact sets, $u\in L^2_{\mathrm{loc}}(W,J\,d\mathcal L^{2m+d};\mathbb C)$ is CR in the distribution sense on a neighbourhood of $K_1$ after transfer by $\Phi$, and $\theta:W\to[0,1]$ is a $C_c^\infty$ function equal to $1$ on a neighbourhood of $K_1$, then there are kernels $B_\varepsilon:\mathbb C^N\times W\to\mathbb C$, defined for $0<\varepsilon<\varepsilon_0$, such that $B_\varepsilon(\cdot,q')$ is entire for each $q'\in W$ and the functions $E_\varepsilon:\mathbb C^N\to\mathbb C$ given by \begin{align*} E_\varepsilon(\zeta)=\int_W B_\varepsilon(\zeta,q')\theta(q')u(q')J(q')\,d\mathcal L^{2m+d}(q') \end{align*} are entire and satisfy \begin{align*} \lim_{\varepsilon\downarrow0}\int_{K_0}|E_\varepsilon(Z(q))-u(q)|^2J(q)\,d\mathcal L^{2m+d}(q)=0. \end{align*} The theorem also includes the local domination estimates in $q'$ on compact subsets of $\mathbb C^N$, so differentiating under the integral is justified. This is the classical Baouendi-Treves approximation theorem; its proof is the holomorphic Gaussian-kernel construction, with the weak CR equations used by [integration by parts](/theorems/210). We now verify its hypotheses. The preceding coordinate step gives exactly the required generic graph $Z(z,s)=(z,s+i\varphi(z,s))$ with $\varphi\in C^\infty(W;\mathbb R^d)$, $\varphi(0)=0$, and $D\varphi_0=0$. Define the compact set $K_0\subset W$ by \begin{align*} K_0=\Phi^{-1}(\overline V). \end{align*} Since $V\Subset V_1$ and $\Phi$ is a homeomorphism from $W$ onto $U_0$, there is a compact set $K_1\subset W$ such that $K_0\Subset K_1\Subset\Phi^{-1}(V_1)$. On a neighbourhood of $K_1$, the function $h$ equals $f\circ\Phi$, and $f$ is CR in the distribution sense on $U$. Hence $h$ satisfies the distributional CR hypothesis required by the local theorem on a neighbourhood of $K_1$. Apply the theorem with $u=h$ and with a cutoff $\theta\in C_c^\infty(W;[0,1])$ chosen so that $\theta=1$ on a neighbourhood of $\operatorname{supp}h$. Because $h$ already has compact support, the product $\theta h$ equals $h$. For $0<\varepsilon<\varepsilon_0$, define $F_\varepsilon:\mathbb C^N\to\mathbb C$ by \begin{align*} F_\varepsilon(\zeta)=\int_W B_\varepsilon(\zeta,q')h(q')J(q')\,d\mathcal L^{2m+d}(q'). \end{align*} The local domination estimates in the Baouendi-Treves theorem allow differentiation under the integral with respect to the ambient holomorphic variable $\zeta$, and $B_\varepsilon(\cdot,q')$ is entire for every $q'$. Therefore $F_\varepsilon$ is entire on $\mathbb C^N$. The convergence conclusion gives \begin{align*} \lim_{\varepsilon\downarrow0}\int_{K_0}|F_\varepsilon(Z(q))-h(q)|^2J(q)\,d\mathcal L^{2m+d}(q)=0. \end{align*}[/step]

