2.1 Asymptotic density

We can now generalize the notion of asymptotic upper and lower densities as follows.

In other words, the upper density $D_{\unicode[STIX]{x1D711}}^{+}(Z)$ is the infimum of all positive numbers $a$ such that

$$\begin{eqnarray}dd^{c}\unicode[STIX]{x1D711}_{r}-\frac{1}{a}\unicode[STIX]{x1D6F6}_{r}^{Z}>0,\end{eqnarray}$$

while the lower density $D_{\unicode[STIX]{x1D711}}^{-}(Z)$ is the supremum of all numbers $c$ such that there exists $z,v\in \mathbb{C}^{n}$ satisfying

$$\begin{eqnarray}dd^{c}\unicode[STIX]{x1D711}_{r}(z)(v,\bar{v})-\frac{1}{c}\unicode[STIX]{x1D6F6}_{r}^{Z}(z)(v,\bar{v})<0.\end{eqnarray}$$

Note that

$$\begin{eqnarray}D_{\unicode[STIX]{x1D711}}^{-}(Z)\leqslant D_{\unicode[STIX]{x1D711}}^{+}(Z)\end{eqnarray}$$

and that either of the densities can be infinite.

In the present article, the following simple proposition is relevant.

Proposition 2.2. If $Z$ is an algebraic hypersurface in $\mathbb{C}^{n}$ then

$$\begin{eqnarray}D_{\unicode[STIX]{x1D711}}^{+}(Z)=0.\end{eqnarray}$$

Proof. Since an algebraic hypersurface $Z$ of degree $d$ is locally a $d$-sheeted cover of a complex hyperplane, the area of $Z\cap B_{r}(z)$ is

$$\begin{eqnarray}\int _{B_{r}(z)}[Z]\wedge \frac{\unicode[STIX]{x1D714}_{o}^{n-1}}{(n-1)!}=O(r^{2n-2})\end{eqnarray}$$

uniformly in $z$. Since $dd^{c}\unicode[STIX]{x1D711}_{r}(v,\bar{v})\geqslant mr^{2n}\unicode[STIX]{x1D714}_{o}(v,\bar{v})$ the result follows from (3).◻

2.2 Uniform flatness

As we already recalled in the introduction, a smooth hypersurface $Z\subset \mathbb{C}^{n}$ is uniformly flat if there is a positive constant $\unicode[STIX]{x1D700}>0$ such that

$$\begin{eqnarray}N_{\unicode[STIX]{x1D700}}(Z)=\{x\in \mathbb{C}^{n};B_{\unicode[STIX]{x1D700}}(x)\cap Z\neq \emptyset \}\end{eqnarray}$$

(the $\unicode[STIX]{x1D700}$-neighborhood of $Z$) is a tubular neighborhood of $Z$. In our previous article [Reference Pingali and VarolinPV-2016] we established the following result.

Proposition 2.3. [Reference Pingali and VarolinPV-2016, Lemma 4.11]

If a smooth hypersurface $Z\subset \mathbb{C}^{n}$ is uniformly flat for each $r>0$ the separation function

$$\begin{eqnarray}S_{r}^{Z}:=|dT|^{2}e^{-\unicode[STIX]{x1D706}_{r}^{T}}:Z\rightarrow \mathbb{R}_{+}\end{eqnarray}$$

is bounded below by a positive constant $C_{r}$.

In dimension $n=1$ the converse of Proposition 2.3 is true as well. And although we suspect it is the case, we do not know if the converse is also true in higher dimensions.

2.3 Interpolation and sampling

In connection with interpolation, we have already mentioned Theorem 1.2 for hypersurfaces. For $k<n-1$ very little is known about interpolation. The most interesting case is $k=0$, which would be most useful in applications. (There are some partial results in [Reference Ortega-Cerdà, Schuster and VarolinOSV-2006], but these results are not decisive.) It is known to experts that if $k=0$ and $n>1$ then it is certainly not density that governs whether or not a sequence of points is interpolating (or for that matter, sampling). Nevertheless, the density does have to be somewhat constrained. An interesting necessary condition was introduced in [Reference LindholmL-1997], and very recently improved in [Reference Gröchenig, Haimi, Ortega-Cerdá and RomeroGHOR-2018].

The following simple proposition is very useful.

Proposition 2.5. (Bounded interpolation operators)

Let $Z\subset \mathbb{C}^{n}$ be a complex subvariety and let $\unicode[STIX]{x1D711}\in \mathscr{C}^{2}(\mathbb{C}^{n})$. If the restriction

$$\begin{eqnarray}\mathscr{R}_{Z}:\mathscr{B}_{n}(\unicode[STIX]{x1D711})\rightarrow \mathfrak{B}_{n-1}(Z,\unicode[STIX]{x1D711})\end{eqnarray}$$

is onto then there is a bounded section $I:\mathfrak{B}_{n-1}(Z,\unicode[STIX]{x1D711})\rightarrow \mathscr{B}_{n}(\unicode[STIX]{x1D711})$ of $\mathscr{R}_{Z}$.

Proof. We define

$$\begin{eqnarray}I:\mathfrak{B}_{n-1}(\unicode[STIX]{x1D711},Z)\rightarrow \mathscr{B}_{n}(\unicode[STIX]{x1D711})\end{eqnarray}$$

by letting $I(f)$ be the extension of $f$ having minimal norm in $\mathscr{B}_{n}(\unicode[STIX]{x1D711})$. Equivalently, if we let $\mathscr{I}_{Z}$ denote the sheaf of germs of holomorphic functions vanishing on $Z$, and write

$$\begin{eqnarray}\mathfrak{J}_{Z}(\unicode[STIX]{x1D711}):=H^{0}(\mathbb{C}^{n},\mathscr{I}_{Z})\cap \mathscr{B}_{n}(\unicode[STIX]{x1D711}),\end{eqnarray}$$

then $I(f)$ is the unique extension of $f$ to $\mathscr{B}_{n}(\unicode[STIX]{x1D711})$ such that

$$\begin{eqnarray}\int _{\mathbb{C}^{n}}I(f)\overline{G}e^{-\unicode[STIX]{x1D711}}\,dV=0\quad \text{for all}~G\in \mathfrak{J}_{Z}(\unicode[STIX]{x1D711}).\end{eqnarray}$$

By the Closed Graph Theorem the section $I:\mathfrak{B}_{n-1}(Z,\unicode[STIX]{x1D711})\rightarrow \mathscr{B}_{n}(\unicode[STIX]{x1D711})$ is bounded if it has closed graph. To show the latter, let $f_{j}\rightarrow f$ in $\mathfrak{B}_{n-1}(Z,\unicode[STIX]{x1D711})$ and let $I(f_{j})\rightarrow F$ in $\mathscr{B}_{n}(\unicode[STIX]{x1D711})$. Then for each $G\in \mathfrak{J}_{Z}$ one has

$$\begin{eqnarray}\int _{\mathbb{C}^{n}}F\overline{G}e^{-\unicode[STIX]{x1D711}}\,dV=\lim _{j\rightarrow \infty }\int _{\mathbb{C}^{n}}I(f_{j})\overline{G}e^{-\unicode[STIX]{x1D711}}\,dV=0.\end{eqnarray}$$

By the weighted Bergman inequality (Proposition 3.3) the $L^{2}$ norm controls the $L_{\ell oc}^{\infty }$ norm for holomorphic functions, and hence by Montel’s theorem the two limits are, perhaps after passing to subsequences, locally uniform. It follows immediately that $F$ is an extension of $f$. Hence $F=I(f)$, and the proof is complete.◻

We end this subsection by noting that a subvariety is a sampling set if and only if

$$\begin{eqnarray}\int _{\mathbb{C}^{n}}|F|^{2}e^{-\unicode[STIX]{x1D711}}\unicode[STIX]{x1D714}_{o}^{n}\lesssim \int _{Z_{\text{reg}}}|F|^{2}e^{-\unicode[STIX]{x1D711}}\unicode[STIX]{x1D714}_{o}^{k}\lesssim \int _{\mathbb{ C}^{n}}|F|^{2}e^{-\unicode[STIX]{x1D711}}\unicode[STIX]{x1D714}_{o}^{n}\end{eqnarray}$$

holds for all $F\in \mathscr{B}_{n}(\unicode[STIX]{x1D711})$. In the case of a smooth hypersurface, [Reference Ortega-Cerdà, Schuster and VarolinOSV-2006] establishes a companion result to Theorem 1.2 that generalizes the positive direction of the sampling theorems established in generalized Bargmann–Fock spaces over $\mathbb{C}$ established by Berndtsson, Ortega-Cerdà and Seip. Surely there is also an analogue for singular, uniformly flat hypersurfaces, but the details have not been worked out.

2.4 $L^{2}$ extension after Ohsawa and Takegoshi

Among the most sophisticated and useful set of results in complex analysis and geometry is the collection of theorems on $L^{2}$ extension that have come to be known as extension theorems of Ohsawa–Takegoshi type. The name derives from the first fundamental result regarding $L^{2}$ extension in several complex variables, which was established by Ohsawa and Takegoshi in their celebrated article [Reference Ohsawa and TakegoshiOT-1987]. Since that time, new proofs and extensions of the original result have been established by many authors, too numerous to state here. The following version, established by the second-named author in [Reference VarolinV-2008], will be a convenient version for our purposes.

Theorem 2.6. Let $X$ be a Stein manifold with Kähler metric $\unicode[STIX]{x1D714}$, and let $Z\subset X$ be a smooth hypersurface. Assume there exists a section $T\in H^{0}(X,{\mathcal{O}}_{X}(L_{Z}))$ and a metric $e^{-\unicode[STIX]{x1D706}}$ for the line bundle $L_{Z}\rightarrow X$ associated to the smooth divisor $Z$, such that $e^{-\unicode[STIX]{x1D706}}|_{Z}$ is still a singular Hermitian metric, and

$$\begin{eqnarray}\sup _{X}|T|^{2}e^{-\unicode[STIX]{x1D706}}\leqslant 1.\end{eqnarray}$$

Let $H\rightarrow X$ be a holomorphic line bundle with singular Hermitian metric $e^{-\unicode[STIX]{x1D713}}$ such that $e^{-\unicode[STIX]{x1D713}}|_{Z}$ is still a singular Hermitian metric. Assume there exists $s\in (0,1]$ such that

(4)$$\begin{eqnarray}\sqrt{-1}(\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}\unicode[STIX]{x1D713}+\text{Ricci}(\unicode[STIX]{x1D714}))\geqslant (1+ts)\sqrt{-1}\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}\unicode[STIX]{x1D706}_{Z}\end{eqnarray}$$

for all $t\in [0,1]$. Then for any section $f\in H^{0}(Z,{\mathcal{O}}_{Z}(H))$ satisfying

$$\begin{eqnarray}\int _{Z}\frac{|f|^{2}e^{-\unicode[STIX]{x1D713}}}{|dT|_{\unicode[STIX]{x1D714}}^{2}e^{-\unicode[STIX]{x1D706}}}\,dA_{\unicode[STIX]{x1D714}}<+\infty\end{eqnarray}$$

there exists a section $F\in H^{0}(X,{\mathcal{O}}_{X}(H))$ such that

$$\begin{eqnarray}F|_{Z}=f\qquad \text{and}\qquad \int _{X}|F|^{2}e^{-\unicode[STIX]{x1D713}}\,dV_{\unicode[STIX]{x1D714}}\leqslant \frac{24\unicode[STIX]{x1D70B}}{s}\int _{Z}\frac{|f|^{2}e^{-\unicode[STIX]{x1D713}}}{|dT|_{\unicode[STIX]{x1D714}}^{2}e^{-\unicode[STIX]{x1D706}}}\,dA_{\unicode[STIX]{x1D714}}.\end{eqnarray}$$

$L^{2}$ extension theorems for higher codimension subvarieties also exist. If the subvariety is cut out by a section of some vector bundle whose rank is equal to the codimension, with the section being generically transverse to the zero section, then the result is very much analogous to Theorem 2.6. For general submanifolds or subvarieties the result requires more normalization.