[guided]The central point is to use a local theorem whose conclusion is already an $L^2$ approximation statement for distributional CR functions, not merely a construction for smooth CR functions. We use the standard Baouendi-Treves local approximation theorem in the following precise form. Suppose $Z:W\to\mathbb C^N$ is a $C^\infty$ generic graph $Z(z,s)=(z,s+i\varphi(z,s))$, where $W\subset\mathbb C^m\times\mathbb R^d$ is open, $\varphi:W\to\mathbb R^d$ is $C^\infty$, $\varphi(0)=0$, and $D\varphi_0=0$. Suppose $K_0\Subset K_1\Subset W$ are compact sets, $u\in L^2_{\mathrm{loc}}(W,J\,d\mathcal L^{2m+d};\mathbb C)$ is CR in the distribution sense on a neighbourhood of $K_1$, and $\theta:W\to[0,1]$ is a $C_c^\infty$ cutoff equal to $1$ near $K_1$. Then the theorem supplies holomorphic Gaussian kernels $B_\varepsilon:\mathbb C^N\times W\to\mathbb C$ such that $B_\varepsilon(\cdot,q')$ is entire and \begin{align*} E_\varepsilon(\zeta)=\int_W B_\varepsilon(\zeta,q')\theta(q')u(q')J(q')\,d\mathcal L^{2m+d}(q') \end{align*} defines an entire function $E_\varepsilon:\mathbb C^N\to\mathbb C$. The same theorem asserts the convergence estimate \begin{align*} \lim_{\varepsilon\downarrow0}\int_{K_0}|E_\varepsilon(Z(q))-u(q)|^2J(q)\,d\mathcal L^{2m+d}(q)=0. \end{align*} The theorem applies to $L^2$ distributional CR functions because the Gaussian-kernel proof uses the CR equation only through [integration by parts](/theorems/2098) against compactly supported smooth test functions; this is exactly the distributional hypothesis in the statement of the theorem. We verify the hypotheses one by one. The first step of the proof produced a graph map $Z:W\to\mathbb C^N$ of the required form, namely $Z(z,s)=(z,s+i\varphi(z,s))$, with $\varphi\in C^\infty(W;\mathbb R^d)$, $\varphi(0)=0$, and $D\varphi_0=0$. Next define \begin{align*} K_0=\Phi^{-1}(\overline V). \end{align*} The set $\overline V$ is compact in $U_0$ because $V\Subset U$, and $\Phi^{-1}:U_0\to W$ is continuous, so $K_0$ is compact in $W$. Since $V\Subset V_1$, the compact set $K_0$ is contained in the open set $\Phi^{-1}(V_1)$. We therefore choose a compact set $K_1\subset W$ with $K_0\Subset K_1\Subset\Phi^{-1}(V_1)$. On $\Phi^{-1}(V_1)$ the localized function $h$ agrees with $f\circ\Phi$. Since $f$ is CR in the distribution sense on $U$ and $\Phi(K_1)\subset V_1\Subset U$, the pulled-back function $h$ satisfies the distributional CR equations on a neighbourhood of $K_1$. This is precisely the CR hypothesis required by the local Baouendi-Treves theorem. Finally, because $h$ has compact support in $W$, we may choose $\theta\in C_c^\infty(W;[0,1])$ with $\theta=1$ on a neighbourhood of $\operatorname{supp}h$. Then $\theta h=h$, so the function produced by the theorem is exactly \begin{align*} F_\varepsilon(\zeta)=\int_W B_\varepsilon(\zeta,q')h(q')J(q')\,d\mathcal L^{2m+d}(q'). \end{align*} The theorem's domination estimates justify differentiating this integral with respect to the ambient holomorphic variable $\zeta$, and the integrand is entire in $\zeta$ for each fixed $q'$. Hence $F_\varepsilon$ is entire on $\mathbb C^N$. Applying the convergence part of the same theorem with $u=h$ gives \begin{align*} \lim_{\varepsilon\downarrow0}\int_{K_0}|F_\varepsilon(Z(q))-h(q)|^2J(q)\,d\mathcal L^{2m+d}(q)=0. \end{align*}[/guided]

[step:Transfer the local convergence back to $M$] For every $q\in K_0=\Phi^{-1}(\overline V)$, the point $\Phi(q)$ belongs to $\overline V\subset V_1$, and hence $h(q)=f(\Phi(q))$. Since $Z=\Phi$, the convergence obtained above becomes \begin{align*} \lim_{\varepsilon\downarrow0}\int_{\Phi^{-1}(V)}|F_\varepsilon(\Phi(q))-f(\Phi(q))|^2J(q)\,d\mathcal L^{2m+d}(q)=0. \end{align*} Using the change-of-variables formula for the density $J\,d\mathcal L^{2m+d}$ under $\Phi$, this is equivalent to \begin{align*} \lim_{\varepsilon\downarrow0}\int_V|F_\varepsilon(\zeta)-f(\zeta)|^2\,d\mu_M(\zeta)=0. \end{align*} Choose a sequence of positive numbers $(\varepsilon_k)_{k=1}^\infty$ with $\varepsilon_k\downarrow0$ and \begin{align*} \|F_{\varepsilon_k}|_M-f\|_{L^2(V,\mu_M)}\to0. \end{align*} [/step]

[step:Approximate the entire functions by holomorphic polynomials] Since $\overline V$ is compact in $M$ and the embedding $M\hookrightarrow\mathbb C^N$ is continuous, $\overline V$ is compact as a subset of $\mathbb C^N$. Choose a compact Euclidean ball $A\subset\mathbb C^N$ with $\overline V\subset A$. For each $k\in\mathbb N$, the function $F_{\varepsilon_k}:\mathbb C^N\to\mathbb C$ is entire. Its Taylor series at $0$ converges uniformly on the compact set $A$, so there is a holomorphic polynomial $P_k:\mathbb C^N\to\mathbb C$ such that \begin{align*} \sup_{\zeta\in A}|P_k(\zeta)-F_{\varepsilon_k}(\zeta)|\le \frac{1}{k(1+\mu_M(V))^{1/2}}. \end{align*} Because $V\subset A$, the definition of the $L^2(V,\mu_M)$ norm gives \begin{align*} \|P_k|_M-F_{\varepsilon_k}|_M\|_{L^2(V,\mu_M)}\le \frac{\mu_M(V)^{1/2}}{k(1+\mu_M(V))^{1/2}}. \end{align*} Since $\mu_M(V)^{1/2}\le(1+\mu_M(V))^{1/2}$, this yields \begin{align*} \|P_k|_M-F_{\varepsilon_k}|_M\|_{L^2(V,\mu_M)}\le \frac{1}{k}. \end{align*} By the triangle inequality in $L^2(V,\mu_M)$, \begin{align*} \|P_k|_M-f\|_{L^2(V,\mu_M)}\le \|P_k|_M-F_{\varepsilon_k}|_M\|_{L^2(V,\mu_M)}+\|F_{\varepsilon_k}|_M-f\|_{L^2(V,\mu_M)}. \end{align*} The first term tends to $0$ by construction, and the second term tends to $0$ by the preceding step. Therefore \begin{align*} \lim_{k\to\infty}\|P_k|_M-f\|_{L^2(V,\mu_M)}=0. \end{align*} This proves the asserted polynomial approximation on the arbitrary relatively compact open set $V\Subset U$. [/step]