The reader will notice that Theorem 2.6 does not mention uniform flatness, and that density is not explicitly stated here. However, the result does address both issues in a slightly more hidden way. The issue of density is captured by the curvature conditions, while uniform flatness, or rather the absence of requiring uniform flatness, is dealt with by introducing the denominator $|dT|_{\unicode[STIX]{x1D714}}^{2}e^{-\unicode[STIX]{x1D706}}$ in the norm on the hypersurface. The following proof of Theorem 1.2 provides a nice illustration.

Proof of Theorem 1.2.

In Theorem 2.6 let $X=\mathbb{C}^{n}$, $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{o}$. Fix any $T\in {\mathcal{O}}(\mathbb{C}^{n})$ whose zero locus is $Z$, such that $dT(z)\neq 0$ for all $z\in Z$. Set

$$\begin{eqnarray}\unicode[STIX]{x1D706}(z)=\frac{1}{\text{Vol}\,B_{r}(0)}\int _{B_{r}(0)}\log |T(z-\unicode[STIX]{x1D701})|^{2}\,dV(\unicode[STIX]{x1D701}).\end{eqnarray}$$

Choose $s\in (0,1]$ such that $D_{\unicode[STIX]{x1D711}}^{+}(Z)<1/(1+s)$. Then by Definition 2.1 the curvature hypothesis (4) is satisfied, and thus we see that for any $f\in {\mathcal{O}}(Z)$ such that

(5)$$\begin{eqnarray}\int _{Z}\frac{|f|^{2}e^{-\unicode[STIX]{x1D711}}}{S_{r}^{Z}}\,dA_{\unicode[STIX]{x1D714}_{o}}<+\infty\end{eqnarray}$$

there exists $F\in {\mathcal{O}}(\mathbb{C}^{n})$ such that

$$\begin{eqnarray}F|_{Z}=f\qquad \text{and}\qquad \int _{\mathbb{C}^{n}}|F|^{2}e^{-\unicode[STIX]{x1D711}}\,dA_{\unicode[STIX]{x1D714}_{o}}<+\infty .\end{eqnarray}$$

Since $Z$ is uniformly flat, Proposition 2.3 implies that every $f\in \mathfrak{B}_{n-1}(Z,\unicode[STIX]{x1D711})$ satisfies (5), and thus Theorem 1.2 is proved.◻

Remark 2.7. Note that something slightly stronger than Theorem 1.2 is proved. In fact, the bounded extension operator guaranteed by Proposition 2.5 is rather uniformly bounded. Its norm is bounded by a constant that depends only on the density $D_{\unicode[STIX]{x1D711}}^{+}(Z)$ and on the separation constant

$$\begin{eqnarray}\sup \{\unicode[STIX]{x1D700}>0;U_{\unicode[STIX]{x1D700}}(Z)~\text{is a tubular neighborhood}\},\end{eqnarray}$$

or equivalently, the lower bound on the separation function.

3.1 Normalization of the weights

Lemma 3.2. There exists a constant $C>0$ with the following property. Let $\unicode[STIX]{x1D711}\in \mathscr{C}^{2}(\mathbb{C}^{k}\times \mathbb{C})$ satisfy

$$\begin{eqnarray}mdd^{c}|z|^{2}\leqslant dd^{c}\unicode[STIX]{x1D711}\leqslant Mdd^{c}|z|^{2}\end{eqnarray}$$

for some positive constants $m$ and $M$. Then for each $r\in (0,1]$ and each polydisk $D_{R}^{k}(0)\Subset \mathbb{C}^{k}$ with polyradius $R=(R_{1},\ldots ,R_{k})\in (0,\infty ]^{k}$ and center $0$ there exists a plurisubharmonic function $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D713}_{R,r}\in \mathscr{C}^{2}(D_{R}^{k}(0)\times D_{r}(0))$ and a holomorphic function $g=g_{R,r}\in {\mathcal{O}}(D_{R}^{k}(0)\times D_{r}(0))$ satisfying

$$\begin{eqnarray}\unicode[STIX]{x1D711}=m|\cdot |^{2}+\unicode[STIX]{x1D713}+2\operatorname{Re}g\end{eqnarray}$$

and

$$\begin{eqnarray}\sup _{D_{R}^{k}(0)\times D_{r}(0)}|\unicode[STIX]{x1D713}|+|d\unicode[STIX]{x1D713}|\leqslant C\cdot (M-m)\left(r\log \frac{e}{r}+r\right),\end{eqnarray}$$

where $C$ is a universal constant independent of the weight $\unicode[STIX]{x1D711}$, the radius $r$ and the polyradius $R$.

Proof. Let $\unicode[STIX]{x1D714}=dd^{c}(\unicode[STIX]{x1D711}-m|\cdot |^{2})$. Then $0\leqslant \unicode[STIX]{x1D714}\leqslant (M-m)\unicode[STIX]{x1D714}_{o}$. Suppose $\unicode[STIX]{x1D712}:\mathbb{R}\rightarrow \mathbb{R}_{{\geqslant}0}$ is a smooth function equal to 1 on $[-1,1]$ and 0 outside $(-2,2)$. Now define $\unicode[STIX]{x1D713}:D_{R}^{k}(0)\times D_{r}(0)\rightarrow \mathbb{R}$ by the formula

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}(z^{1},\ldots ,z^{k+1}) & = & \displaystyle \frac{1}{\unicode[STIX]{x1D70B}}\int _{|\unicode[STIX]{x1D701}|<2r}\unicode[STIX]{x1D712}\left(\frac{\unicode[STIX]{x1D701}}{r}\right)\unicode[STIX]{x1D714}_{k+1\overline{k+1}}(z^{1},z^{2},\ldots ,z^{k},\unicode[STIX]{x1D701})\nonumber\\ \displaystyle & & \displaystyle \times \log |z^{k+1}-\unicode[STIX]{x1D701}|^{2}\,dA(\unicode[STIX]{x1D701}).\nonumber\end{eqnarray}$$

It is well-known that $dd^{c}\unicode[STIX]{x1D713}=\unicode[STIX]{x1D714}$ on $D_{R}^{k}(0)\times D_{r}(0)$. It follows that the function $\unicode[STIX]{x1D711}-m|\cdot |^{2}-\unicode[STIX]{x1D713}$ is pluriharmonic on the simply connected set $D_{R}(0)^{k}\times D_{r}(0)$, and thus equals $2\operatorname{Re}g$ for some $g\in {\mathcal{O}}(D_{R}(0)^{k}\times D_{r}(0))$. Hence in particular, $\unicode[STIX]{x1D713}\in \mathscr{C}^{2}(D_{R}^{k}(0)\times D_{r}(0))$.

The bound on $|\unicode[STIX]{x1D713}|$ follows from an obvious estimate. As for the bound on $|d\unicode[STIX]{x1D713}|$, notice that

$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}z^{k+1}}(z,z^{k+1})=\frac{1}{\unicode[STIX]{x1D70B}}\int _{|\unicode[STIX]{x1D701}|<2r}\unicode[STIX]{x1D712}\left(\frac{\unicode[STIX]{x1D701}}{r}\right)\unicode[STIX]{x1D714}_{k+1\overline{k+1}}(z^{1},z^{2},\ldots ,z^{k},\unicode[STIX]{x1D701})\frac{dA(\unicode[STIX]{x1D701})}{z^{k+1}-\unicode[STIX]{x1D701}},\end{eqnarray}$$

which is estimated using polar coordinates in $\unicode[STIX]{x1D701}$ centered at $z^{k+1}\in D_{r}(0)$. As for the other partial derivatives, for $1\leqslant j\leqslant k$ we have

$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}z^{j}}(z,z^{j}) & = & \displaystyle \frac{1}{\unicode[STIX]{x1D70B}}\int _{|\unicode[STIX]{x1D701}|<2r}\unicode[STIX]{x1D712}\left(\frac{\unicode[STIX]{x1D701}}{r}\right)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z^{j}}\unicode[STIX]{x1D714}_{k+1\overline{k+1}}(z^{1},z^{2},\ldots ,z^{k},\unicode[STIX]{x1D701})\nonumber\\ \displaystyle & & \displaystyle \times \log |z^{k+1}-\unicode[STIX]{x1D701}|^{2}\,dA(\unicode[STIX]{x1D701})\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{\unicode[STIX]{x1D70B}}\int _{|\unicode[STIX]{x1D701}|<2r}\unicode[STIX]{x1D712}\left(\frac{\unicode[STIX]{x1D701}}{r}\right)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}\unicode[STIX]{x1D714}_{j\overline{k+1}}(z^{1},z^{2},\ldots ,z^{k},\unicode[STIX]{x1D701})\nonumber\\ \displaystyle & & \displaystyle \times \log |z^{k+1}-\unicode[STIX]{x1D701}|^{2}\,dA(\unicode[STIX]{x1D701})\nonumber\\ \displaystyle & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D70B}}\int _{|\unicode[STIX]{x1D701}|<2r}\unicode[STIX]{x1D712}\left(\frac{\unicode[STIX]{x1D701}}{r}\right)\unicode[STIX]{x1D714}_{j\overline{k+1}}(z^{1},z^{2},\ldots ,z^{k},\unicode[STIX]{x1D701})\frac{dA(\unicode[STIX]{x1D701})}{z^{k+1}-\unicode[STIX]{x1D701}}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{\unicode[STIX]{x1D70B}}\int _{|\unicode[STIX]{x1D701}|<2r}\frac{1}{r}\unicode[STIX]{x1D712}^{\prime }\left(\frac{\unicode[STIX]{x1D701}}{r}\right)\unicode[STIX]{x1D714}_{j\overline{k+1}}(z^{1},z^{2},\ldots ,z^{k},\unicode[STIX]{x1D701})\nonumber\\ \displaystyle & & \displaystyle \times \log |z^{k+1}-\unicode[STIX]{x1D701}|^{2}\,dA(\unicode[STIX]{x1D701}),\nonumber\end{eqnarray}$$

where the second equality follows because $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}=0$ and the third equality is obtained via integration by parts. Since $\unicode[STIX]{x1D714}\leqslant (M-m)\unicode[STIX]{x1D714}_{0}$, $|\unicode[STIX]{x1D714}_{j\overline{k+1}}|\leqslant M-m$, and the proof is complete.◻

Proposition 3.3. Let $\unicode[STIX]{x1D711}\in \mathscr{C}^{2}(\mathbb{C}^{n})$ be a weight function such that

(6)$$\begin{eqnarray}-M\unicode[STIX]{x1D714}_{o}\leqslant dd^{c}\unicode[STIX]{x1D711}\leqslant M\unicode[STIX]{x1D714}_{o}\end{eqnarray}$$

for some positive constant $M$. Then for each $r>0$ there exists a constant $C_{r}$ depending on $r$ and $M$ such that if $F\in \mathscr{B}_{n}(\unicode[STIX]{x1D711})$ then for any $z\in \mathbb{C}^{n}$

(7)$$\begin{eqnarray}(|F|^{2}e^{-\unicode[STIX]{x1D711}})(z)\leqslant C_{r}\int _{B_{r}^{n}(z)}|F|^{2}e^{-\unicode[STIX]{x1D711}}\,dV\end{eqnarray}$$

and

(8)$$\begin{eqnarray}|d(|F|^{2}e^{-\unicode[STIX]{x1D711}})|(z)\leqslant C_{r}\int _{B_{r}^{n}(z)}|F|^{2}e^{-\unicode[STIX]{x1D711}}\,dV.\end{eqnarray}$$

Proof. By rescaling and translating, we may assume that $r=1$ and $z=0$. By Lemma 3.1 applied to the form $\unicode[STIX]{x1D714}=dd^{c}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}(0))=dd^{c}\unicode[STIX]{x1D711}$ there exists a function $\unicode[STIX]{x1D713}$ such that

$$\begin{eqnarray}dd^{c}\unicode[STIX]{x1D713}=dd^{c}\unicode[STIX]{x1D711}\qquad \text{and}\qquad \sup _{\mathbb{ B}}|\unicode[STIX]{x1D713}|+|d\unicode[STIX]{x1D713}|\leqslant C_{o}\end{eqnarray}$$

for some positive constant $C_{o}$. It follows from the equation that $\unicode[STIX]{x1D713}-\unicode[STIX]{x1D713}(0)-\unicode[STIX]{x1D711}+\unicode[STIX]{x1D711}(0)=2\operatorname{Re}G$ for some holomorphic function $G$ whose real part vanishes at 0. The imaginary part of $G$ can be chosen arbitrarily; for example we can take it to be $\int _{0}^{z}d^{c}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D711})$, where the integral is over any curve in $\mathbb{B}$ originating at $0$ and terminating at $z$. This choice yields the property $G(0)=0$. Thus we have

$$\begin{eqnarray}\sup _{\mathbb{B}}|\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}(0)+2\operatorname{Re}G|+|d(\unicode[STIX]{x1D711}+2\operatorname{Re}G)|\leqslant |\unicode[STIX]{x1D713}(0)|+\sup _{\mathbb{B}}|\unicode[STIX]{x1D713}|+|d\unicode[STIX]{x1D713}|\leqslant 2C_{o}.\end{eqnarray}$$

We therefore have

$$\begin{eqnarray}|F|^{2}e^{-\unicode[STIX]{x1D711}}=|Fe^{G}|^{2}e^{-\unicode[STIX]{x1D711}(0)}e^{-\unicode[STIX]{x1D711}+\unicode[STIX]{x1D711}(0)-2\operatorname{Re}G}.\end{eqnarray}$$

Since the last factor is bounded, it can be eliminated from consideration, and we are reduced to the unweighted case (for the holomorphic function $Fe^{G}$). The unweighted case is an elementary exercise in complex analysis (with a number of solutions), and is left to the reader.◻

Remark 3.4. Note that the proof of Proposition 3.3 yields a slightly more general fact: if $\unicode[STIX]{x1D6FA}\subset \mathbb{C}^{n}$ is an open set and $F\in {\mathcal{O}}(\unicode[STIX]{x1D6FA})$ satisfies

$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}|F|^{2}e^{-\unicode[STIX]{x1D711}}\,dV<+\infty\end{eqnarray}$$

where $\unicode[STIX]{x1D711}\in \mathscr{C}^{2}(\unicode[STIX]{x1D6FA})$ satisfies $-M\unicode[STIX]{x1D714}_{o}\leqslant dd^{c}\unicode[STIX]{x1D711}\leqslant M\unicode[STIX]{x1D714}_{o}$only in $\unicode[STIX]{x1D6FA}$, then (7) and (8) hold for any $z\in \unicode[STIX]{x1D6FA}$ and $r\in (0,\infty )$ such that $B_{r}(z)\subset \unicode[STIX]{x1D6FA}$.

3.2 Interpolation sequences in $\mathbb{C}$ are uniformly separated

In Section 2 we noted that interpolation sequences in Bargmann–Fock spaces are uniformly separated. Let us recall the proof from [Reference Ortega-Cerdà and SeipOS-1998].

Let $\unicode[STIX]{x1D711}$ be a Bargmann–Fock weight on $\mathbb{C}$ and let $\unicode[STIX]{x1D6E4}\subset \mathbb{C}$ be a closed discrete subset such that

$$\begin{eqnarray}\mathscr{R}_{\unicode[STIX]{x1D6E4}}:\mathscr{B}_{1}(\unicode[STIX]{x1D711})\rightarrow \mathfrak{B}_{0}(\unicode[STIX]{x1D6E4}):=\left\{f:\unicode[STIX]{x1D6E4}\rightarrow \mathbb{C};\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}}|f(\unicode[STIX]{x1D6FE})|^{2}e^{-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE})}<+\infty \right\}\end{eqnarray}$$

is surjective.

Now choose $\unicode[STIX]{x1D6FE}_{o},\unicode[STIX]{x1D6FE}_{1}\in \unicode[STIX]{x1D6E4}$ distinct. The function $f:\unicode[STIX]{x1D6E4}\ni \unicode[STIX]{x1D6FE}\mapsto e^{\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{o})/2}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FE}_{o},\unicode[STIX]{x1D6FE}}$ has norm

$$\begin{eqnarray}\Vert f\Vert ^{2}=|f(\unicode[STIX]{x1D6FE}_{o})|^{2}e^{-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{o})}=1.\end{eqnarray}$$

By Proposition 2.5 there exists $F\in \mathscr{B}_{1}(\unicode[STIX]{x1D711})$ such that $\Vert F\Vert ^{2}\leqslant C$ where $C$ is independent of $f$ (hence of $\unicode[STIX]{x1D6FE}_{o}$). It follows that

$$\begin{eqnarray}\displaystyle \frac{1}{|\unicode[STIX]{x1D6FE}_{o}-\unicode[STIX]{x1D6FE}_{1}|} & = & \displaystyle \left|\frac{|f(\unicode[STIX]{x1D6FE}_{o})|^{2}e^{-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{o})}-|f(\unicode[STIX]{x1D6FE}_{1})|^{2}e^{-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{1})}}{|\unicode[STIX]{x1D6FE}_{o}-\unicode[STIX]{x1D6FE}_{1}|}\right|\nonumber\\ \displaystyle & = & \displaystyle \left|\frac{1}{\unicode[STIX]{x1D6FE}_{1}-\unicode[STIX]{x1D6FE}_{o}}\int _{0}^{1}\frac{d}{dt}|F(\unicode[STIX]{x1D6FE}_{o}+t(\unicode[STIX]{x1D6FE}_{1}-\unicode[STIX]{x1D6FE}_{o}))|^{2}e^{-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{o}+t(\unicode[STIX]{x1D6FE}_{1}-\unicode[STIX]{x1D6FE}_{o}))}\,dt\right|\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \sup _{\mathbb{C}}|d(|F|^{2}e^{-\unicode[STIX]{x1D711}})|.\nonumber\end{eqnarray}$$

By (8) of Proposition 3.3, $|\unicode[STIX]{x1D6FE}_{o}-\unicode[STIX]{x1D6FE}_{1}|\gtrsim \Vert F\Vert ^{-2}\geqslant C^{-1}$, which is what we wanted to show.

4.1 The standard Bargmann–Fock space

4.1.1 Reduction

The strategy of our proof consists in seeking a function $f\in \mathfrak{B}_{1}(C_{1})$ for which any holomorphic extension would violate (8) of Proposition 3.3. (See Section 3.2 for the case of sequences in $\mathbb{C}$.) With this general goal in mind, let $T_{1}^{\pm }$, $T_{1}\in {\mathcal{O}}(\mathbb{C}^{2})$ be defined by

$$\begin{eqnarray}T_{1}^{\pm }(x,y)=xy\mp 1,\qquad T_{1}(x,y)=T_{1}^{-}(x,y)T_{1}^{+}(x,y)=x^{2}y^{2}-1,\end{eqnarray}$$

which cut out the curves $C_{1+}$, $C_{1-}$ and $C_{1}=C_{1+}\cup C_{1-}$:

$$\begin{eqnarray}C_{1\pm }:=\{T_{1\pm }=0\}\qquad \text{and}\qquad C_{1}=\{T_{1}=0\}.\end{eqnarray}$$

Then $C_{1+}\cap C_{1-}$  = Ø.

We shall construct a function $g_{\unicode[STIX]{x1D6FF}}\in {\mathcal{O}}(C_{1+})$ such that

(9)$$\begin{eqnarray}|g_{\unicode[STIX]{x1D6FF}}(\unicode[STIX]{x1D6FF}^{-1},\unicode[STIX]{x1D6FF})|e^{-(\unicode[STIX]{x1D6FF}^{-2}+\unicode[STIX]{x1D6FF}^{2})}\sim 1\qquad \text{and}\qquad \int _{C_{1+}}|g_{\unicode[STIX]{x1D6FF}}|^{2}e^{-|\cdot |^{2}}\unicode[STIX]{x1D714}_{o}\leqslant C/\sqrt{\unicode[STIX]{x1D6FF}}\end{eqnarray}$$

for some constant $C>0$ independent of $\unicode[STIX]{x1D6FF}$. Assuming for the moment that such a function has been found, if we define the function $f_{\unicode[STIX]{x1D6FF}}\in {\mathcal{O}}(C_{1})$ by

$$\begin{eqnarray}f_{\unicode[STIX]{x1D6FF}}(z)=\left\{\begin{array}{@{}ll@{}}g_{\unicode[STIX]{x1D6FF}}(z), & z\in C_{1+}\\ 0, & z\in C_{1-},\end{array}\right.\end{eqnarray}$$

then

$$\begin{eqnarray}\left|f_{\unicode[STIX]{x1D6FF}}\left(\frac{1}{\unicode[STIX]{x1D6FF}},\unicode[STIX]{x1D6FF}\right)\right|e^{-(1/\unicode[STIX]{x1D6FF}^{2}+\unicode[STIX]{x1D6FF}^{2})}\sim 1,\qquad \left|f_{\unicode[STIX]{x1D6FF}}\left(\frac{1}{\unicode[STIX]{x1D6FF}},-\unicode[STIX]{x1D6FF}\right)\right|^{-(1/\unicode[STIX]{x1D6FF}^{2}+\unicode[STIX]{x1D6FF}^{2})}=0\end{eqnarray}$$

and

$$\begin{eqnarray}\int _{C_{1}}|f_{\unicode[STIX]{x1D6FF}}|^{2}e^{-|\cdot |^{2}}\unicode[STIX]{x1D714}_{o}\leqslant C/\sqrt{\unicode[STIX]{x1D6FF}},\end{eqnarray}$$

and in particular $f_{\unicode[STIX]{x1D6FF}}\in \mathfrak{B}_{1}(C_{1})$. To prove Theorem 1 by contradiction, suppose there exists $F\in \mathscr{B}_{2}$ extending $f_{\unicode[STIX]{x1D6FF}}$. Since the square norm of $f_{\unicode[STIX]{x1D6FF}}$ is bounded by $C/\sqrt{\unicode[STIX]{x1D6FF}}$, Proposition 2.5 says one can find $F_{\unicode[STIX]{x1D6FF}}\in \mathscr{B}_{2}$ such that $\Vert F_{\unicode[STIX]{x1D6FF}}\Vert ^{2}\leqslant \tilde{C}/\sqrt{\unicode[STIX]{x1D6FF}}$ for some constant $\tilde{C}$ independent of $\unicode[STIX]{x1D6FF}$. But then by (8) of Proposition 3.3

$$\begin{eqnarray}\displaystyle \frac{1}{2\unicode[STIX]{x1D6FF}} & = & \displaystyle \frac{|F(\unicode[STIX]{x1D6FF}^{-1},\unicode[STIX]{x1D6FF})|^{2}e^{-(\unicode[STIX]{x1D6FF}^{2}+\unicode[STIX]{x1D6FF}^{-2})}-F(\unicode[STIX]{x1D6FF}^{-1},-\unicode[STIX]{x1D6FF})|^{2}e^{-(\unicode[STIX]{x1D6FF}^{2}+\unicode[STIX]{x1D6FF}^{-2})}}{2\unicode[STIX]{x1D6FF}}\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2\unicode[STIX]{x1D6FF}}\int _{-1}^{1}\frac{d}{ds}(|F_{\unicode[STIX]{x1D6FF}}(\unicode[STIX]{x1D6FF}^{-1},s\unicode[STIX]{x1D6FF})|^{2}e^{-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FF}^{-1},s\unicode[STIX]{x1D6FF})})\,ds\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \sup _{\mathbb{C}^{2}}|d(|F_{\unicode[STIX]{x1D6FF}}|^{2}e^{-\unicode[STIX]{x1D711}})|\leqslant {\hat{C}}/\sqrt{\unicode[STIX]{x1D6FF}},\nonumber\end{eqnarray}$$

where the constant ${\hat{C}}$ is independent of $\unicode[STIX]{x1D6FF}$. The desired contradiction is thus obtained by taking $\unicode[STIX]{x1D6FF}$ sufficiently small.

4.1.2 Conclusion of the proof in the standard Bargmann–Fock space

It remains only to produce the $g=g_{\unicode[STIX]{x1D6FF}}$ on $C_{1+}$ satisfying (9). We shall define a function close to $g_{\unicode[STIX]{x1D6FF}}$ on a large but finite open subset of $C_{1+}$, and then approximately extend this function to all of $C_{1+}$ using Hörmander’s theorem, thus obtaining $g_{\unicode[STIX]{x1D6FF}}$.

We work on $\mathbb{C}^{\ast }$, after using the parametrization

(10)$$\begin{eqnarray}\unicode[STIX]{x1D708}:\mathbb{C}^{\ast }\ni t\mapsto (t,t^{-1})\in C_{1+}\end{eqnarray}$$

of $C_{1+}$. Our $L^{2}$ norm is then

$$\begin{eqnarray}\int _{C_{1+}}|g|^{2}e^{-|\cdot |^{2}}\unicode[STIX]{x1D714}_{o}=\int _{\mathbb{C}^{\ast }}|f(t)|^{2}e^{-|t|^{2}-|t|^{-2}}(1+|t|^{-4})\,dA(t),\end{eqnarray}$$

where $f=\unicode[STIX]{x1D708}^{\ast }g$. Note that for the weight $\unicode[STIX]{x1D711}_{o}(t):=|t|^{2}+|t|^{-2}-\log (1+|t|^{-4})$,

$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}_{o}}{\unicode[STIX]{x2202}t\unicode[STIX]{x2202}\bar{t}}=1+|t|^{-4}-\frac{4|t|^{2}}{(1+|t|^{4})^{2}}=\frac{|t|^{12}+3|t|^{4}(|t|^{2}-1)^{2}+2|t|^{6}+1}{|t|^{4}(1+|t|^{4})^{2}},\end{eqnarray}$$

which is positive, $\rightarrow 1$ as $|t|\rightarrow \infty$ and $\rightarrow \infty$ as $|t|\rightarrow 0$. Thus

$$\begin{eqnarray}dd^{c}\unicode[STIX]{x1D711}_{o}\geqslant c_{o}dd^{c}|t|^{2}\end{eqnarray}$$

for some positive constant $c_{o}$. Moreover, there exists a compact subset $K\Subset \mathbb{C}$ (necessarily containing the origin) such that

(11)$$\begin{eqnarray}dd^{c}\unicode[STIX]{x1D711}_{o}\leqslant 2dd^{c}|t|^{2}\quad \text{for}~t\in \mathbb{C}^{\ast }-K.\end{eqnarray}$$

Now fix $\unicode[STIX]{x1D6FF}\in (0,1)$, keeping in mind that we let $\unicode[STIX]{x1D6FF}\rightarrow 0$. To find a function $f$ such that $|f(1/\unicode[STIX]{x1D6FF})|^{2}e^{-\unicode[STIX]{x1D6FF}^{2}}e^{-\unicode[STIX]{x1D6FF}^{-2}}\sim 1$ for some $\unicode[STIX]{x1D6FF}\ll 1$, one need only worry about the factor $e^{-\unicode[STIX]{x1D6FF}^{-2}}$, which is extremely small. A natural choice is

$$\begin{eqnarray}f_{o}(t)=e^{t^{2}/2},\end{eqnarray}$$

which satisfies $|f_{o}(1/\unicode[STIX]{x1D6FF})|^{2}e^{-\unicode[STIX]{x1D6FF}^{2}}e^{-\unicode[STIX]{x1D6FF}^{-2}}=e^{-\unicode[STIX]{x1D6FF}^{2}}$. Unfortunately the function

$$\begin{eqnarray}|f_{o}(t)|^{2}e^{-\unicode[STIX]{x1D711}_{o}(t)}=e^{\operatorname{Re}t^{2}-|t|^{2}-|t|^{-2}}(1+|t|^{-4})=e^{-2(\operatorname{Im}t)^{2}}e^{-|t|^{-2}}(1+|t|^{-4}),\end{eqnarray}$$

while locally integrable near the origin in $\mathbb{C}$, is not integrable in a neighborhood of $\{\infty \}$, where it is asymptotically $e^{-2(\operatorname{Im}t)^{2}}$. Thus we are going to take the function $\unicode[STIX]{x1D712}f_{o}$, where $\unicode[STIX]{x1D712}$ is a cut-off function to be described shortly, and then correct this function using Hörmander’s theorem.

Consider the vertical strip

(12)$$\begin{eqnarray}S_{\unicode[STIX]{x1D6FF}}:=\left\{t\in \mathbb{C};\left|\operatorname{Re}t-\frac{1}{\unicode[STIX]{x1D6FF}}\right|\leqslant \frac{3}{4\sqrt{\unicode[STIX]{x1D6FF}}}\right\}\subset \mathbb{C}^{\ast }.\end{eqnarray}$$

Take a function $\unicode[STIX]{x1D712}\in \mathscr{C}_{o}^{\infty }(S_{\unicode[STIX]{x1D6FF}})$ such that

$$\begin{eqnarray}0\leqslant \unicode[STIX]{x1D712}\leqslant 1,\qquad \unicode[STIX]{x1D712}(t)\equiv 1~\text{for}~\left|\operatorname{Re}t-\frac{1}{\unicode[STIX]{x1D6FF}}\right|\leqslant \frac{1}{2\sqrt{\unicode[STIX]{x1D6FF}}}\qquad \text{and}\qquad \sup _{S_{\unicode[STIX]{x1D6FF}}}|d\unicode[STIX]{x1D712}|\leqslant 5\sqrt{\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

(Such a function can be chosen to depend only on $\operatorname{Re}t$, for instance.)

The function $\unicode[STIX]{x1D712}f_{o}$ satisfies

$$\begin{eqnarray}\int _{\mathbb{C}^{\ast }}|\unicode[STIX]{x1D712}f_{o}|^{2}e^{-\unicode[STIX]{x1D711}_{o}}\,dA\leqslant C\unicode[STIX]{x1D6FF}^{-1/2}\end{eqnarray}$$

for some constant $C$ that does not depend on $\unicode[STIX]{x1D6FF}$. Moreover,

$$\begin{eqnarray}\int _{\mathbb{C}^{\ast }}|\bar{\unicode[STIX]{x2202}}\unicode[STIX]{x1D712}f_{o}|^{2}e^{-\unicode[STIX]{x1D711}_{o}}\,dA\leqslant \sup |d\unicode[STIX]{x1D712}|^{2}\int _{\text{Supp}(\unicode[STIX]{x1D712})}|f_{o}|^{2}e^{-\unicode[STIX]{x1D711}_{o}}\,dA\leqslant C\sqrt{\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

By Hörmander’s theorem there exists a function $u$ such that

$$\begin{eqnarray}\bar{\unicode[STIX]{x2202}}u=\bar{\unicode[STIX]{x2202}}\unicode[STIX]{x1D712}f_{o}\qquad \text{and}\qquad \int _{\mathbb{C}^{\ast }}|u|^{2}e^{-\unicode[STIX]{x1D711}_{o}}\,dA\leqslant A\sqrt{\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

Note in particular that

$$\begin{eqnarray}u\in {\mathcal{O}}(\{t;|\operatorname{Re}t-\unicode[STIX]{x1D6FF}^{-1}|<1/(2\sqrt{\unicode[STIX]{x1D6FF}})\}).\end{eqnarray}$$

Moreover, if $\unicode[STIX]{x1D6FF}$ is small enough then by (11) and the proof of Proposition 3.3 (cf. Remark 3.4)

$$\begin{eqnarray}|u(\unicode[STIX]{x1D6FF}^{-1})|^{2}e^{-\unicode[STIX]{x1D6FF}^{2}-\unicode[STIX]{x1D6FF}^{-2}}=|u(\unicode[STIX]{x1D6FF}^{-1})|^{2}e^{-\unicode[STIX]{x1D711}_{o}(\unicode[STIX]{x1D6FF}^{-1})}(1+\unicode[STIX]{x1D6FF}^{4})^{-1}\leqslant C\sqrt{\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

(This estimate is established by estimating $|u(\unicode[STIX]{x1D6FF}^{-1})|^{2}e^{-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FF}^{-1})}$ by its $L^{2}$ norm over the disk of radius $1$ and center $1/\unicode[STIX]{x1D6FF}$ using the QuimBo Trick.) Hence the function

$$\begin{eqnarray}f:=\unicode[STIX]{x1D712}f_{o}-u\end{eqnarray}$$

satisfies

$$\begin{eqnarray}\int _{\mathbb{C}^{\ast }}|f|^{2}e^{-\unicode[STIX]{x1D711}_{o}}\lesssim \unicode[STIX]{x1D6FF}^{-1/2}\qquad \text{and}\qquad |f(\unicode[STIX]{x1D6FF}^{-1})|^{2}e^{-\unicode[STIX]{x1D6FF}^{2}-\unicode[STIX]{x1D6FF}^{-2}}\sim (1+\sqrt{\unicode[STIX]{x1D6FF}})e^{-\unicode[STIX]{x1D6FF}^{2}}\sim 1.\end{eqnarray}$$

Letting

$$\begin{eqnarray}g_{\unicode[STIX]{x1D6FF}}(t,t^{-1}):=f(t)\end{eqnarray}$$

provides the function satisfying (9), and hence proves Theorem 1 in the standard case.◻

4.2 The general case

The passage to the general case involves using the QuimBo Trick in the form of Lemma 3.2 to reduce to a situation that is very similar to the standard case. In particular, we will be brief when stating estimates in this setting that are very similar to those of the standard case.

First we normalize the weight $\unicode[STIX]{x1D711}$ in the bidisk

$$\begin{eqnarray}D_{\unicode[STIX]{x1D6FF}}^{2}:=\{(x,y)\in \mathbb{C}^{2};|x|<2/\unicode[STIX]{x1D6FF},|y|<1\}\end{eqnarray}$$

via Lemma 3.2. Thus we have functions $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FF}}\in \mathscr{C}^{2}(D_{\unicode[STIX]{x1D6FF}}^{2})$ and $h_{\unicode[STIX]{x1D6FF}}\in {\mathcal{O}}(D_{\unicode[STIX]{x1D6FF}}^{2})$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D711}=m|\cdot |^{2}+\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FF}}+2\operatorname{Re}h_{\unicode[STIX]{x1D6FF}}\qquad \text{and}\qquad \Vert \unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FF}}\Vert _{\mathscr{ C}^{1}}\leqslant C\end{eqnarray}$$

for some constant $C$ independent of $\unicode[STIX]{x1D6FF}$. In particular, this relation holds on

$$\begin{eqnarray}\tilde{S}_{\unicode[STIX]{x1D6FF}}\subset D_{\unicode[STIX]{x1D6FF}}^{2},\end{eqnarray}$$

where (compare (12))

$$\begin{eqnarray}\tilde{S}_{\unicode[STIX]{x1D6FF}}:=\unicode[STIX]{x1D708}(S_{\unicode[STIX]{x1D6FF}})=\left\{(t,1/t);\left|\operatorname{Re}t-\frac{1}{\unicode[STIX]{x1D6FF}}\right|\leqslant \frac{3}{4\sqrt{\unicode[STIX]{x1D6FF}}}\right\}.\end{eqnarray}$$

Again, pulling back by $\unicode[STIX]{x1D708}$, we work on $\mathbb{C}^{\ast }$, where the $L^{2}$ norm is

$$\begin{eqnarray}\int _{\mathbb{C}^{\ast }}|f(t)|^{2}e^{-\unicode[STIX]{x1D711}(t,t^{-1})}(1+|t|^{-4})\,dA(t).\end{eqnarray}$$

This time, however, the weight $\unicode[STIX]{x1D711}_{o}(t)=\unicode[STIX]{x1D711}(t,t^{-1})-\log (1+|t|^{-4})$ could fail to be positively curved if $m$ is sufficiently small. We therefore need to choose a weight for which Hörmander’s theorem can be applied, and that still provides the right estimates. With this in mind, we let

$$\begin{eqnarray}\unicode[STIX]{x1D702}(t):=\unicode[STIX]{x1D711}(t,t^{-1})-\frac{m}{2}|t|^{-2}.\end{eqnarray}$$

Then

$$\begin{eqnarray}dd^{c}\unicode[STIX]{x1D702}(t)\geqslant \frac{m}{2}\unicode[STIX]{x1D708}^{\ast }\unicode[STIX]{x1D714}_{o}\geqslant \frac{m}{2}dd^{c}|t|^{2}\end{eqnarray}$$

and

$$\begin{eqnarray}e^{-\unicode[STIX]{x1D711}(t,t^{-1})}(1+|t|^{-4})=e^{-\unicode[STIX]{x1D702}(t)}(1+|t|^{-4})e^{-(m/2)|t|^{-2}}\leqslant C_{m}e^{-\unicode[STIX]{x1D702}(t)}\end{eqnarray}$$

for some constant $C_{m}$.

Now let $\unicode[STIX]{x1D712}\in \mathscr{C}_{o}^{\infty }([0,3/4))$ have the property that $0\leqslant \unicode[STIX]{x1D712}\leqslant 1$, $\unicode[STIX]{x1D712}(r)\equiv 1$ for $0\leqslant r\leqslant 1/2$ and $|\unicode[STIX]{x1D712}^{\prime }|\leqslant 5$. Define

$$\begin{eqnarray}\tilde{f}(t):=\unicode[STIX]{x1D712}(\sqrt{\unicode[STIX]{x1D6FF}}|\operatorname{Re}t-\unicode[STIX]{x1D6FF}|)\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FF}|\operatorname{Im}t|)e^{g_{\unicode[STIX]{x1D6FF}}(t,t^{-1})+mt^{2}/2}.\end{eqnarray}$$

Then by Lemma 3.2,

$$\begin{eqnarray}\int _{\mathbb{C}^{\ast }}|\tilde{f}|^{2}e^{-\unicode[STIX]{x1D711}_{o}}\,dA\lesssim \int _{\tilde{S}_{\unicode[STIX]{x1D6FF}}}e^{m(\operatorname{Re}t^{2}-|t|^{2})-(m/2)|t|^{-2}}\,dA(t)\lesssim \unicode[STIX]{x1D6FF}^{-1/2},\end{eqnarray}$$

where the last estimate is proved as in the standard case. Also, since

$$\begin{eqnarray}|\bar{\unicode[STIX]{x2202}}(\unicode[STIX]{x1D712}(\sqrt{\unicode[STIX]{x1D6FF}}|\operatorname{Re}t-\unicode[STIX]{x1D6FF}|)\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FF}|\operatorname{Im}t|))|^{2}\lesssim \unicode[STIX]{x1D6FF}\cdot \mathbf{1}_{\tilde{S}_{\unicode[STIX]{x1D6FF}}},\end{eqnarray}$$

we have

$$\begin{eqnarray}\int _{\mathbb{C}^{\ast }}|\bar{\unicode[STIX]{x2202}}\tilde{f}|^{2}e^{-\unicode[STIX]{x1D702}}\,dA\lesssim \unicode[STIX]{x1D6FF}\int _{\tilde{S}_{\unicode[STIX]{x1D6FF}}}e^{m(\operatorname{Re}t^{2}-|t|^{2})-(m/2)|t|^{-2}}\,dA(t)\lesssim \unicode[STIX]{x1D6FF}^{1/2}.\end{eqnarray}$$

By Hörmander’s theorem there exists a smooth function $u$ such that

$$\begin{eqnarray}\bar{\unicode[STIX]{x2202}}u=\bar{\unicode[STIX]{x2202}}\tilde{f}\qquad \text{and}\qquad \int _{\mathbb{C}^{\ast }}|u|^{2}e^{-\unicode[STIX]{x1D711}_{o}}\,dA\lesssim \int _{\mathbb{C}^{\ast }}|u|^{2}e^{-\unicode[STIX]{x1D702}}\,dA=O(\unicode[STIX]{x1D6FF}^{1/2}).\end{eqnarray}$$

Since $u$ is holomorphic in a neighborhood of $\unicode[STIX]{x1D6FF}^{-1}$, as in the standard case we have (via Remark 3.4)

$$\begin{eqnarray}|u(\unicode[STIX]{x1D6FF}^{-1})|^{2}e^{-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FF}^{-1},\unicode[STIX]{x1D6FF})}\lesssim \unicode[STIX]{x1D6FF}^{1/2}.\end{eqnarray}$$

It follows that the function

$$\begin{eqnarray}f(t,t^{-1}):=\unicode[STIX]{x1D712}(\sqrt{\unicode[STIX]{x1D6FF}}|\operatorname{Re}t-\unicode[STIX]{x1D6FF}|)\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FF}|\operatorname{Im}t|)e^{g_{\unicode[STIX]{x1D6FF}}(t,t^{-1})+mt^{2}/2}-u(t)\end{eqnarray}$$

is holomorphic and satisfies

$$\begin{eqnarray}|f(\unicode[STIX]{x1D6FF}^{-1},\unicode[STIX]{x1D6FF})|^{2}e^{-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FF}^{-1},\unicode[STIX]{x1D6FF})}\sim 1\qquad \text{and}\qquad \int _{C_{+}}|f|^{2}e^{-\unicode[STIX]{x1D711}}\unicode[STIX]{x1D714}_{o}\lesssim \unicode[STIX]{x1D6FF}^{-1/2}.\end{eqnarray}$$

As in the standard case, $f$ has no extension in $\mathscr{B}_{2}(\unicode[STIX]{x1D711})$ as soon as $\unicode[STIX]{x1D6FF}$ is small enough. The proof of Theorem 1 is complete.◻

5.1 Asymptotics of the norm from the $L^{2}$ extension theorem

Consider the $L^{2}$ Extension Theorem 2.6 in the following situation. We take $X=\mathbb{C}^{3},Z=S$, $T(x,y,z):=z-xy^{2}$, $\unicode[STIX]{x1D711}$ as in the hypotheses of Theorem 3, that is, satisfying (1), $s=1$ and $\unicode[STIX]{x1D706}$ defined by

(13)$$\begin{eqnarray}\unicode[STIX]{x1D706}(x,y,z):=\frac{1}{\text{Vol}(B_{R}(0))}\int _{B_{R}(x,y,z)}\log |T(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})|^{2}\,dV(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D701}).\end{eqnarray}$$

Notice that $\unicode[STIX]{x1D706}$ is plurisubharmonic, that is, $dd^{c}\unicode[STIX]{x1D706}\geqslant 0$. As we saw in the proof of Proposition 2.2, given any $\unicode[STIX]{x1D700}>0$, there exists $R$ sufficiently large such that

$$\begin{eqnarray}dd^{c}\unicode[STIX]{x1D706}\leqslant \unicode[STIX]{x1D700}\unicode[STIX]{x1D714}_{o}.\end{eqnarray}$$

Hence the curvature hypothesis (4) of Theorem 2.6 is satisfied if we take $R\gg 1$. We fix $R$ from here on.

It follows that for any $f\in {\mathcal{O}}(S)$ satisfying

(14)$$\begin{eqnarray}\int _{S}\frac{|f|^{2}e^{-\unicode[STIX]{x1D711}}}{|dT|_{\unicode[STIX]{x1D714}_{o}}^{2}e^{-\unicode[STIX]{x1D706}}}\unicode[STIX]{x1D714}_{o}^{2}<+\infty\end{eqnarray}$$

there exists $F\in \mathscr{B}_{3}(\unicode[STIX]{x1D711})$ such that $F|_{S}=f$. (There is also an estimate for the $\mathscr{B}_{3}(\unicode[STIX]{x1D711})$-norm of $F$ in terms of the norm (14) of $f$, but this estimate is not important to us.)

Now, $dT=dz-y^{2}\,dx-2xy\,dy$, so

$$\begin{eqnarray}|dT|_{\unicode[STIX]{x1D714}_{o}}^{2}=1+4|xy|^{2}+|y^{2}|^{2}.\end{eqnarray}$$

Lemma 5.1. There exists $C>0$ such that

$$\begin{eqnarray}\displaystyle \frac{1}{C}(1+|x|^{2}+|y|^{2}+4|xy|^{2}+|y|^{4}) & {\leqslant} & \displaystyle e^{\unicode[STIX]{x1D706}(x,y,xy^{2})}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle C(1+|x|^{2}+|y|^{2}+4|xy|^{2}+|y|^{4}).\nonumber\end{eqnarray}$$

Proof. We have

$$\begin{eqnarray}\displaystyle & & \displaystyle \!\!\!\unicode[STIX]{x1D706}(x,y,xy^{2})\nonumber\\ \displaystyle & & \displaystyle \!\!\!\!\!\quad =\frac{6}{\unicode[STIX]{x1D70B}^{3}R^{6}}\!\int _{B_{R}(0)}\!\log |\unicode[STIX]{x1D701}+xy^{2}-(\unicode[STIX]{x1D709}+x)(\unicode[STIX]{x1D702}+y)^{2}|^{2}\,dV(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})\nonumber\\ \displaystyle & & \displaystyle \!\!\!\!\!\quad =\frac{6}{\unicode[STIX]{x1D70B}^{3}R^{6}}\!\int _{B_{R}(0)}\!\log |\unicode[STIX]{x1D701}-\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}^{2}-(x\cdot \unicode[STIX]{x1D702}^{2}+y\cdot 2\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}+xy\cdot 2\unicode[STIX]{x1D702}+y^{2\!}\cdot \unicode[STIX]{x1D709})|^{2}\,dV(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})\nonumber\\ \displaystyle & & \displaystyle \!\!\!\!\!\quad \leqslant \frac{6}{\unicode[STIX]{x1D70B}^{3}R^{6}}\!\int _{B_{R}(0)}\!\log (|\unicode[STIX]{x1D701}|+|\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}^{2}|+|(\unicode[STIX]{x1D702}^{2},2\unicode[STIX]{x1D709}\unicode[STIX]{x1D702},2\unicode[STIX]{x1D702},\unicode[STIX]{x1D709})\cdot (x,y,xy,y^{2})|)^{2}\,dV(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})\nonumber\\ \displaystyle & & \displaystyle \!\!\!\!\!\quad \leqslant \frac{6}{\unicode[STIX]{x1D70B}^{3}R^{6}}\!\int _{B_{R}(0)}\!\log \left(2R^{3}+2R^{2}\sqrt{|x|^{2}+|y|^{2}+4|xy|^{2}+|y^{2}|^{2}}\right)^{2}dV(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})\nonumber\\ \displaystyle & & \displaystyle \!\!\!\!\!\quad =\log (|x|^{2}+|y|^{2}+4|xy|^{2}+|y^{2}|^{2})\nonumber\\ \displaystyle & & \displaystyle \!\!\!\!\!\qquad +\,\frac{6}{\unicode[STIX]{x1D70B}^{3}R^{6}}\!\int _{B_{R}(0)}\!\log \left(\frac{2R^{3}}{\sqrt{|x|^{2}+|y|^{2}+4|xy|^{2}+|y^{2}|^{2}}}+2R^{2}\right)^{2}dV(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D701}).\nonumber\end{eqnarray}$$

Thus if $(x,y)$ is outside a ball of radius $R$ then the second term is small. For $(x,y,xy^{2})$ such that $(x,y)$ is inside the ball of radius $R$, $\unicode[STIX]{x1D706}$ is already bounded.

To obtain the opposite inequality, one proceeds as follows. First, note that

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}(x,y,xy^{2}) & {\geqslant} & \displaystyle \frac{6}{\unicode[STIX]{x1D70B}^{3}R^{6}}\int _{B_{R}(0)}\log (|(x\cdot \unicode[STIX]{x1D702}^{2}+y\cdot 2\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}+xy\cdot 2\unicode[STIX]{x1D702}+y^{2}\cdot \unicode[STIX]{x1D709})|\nonumber\\ \displaystyle & & \displaystyle -\,|\unicode[STIX]{x1D701}-\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}^{2}|)^{2}\,dV(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D701}).\nonumber\end{eqnarray}$$

The region of $B_{R}(0)$ where

$$\begin{eqnarray}(|(x,y,2xy,y^{2})\cdot (\unicode[STIX]{x1D702}^{2},2\unicode[STIX]{x1D709}\unicode[STIX]{x1D702},\unicode[STIX]{x1D702},\unicode[STIX]{x1D709})|-|\unicode[STIX]{x1D701}-\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}^{2}|)^{2}<1\end{eqnarray}$$

is of no concern to us, since $\log |t|$ is locally integrable near $t=0$. Thus we may focus on the region $B_{+}\subset B_{R}(0)$ where $(|(x,y,2xy,y^{2})\cdot (\unicode[STIX]{x1D702}^{2},2\unicode[STIX]{x1D709}\unicode[STIX]{x1D702},\unicode[STIX]{x1D702},\unicode[STIX]{x1D709})|-|\unicode[STIX]{x1D701}-\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}^{2}|)^{2}\geqslant 1$ at the cost of subtracting some possibly large but fixed constant (depending only on $R$). We may also assume that $|x|^{2}+|y|^{2}\geqslant R^{7}$. The aforementioned region contains the set

$$\begin{eqnarray}\displaystyle \mathfrak{C}_{R}(x,y) & := & \displaystyle \left\{(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})\in B_{R}(0);|(x,y,2xy,y^{2})\cdot (\unicode[STIX]{x1D702}^{2},2\unicode[STIX]{x1D709}\unicode[STIX]{x1D702},\unicode[STIX]{x1D702},\unicode[STIX]{x1D709})|\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \left.{\geqslant}{\textstyle \frac{1}{2}}\Vert (x,y,2xy,y^{2})\Vert \right\},\nonumber\end{eqnarray}$$

and $\mathfrak{C}_{R}(x,y)$ has positive volume in $B_{R}(0)$ uniformly in $R$. It follows that

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}(x,y,xy^{2}) & {\geqslant} & \displaystyle M_{R}\int _{\mathfrak{C}_{R}(x,y)}\log \left(\frac{1}{2}\Vert (x,y,2xy,y^{2})\Vert -|\unicode[STIX]{x1D701}-\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}^{2}|\right)^{2}dV(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})\nonumber\\ \displaystyle & = & \displaystyle M_{R}\,\text{Vol}(\mathfrak{C}_{R}(x,y))\log (1+|x|^{2}+|y|^{2}+4|xy|^{2}+|y|^{4})\nonumber\\ \displaystyle & & \displaystyle +\,M_{R}\int _{\mathfrak{C}(x,y)}\log \left(\frac{1}{2}-\frac{1+2|\unicode[STIX]{x1D701}-\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}^{2}|}{\sqrt{1+\Vert (x,y,2xy,y^{2})\Vert ^{2}}}\right)^{2}dV(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})\nonumber\\ \displaystyle & {\geqslant} & \displaystyle C_{o}\log (1+|x|^{2}+|y|^{2}+4|xy|^{2}+|y|^{4}),\nonumber\end{eqnarray}$$

and the proof is complete. ◻

By Lemma 5.1 we have

$$\begin{eqnarray}\int _{S}\frac{|f|^{2}e^{-\unicode[STIX]{x1D711}}}{|dT|_{\unicode[STIX]{x1D714}_{o}}^{2}e^{-\unicode[STIX]{x1D706}}}\unicode[STIX]{x1D714}_{o}^{2}\leqslant \int _{S}\frac{|f|^{2}e^{-\unicode[STIX]{x1D711}}}{(1+4|xy|^{2}+|y|^{4})}(1+|x|^{2}+|y|^{2}+4|xy|^{2}+|y|^{4})\unicode[STIX]{x1D714}_{o}^{2}.\end{eqnarray}$$

If we now use our parametrization $\unicode[STIX]{x1D6F9}_{2}:\mathbb{C}\ni (x,y)\mapsto (x,y,xy^{2})\in S$, we find that

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{2}^{\ast }\frac{\unicode[STIX]{x1D714}_{o}^{2}}{2}=(1+4|xy|^{2}+|y|^{4})\,dV(x,y),\end{eqnarray}$$

and therefore

(15)$$\begin{eqnarray}\displaystyle \int _{S}\frac{|f|^{2}e^{-\unicode[STIX]{x1D711}}}{|dT|_{\unicode[STIX]{x1D714}_{o}}^{2}e^{-\unicode[STIX]{x1D706}}}\unicode[STIX]{x1D714}_{o}^{2} & {\leqslant} & \displaystyle 2\int _{\mathbb{C}^{2}}|f(x,y,xy^{2})|^{2}e^{-\unicode[STIX]{x1D711}(x,y,xy^{2})}\nonumber\\ \displaystyle & & \displaystyle \times \,(1+|x|^{2}+|y|^{2}+4|xy|^{2}+|y|^{4})\,dV(x,y).\end{eqnarray}$$

The norm on the right hand side of (15) is larger than the $\mathfrak{H}_{2}(S,\unicode[STIX]{x1D711})$-norm, obtained when the weight factor $(1+|x|^{2}+|y|^{2}+4|xy|^{2}+|y|^{4})$ is replaced by the weight factor $(1+4|xy|^{2}+|y|^{4})$. The factors become incomparable if $y=0$ and $x$ is very large. Of course, uniform flatness fails only on a neighborhood of the line

$$\begin{eqnarray}L_{o}:=\{(x,y,z)\in S;y=0\}\end{eqnarray}$$

near $\infty$. More precisely, for any $\unicode[STIX]{x1D6FF}>0$ the set

$$\begin{eqnarray}S_{\unicode[STIX]{x1D6FF}}:=S\cap \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}\end{eqnarray}$$

is uniformly flat in $\mathbb{C}^{3}$, where

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}=\{(x,y,z)\in \mathbb{C}^{3};|y|>\unicode[STIX]{x1D6FF}\}.\end{eqnarray}$$

Note that $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}$, being a product of pseudoconvex domains, is pseudoconvex (and hence the $L^{2}$-extension theorems apply to it).

5.2 Reduction

For any $f\in \mathfrak{B}_{2}(S,\unicode[STIX]{x1D711})$ Proposition 5.2 yields a function $H\in \mathscr{B}_{3}(\unicode[STIX]{x1D711})$ such that $H|_{L_{o}}=f|_{L_{o}}$. It follows that the function $g:=f-H|_{S}\in {\mathcal{O}}(S)$ vanishes along $L_{o}$, and by Proposition 5.3 $g\in \mathfrak{B}_{2}(S,\unicode[STIX]{x1D711})$. Thus it suffices to extend functions $g\in \mathfrak{B}_{2}(S,\unicode[STIX]{x1D711})$ that vanish along $L_{o}$.

5.3 Extension of functions in $\mathfrak{B}_{2}(S,\unicode[STIX]{x1D711})$ that vanish along $L_{o}$

5.3.1 Extension away from $L_{o}$

We can apply the $L^{2}$ Extension Theorem 2.6 with $X=\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}$, $Z=S_{\unicode[STIX]{x1D6FF}}$, and with $T$, $s=1$, $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D706}$ as in Section 5.1. In $S_{\unicode[STIX]{x1D6FF}}$ we have

$$\begin{eqnarray}|dT|_{\unicode[STIX]{x1D714}_{o}}^{2}e^{-\unicode[STIX]{x1D706}}\geqslant C\frac{1}{1+\frac{|x|^{2}+|y|^{2}}{1+4|xy|^{2}+|y|^{4}}}\geqslant C\frac{1}{1+\frac{4|x|^{2}+|y|^{2}}{1+\unicode[STIX]{x1D6FF}^{2}(4|x|^{2}+|y|^{2})}}\geqslant \frac{C\unicode[STIX]{x1D6FF}^{2}}{1+\unicode[STIX]{x1D6FF}^{2}}.\end{eqnarray}$$

Hence for any $f\in \mathfrak{B}_{2}(S,\unicode[STIX]{x1D711})$ one has

$$\begin{eqnarray}\int _{S_{\unicode[STIX]{x1D6FF}}}\frac{|f|^{2}e^{-\unicode[STIX]{x1D711}}}{|dT|_{\unicode[STIX]{x1D714}_{o}}^{2}e^{-\unicode[STIX]{x1D706}}}\unicode[STIX]{x1D714}_{o}^{2}\leqslant C_{o}\unicode[STIX]{x1D6FF}^{-2}\int _{S_{\unicode[STIX]{x1D6FF}}}|f|^{2}e^{-\unicode[STIX]{x1D711}}\unicode[STIX]{x1D714}_{o}^{2}.\end{eqnarray}$$

So we find a function $F_{\unicode[STIX]{x1D6FF}}\in {\mathcal{O}}(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}})$ such that

$$\begin{eqnarray}F_{\unicode[STIX]{x1D6FF}}|_{S_{\unicode[STIX]{x1D6FF}}}=f|_{S_{\unicode[STIX]{x1D6FF}}}\qquad \text{and}\qquad \int _{\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}}|F_{\unicode[STIX]{x1D6FF}}|^{2}e^{-\unicode[STIX]{x1D711}}\,dV<+\infty .\end{eqnarray}$$

This argument applies to all $f\in \mathfrak{B}_{2}(S,\unicode[STIX]{x1D711})$, and not just those $f$ that vanish along $L_{o}$.

5.3.2 Division by $y$ in $\mathfrak{B}_{2}(S,\unicode[STIX]{x1D711})$

Consider the domain

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}:=\{(x,y,z)\in \mathbb{C}^{3};|y|<2\unicode[STIX]{x1D6FF}\}\end{eqnarray}$$

and the surface $\widetilde{S}_{\unicode[STIX]{x1D6FF}}:=S\cap \widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}$.

Lemma 5.4. Fix $\unicode[STIX]{x1D6FF}>0$. If $h\in {\mathcal{O}}(S)$ and $yh\in \mathfrak{B}_{2}(S,\unicode[STIX]{x1D711})$ then $h\in \mathfrak{B}_{2}(S,\unicode[STIX]{x1D711})$.

Proof. We work in $\mathbb{C}^{2}$ after pulling back by the embedding $\unicode[STIX]{x1D6F7}$. Then $f\in \mathfrak{B}_{2}(S,\unicode[STIX]{x1D711})$ means that the $L^{2}$ norm

$$\begin{eqnarray}\int _{\mathbb{C}^{2}}|f(x,y)|^{2}e^{-\unicode[STIX]{x1D711}(x,y,xy^{2})}(1+4|xy|^{2}+|y|^{4})\,dV(x,y)\end{eqnarray}$$

is finite. Thus the hypothesis $yh\in \mathfrak{B}_{2}(S,\unicode[STIX]{x1D711})$ means that

$$\begin{eqnarray}\int _{\mathbb{C}^{2}}|y\unicode[STIX]{x1D6F7}^{\ast }h|^{2}e^{-\unicode[STIX]{x1D6F7}^{\ast }\unicode[STIX]{x1D711}}(1+4|xy|^{2}+|y|^{4})\,dV(x,y)<+\infty\end{eqnarray}$$

and thus there exists $A>0$ such that for any $R>2\unicode[STIX]{x1D6FF}$

$$\begin{eqnarray}\int _{D_{R}(0)}\int _{|y|\leqslant 2\unicode[STIX]{x1D6FF}}|y\unicode[STIX]{x1D6F7}^{\ast }h(x,y)|^{2}e^{-\unicode[STIX]{x1D6F7}^{\ast }\unicode[STIX]{x1D711}(x,y)}(1+4|xy|^{2}+|y|^{4})\,dA(y)\,dA(x)\leqslant A.\end{eqnarray}$$

By Lemma 3.2 (with $m=0$) there exist functions

$$\begin{eqnarray}g\in {\mathcal{O}}(D_{R}(0)\times D_{2\unicode[STIX]{x1D6FF}}(0)\times D_{4\unicode[STIX]{x1D6FF}^{2}R}(0))\end{eqnarray}$$

and

$$\begin{eqnarray}\unicode[STIX]{x1D713}\in \mathscr{C}^{2}(D_{R}(0)\times D_{2\unicode[STIX]{x1D6FF}}(0)\times D_{4\unicode[STIX]{x1D6FF}^{2}R}(0))\end{eqnarray}$$

such that

$$\begin{eqnarray}\unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}+2\operatorname{Re}g\qquad \text{and}\qquad |\unicode[STIX]{x1D713}|\leqslant C\end{eqnarray}$$

on $D_{R}(0)\times D_{2\unicode[STIX]{x1D6FF}}(0)\times D_{4\unicode[STIX]{x1D6FF}^{2}R}(0)$. Note that the constant $C$ depends on $\unicode[STIX]{x1D6FF}$ but not on $R$.

Now, by elementary complex analysis there exists a universal constant $c_{o}$ so that for any entire holomorphic function $f\in {\mathcal{O}}(\mathbb{C}^{2})$

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{|y|\leqslant 2\unicode[STIX]{x1D6FF}}|\unicode[STIX]{x1D6F7}^{\ast }(he^{-g})(x,y)f(x,y)|^{2}\,dA(y)\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant c_{o}\int _{\unicode[STIX]{x1D6FF}\leqslant |y|\leqslant 2\unicode[STIX]{x1D6FF}}|\unicode[STIX]{x1D6F7}^{\ast }(he^{-g})(x,y)f(x,y)|^{2}\,dA(y).\nonumber\end{eqnarray}$$

Applying this estimate with $f(x,y)$ equal to 1, $2xy$ and $y^{2}$ yields

$$\begin{eqnarray}\displaystyle & & \displaystyle \!\!\!\int _{D_{R}(0)}\int _{|y|\leqslant 2\unicode[STIX]{x1D6FF}}|\unicode[STIX]{x1D6F7}^{\ast }h(x,y)|^{2}e^{-\unicode[STIX]{x1D6F7}^{\ast }\unicode[STIX]{x1D711}(x,y)}(1+4|xy|^{2}+|y|^{4})\,dA(y)\,dA(x)\nonumber\\ \displaystyle & & \displaystyle \!\!\!\quad \leqslant e^{C}\int _{D_{R}(0)}\int _{|y|\leqslant 2\unicode[STIX]{x1D6FF}}|\unicode[STIX]{x1D6F7}^{\ast }(e^{-g}h)(x,y)|^{2}(1+4|xy|^{2}+|y|^{4})\,dA(y)\,dA(x)\nonumber\\ \displaystyle & & \displaystyle \!\!\!\quad \leqslant c_{o}e^{C}\int _{D_{R}(0)}\int _{\unicode[STIX]{x1D6FF}\leqslant |y|\leqslant 2\unicode[STIX]{x1D6FF}}|\unicode[STIX]{x1D6F7}^{\ast }(e^{-g}h)(x,y)|^{2}(1+4|xy|^{2}+|y|^{4})\,dA(y)\,dA(x)\nonumber\\ \displaystyle & & \displaystyle \!\!\!\quad \leqslant \frac{c_{o}e^{C}}{\unicode[STIX]{x1D6FF}^{2}}\int _{D_{R}(0)}\int _{\unicode[STIX]{x1D6FF}\leqslant |y|\leqslant 2\unicode[STIX]{x1D6FF}}|y\unicode[STIX]{x1D6F7}^{\ast }(e^{-g}h)(x,y)|^{2}(1+4|xy|^{2}+|y|^{4})\,dA(y)\,dA(x)\nonumber\\ \displaystyle & & \displaystyle \!\!\!\quad \leqslant \frac{c_{o}e^{2C}}{\unicode[STIX]{x1D6FF}^{2}}\int _{D_{R}(0)}\int _{|y|\leqslant 2\unicode[STIX]{x1D6FF}}|y\unicode[STIX]{x1D6F7}^{\ast }h(x,y)|^{2}e^{-\unicode[STIX]{x1D6F7}^{\ast }\unicode[STIX]{x1D711}(x,y)}(1+4|xy|^{2}+|y|^{4})\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,dA(y)\,dA(x)\nonumber\\ \displaystyle & & \displaystyle \!\!\!\quad \leqslant \frac{c_{o}e^{2C}A}{\unicode[STIX]{x1D6FF}^{2}}.\nonumber\end{eqnarray}$$

Since $A$, $C$ and $c_{o}$ are independent of $R$, $yh\in \mathfrak{B}_{2}(\widetilde{S}_{\unicode[STIX]{x1D6FF}},\unicode[STIX]{x1D711})$, as claimed.◻

5.3.3 Extension near $L_{o}$

Let $g\in \mathfrak{B}_{2}(S,\unicode[STIX]{x1D711})$ vanish along $L_{o}$. Then by the Nullstellensatz and Lemma 5.4 $g=y\tilde{g}$ for some $\tilde{g}\in \mathfrak{B}_{2}(S_{\unicode[STIX]{x1D6FF}},\unicode[STIX]{x1D711})$.

We can apply the $L^{2}$ Extension Theorem 2.6 in this setting as well, and to obtain an extension of $f|_{\widetilde{S}_{\unicode[STIX]{x1D6FF}}}$ to $\widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}$ we must show that

$$\begin{eqnarray}\int _{\widetilde{S}_{\unicode[STIX]{x1D6FF}}}\frac{|g|^{2}e^{-\unicode[STIX]{x1D711}}}{|dT|_{\unicode[STIX]{x1D714}_{o}}^{2}e^{-\unicode[STIX]{x1D706}}}\unicode[STIX]{x1D714}_{o}^{2}<+\infty .\end{eqnarray}$$

Pulling back to $\mathbb{C}^{2}$ via the parametrization $\unicode[STIX]{x1D6F7}$, we need to show that

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\mathbb{C}}\int _{D_{2\unicode[STIX]{x1D6FF}}(0)}|g(x,y,xy^{2})|^{2}e^{-\unicode[STIX]{x1D711}(x,y,xy^{2})}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,(1+|x|^{2}+|y|^{2}+4|xy|^{2}+|y|^{4})\,dA(y)\,dA(x)<+\infty .\nonumber\end{eqnarray}$$

But

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\mathbb{C}}\int _{D_{2\unicode[STIX]{x1D6FF}}(0)}|g(x,y,xy^{2})|^{2}e^{-\unicode[STIX]{x1D711}(x,y,xy^{2})}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,(1+|x|^{2}+|y|^{2}+4|xy|^{2}+|y|^{4})\,dA(y)\,dA(x)\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{\mathbb{C}}\int _{D_{2\unicode[STIX]{x1D6FF}}(0)}|g(x,y,xy^{2})|^{2}e^{-\unicode[STIX]{x1D711}(x,y,xy^{2})}(1+4|xy|^{2}+|y|^{4})\,dA(y)\,dA(x)\nonumber\\ \displaystyle & & \displaystyle \qquad +\int _{\mathbb{C}}\int _{D_{2\unicode[STIX]{x1D6FF}}(0)}|\tilde{g}(x,y,xy^{2})|^{2}e^{-\unicode[STIX]{x1D711}(x,y,xy^{2})}(|xy|^{2}+|y|^{4})\,dA(y)\,dA(x).\nonumber\end{eqnarray}$$

Since $g$ and (by Lemma 5.4) $\tilde{g}$ are both in $\mathfrak{B}_{2}(\widetilde{S}_{\unicode[STIX]{x1D6FF}},\unicode[STIX]{x1D711})$, the last two integrals are finite. Consequently Theorem 2.6 yields $\tilde{F}_{\unicode[STIX]{x1D6FF}}\in {\mathcal{O}}(\widetilde{S}_{\unicode[STIX]{x1D6FF}})$ such that

$$\begin{eqnarray}\tilde{F}_{\unicode[STIX]{x1D6FF}}|_{\widetilde{S}_{\unicode[STIX]{x1D6FF}}}=f|_{\widetilde{S}_{\unicode[STIX]{x1D6FF}}}\qquad \text{and}\qquad \int _{\widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}}|\tilde{F}_{\unicode[STIX]{x1D6FF}}|^{2}e^{-\unicode[STIX]{x1D711}}\,dV<+\infty .\end{eqnarray}$$

5.3.4 Patching together the two extensions $F_{\unicode[STIX]{x1D6FF}}$ and $\tilde{F}_{\unicode[STIX]{x1D6FF}}$

Fix a function $\unicode[STIX]{x1D712}\in \mathscr{C}_{o}^{\infty }([0,\infty ))$ such that $\unicode[STIX]{x1D712}(t)=1$ if $0\leqslant t\leqslant \unicode[STIX]{x1D6FF}$, $\unicode[STIX]{x1D712}(t)=0$ if $t\geqslant 2\unicode[STIX]{x1D6FF}$, and $|\unicode[STIX]{x1D712}^{\prime }|\leqslant 2/\unicode[STIX]{x1D6FF}$. Let

$$\begin{eqnarray}\displaystyle \tilde{F}(x,y,z) & := & \displaystyle (1-\unicode[STIX]{x1D712}(|y|))F_{\unicode[STIX]{x1D6FF}}(x,y,z)+\unicode[STIX]{x1D712}(|y|)\tilde{F}_{\unicode[STIX]{x1D6FF}}(x,y,z)\nonumber\\ \displaystyle & = & \displaystyle F_{\unicode[STIX]{x1D6FF}}(x,y,z)+\unicode[STIX]{x1D712}(|y|)(\tilde{F}_{\unicode[STIX]{x1D6FF}}(x,y,z)-F_{\unicode[STIX]{x1D6FF}}(x,y,z)).\nonumber\end{eqnarray}$$

Then $\tilde{F}\in \mathscr{C}^{\infty }(\mathbb{C}^{3})\cap {\mathcal{O}}(\{(x,y,z);\unicode[STIX]{x1D6FF}\leqslant |y|\leqslant 2\unicode[STIX]{x1D6FF}\})$, $\tilde{F}|_{S}=f$, and

$$\begin{eqnarray}\int _{\mathbb{C}^{3}}|\tilde{F}|^{2}e^{-\unicode[STIX]{x1D711}}\,dV<+\infty .\end{eqnarray}$$

We wish to correct $\tilde{F}$ by subtracting from it a function $u\in \mathscr{C}^{\infty }(\mathbb{C}^{3})$ such that

$$\begin{eqnarray}\bar{\unicode[STIX]{x2202}}u=\bar{\unicode[STIX]{x2202}}\tilde{F},\qquad u|_{S}\equiv 0\qquad \text{and}\qquad \int _{\mathbb{C}^{3}}|u|^{2}e^{-\unicode[STIX]{x1D711}}\,dV<+\infty .\end{eqnarray}$$

Consider the weight

$$\begin{eqnarray}\unicode[STIX]{x1D713}=\unicode[STIX]{x1D711}+\log |T|^{2}-\unicode[STIX]{x1D706},\end{eqnarray}$$

where $\unicode[STIX]{x1D706}$ is given by (13). Then

(16)$$\begin{eqnarray}-\unicode[STIX]{x1D711}\leqslant -\unicode[STIX]{x1D713}\qquad \text{and}\qquad \unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}\unicode[STIX]{x1D713}\geqslant \unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}\unicode[STIX]{x1D711}-\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}\unicode[STIX]{x1D706}\geqslant \frac{m}{2}\unicode[STIX]{x1D714}_{o}\end{eqnarray}$$

for $R>0$ sufficiently large. Note also that on $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}\cap \widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}$ the difference $\tilde{F}_{\unicode[STIX]{x1D6FF}}-F_{\unicode[STIX]{x1D6FF}}$ is holomorphic and vanishes on $S$. It follows that

$$\begin{eqnarray}\tilde{F}_{\unicode[STIX]{x1D6FF}}-F_{\unicode[STIX]{x1D6FF}}=T\cdot h\end{eqnarray}$$

for some $h\in {\mathcal{O}}(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}\cap \widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}})$ which evidently satisfies

(17)$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}\cap \widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}}|\tilde{F}_{\unicode[STIX]{x1D6FF}}-F_{\unicode[STIX]{x1D6FF}}|^{2}e^{-\unicode[STIX]{x1D713}}\,dV=\int _{\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}\cap \widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}}|h|^{2}e^{-\unicode[STIX]{x1D711}+\unicode[STIX]{x1D706}}\,dV.\end{eqnarray}$$

We claim that the integral (17) is finite. To establish this claim, note that in $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}$ the surface $S$ is uniformly flat. Thus by the proof of Lemma 3.2 in [Reference Ortega-Cerdà, Schuster and VarolinOSV-2006] there exist $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FF}}>0$ and $M_{\unicode[STIX]{x1D6FF}}>0$ such that $U_{\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FF}}}(S_{\unicode[STIX]{x1D6FF}}\cap \widetilde{S}_{\unicode[STIX]{x1D6FF}})$ is a tubular neighborhood of $S_{\unicode[STIX]{x1D6FF}}\cap \widetilde{S}_{\unicode[STIX]{x1D6FF}}$ and

(18)$$\begin{eqnarray}\unicode[STIX]{x1D706}(x,y,z)\geqslant \log |T(x,y,z)|^{2}-M_{\unicode[STIX]{x1D6FF}}\end{eqnarray}$$

for all $(x,y,z)\in \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}\cap \widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}-U_{\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FF}}/2}$. Now,

$$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}\cap \widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}}|h|^{2}e^{-\unicode[STIX]{x1D711}+\unicode[STIX]{x1D706}}\,dV & {\leqslant} & \displaystyle \int _{\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}\cap \widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}-U_{\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FF}}}(S_{\unicode[STIX]{x1D6FF}}\cap \widetilde{S}_{\unicode[STIX]{x1D6FF}})}|h|^{2}e^{-\unicode[STIX]{x1D711}+\unicode[STIX]{x1D706}}\,dV\nonumber\\ \displaystyle & & \displaystyle +\int _{U_{\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FF}}}(S_{\unicode[STIX]{x1D6FF}}\cap \widetilde{S}_{\unicode[STIX]{x1D6FF}})}|h|^{2}e^{-\unicode[STIX]{x1D711}+\unicode[STIX]{x1D706}}\,dV.\nonumber\end{eqnarray}$$

From (18) we see that

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}\cap \widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}-U_{\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FF}}}(S_{\unicode[STIX]{x1D6FF}}\cap \widetilde{S}_{\unicode[STIX]{x1D6FF}})}|h|^{2}e^{-\unicode[STIX]{x1D711}+\unicode[STIX]{x1D706}}\,dV\leqslant e^{M_{\unicode[STIX]{x1D6FF}}}\int _{\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}\cap \widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}}|Th|^{2}e^{-\unicode[STIX]{x1D711}}\,dV\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant 2e^{M_{\unicode[STIX]{x1D6FF}}}\left(\int _{\widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}}|\tilde{F}_{\unicode[STIX]{x1D6FF}}|^{2}e^{-\unicode[STIX]{x1D711}}\,dV+\int _{\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}}|F_{\unicode[STIX]{x1D6FF}}|^{2}e^{-\unicode[STIX]{x1D711}}\,dV\right)<+\infty .\nonumber\end{eqnarray}$$

On the other hand, once again by the method of proof of [Reference Ortega-Cerdà, Schuster and VarolinOSV-2006, Lemma 5.3]

$$\begin{eqnarray}\displaystyle \int _{U_{\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FF}}}(S_{\unicode[STIX]{x1D6FF}}\cap \widetilde{S}_{\unicode[STIX]{x1D6FF}})}|h|^{2}e^{-\unicode[STIX]{x1D711}+\unicode[STIX]{x1D706}}\,dV & {\lesssim} & \displaystyle \int _{U_{\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FF}}}(S_{\unicode[STIX]{x1D6FF}}\cap \widetilde{S}_{\unicode[STIX]{x1D6FF}})-U_{\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FF}}/2}(S_{\unicode[STIX]{x1D6FF}}\cap \widetilde{S}_{\unicode[STIX]{x1D6FF}})}|h|^{2}e^{-\unicode[STIX]{x1D711}+\unicode[STIX]{x1D706}}\,dV\nonumber\\ \displaystyle & {\leqslant} & \displaystyle e^{M_{\unicode[STIX]{x1D6FF}}}\int _{U_{\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FF}}}(S_{\unicode[STIX]{x1D6FF}}\cap \widetilde{S}_{\unicode[STIX]{x1D6FF}})-U_{\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FF}}/2}(S_{\unicode[STIX]{x1D6FF}}\cap \widetilde{S}_{\unicode[STIX]{x1D6FF}})}|Th|^{2}e^{-\unicode[STIX]{x1D711}}\,dV\nonumber\\ \displaystyle & {\leqslant} & \displaystyle 2e^{M_{\unicode[STIX]{x1D6FF}}}\left(\int _{\widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}}|\tilde{F}_{\unicode[STIX]{x1D6FF}}|^{2}e^{-\unicode[STIX]{x1D711}}\,dV+\int _{\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}}|F_{\unicode[STIX]{x1D6FF}}|^{2}e^{-\unicode[STIX]{x1D711}}\,dV\right)\nonumber\\ \displaystyle & {<} & \displaystyle +\infty ,\nonumber\end{eqnarray}$$

which establishes the finiteness of (17).

Next, observe that

$$\begin{eqnarray}\displaystyle \int _{\mathbb{C}^{3}}|\bar{\unicode[STIX]{x2202}}\tilde{F}|^{2}e^{-\unicode[STIX]{x1D713}}\,dV & = & \displaystyle \int _{\mathbb{C}^{3}}|\bar{\unicode[STIX]{x2202}}\unicode[STIX]{x1D712}(|y|)|^{2}|F_{\unicode[STIX]{x1D6FF}}-\tilde{F}_{\unicode[STIX]{x1D6FF}}|^{2}e^{-\unicode[STIX]{x1D713}}\,dV\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{C}{\unicode[STIX]{x1D6FF}^{2}}\int _{\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FF}}\cap \widetilde{\unicode[STIX]{x1D6FA}}_{\unicode[STIX]{x1D6FF}}}|\tilde{F}_{\unicode[STIX]{x1D6FF}}-F_{\unicode[STIX]{x1D6FF}}|^{2}e^{-\unicode[STIX]{x1D713}}\,dV<+\infty .\nonumber\end{eqnarray}$$

By Hörmander’s theorem, (16) and the regularity of $\bar{\unicode[STIX]{x2202}}$ there exists $u\in \mathscr{C}^{\infty }(\mathbb{C}^{3})$ such that

$$\begin{eqnarray}\bar{\unicode[STIX]{x2202}}u=\bar{\unicode[STIX]{x2202}}\tilde{F}\qquad \text{and}\qquad \int _{\mathbb{C}^{3}}|u|^{2}e^{-\unicode[STIX]{x1D711}}\,dV\leqslant \int _{\mathbb{ C}^{3}}|u|^{2}e^{-\unicode[STIX]{x1D713}}\,dV<+\infty .\end{eqnarray}$$

The finiteness of the second integral together with the smoothness of $u$ implies that $u|_{S}\equiv 0$. Thus we have a function

$$\begin{eqnarray}F:=\tilde{F}-u\in \mathscr{B}_{3}(\unicode[STIX]{x1D711})\end{eqnarray}$$

such that $F|_{S}=\tilde{F}|_{S}=f$. The proof of Theorem 3 is complete.

6.1 Extension from $C_{2}$ to $S$

Without necessarily explicitly mentioning it, we shall identify $C_{2}$ with $C_{2}\times \{1\}=S\cap \{z=1\}$. In this section we prove the following result.

Theorem 6.1. Let $\unicode[STIX]{x1D711}\in \mathscr{C}^{2}(\mathbb{C}^{2})$ satisfy the Bargmann–Fock curvature condition (1), and define

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D711}}(x,y,z):=\unicode[STIX]{x1D711}(x,y)+\unicode[STIX]{x1D700}|z-1|^{2}.\end{eqnarray}$$

Then for each $f\in \mathfrak{B}_{1}(C_{2},\unicode[STIX]{x1D711})$ there exists $g\in \mathfrak{B}_{2}(S,\tilde{\unicode[STIX]{x1D711}})$ such that $g|_{C_{2}}=f$.

The approach is to apply the $L^{2}$ Extension Theorem 2.6. However, there is some work to be done before the application of Theorem 2.6 is possible. Of course, we take $X=S$, $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{o}|_{S}$ and $Z=C_{2}$, but the rest of the data is perhaps not as obvious.

6.1.1 Step 1: (Defining section and metric for the defining line bundle)

Let us start with the function $T\in {\mathcal{O}}(S)$ that cuts out $C_{2}$. The premise is that $C_{2}$ is uniformly flat in $S$, and the rationale is that this is so because $C_{2}\subset \mathbb{C}^{2}\times \{1\}$ is obtained from $S\subset \mathbb{C}^{3}$ by intersection with the hyperplane $\{z=1\}$. If we pursue this clue, we should try

$$\begin{eqnarray}T(x,y,z):=z-1.\end{eqnarray}$$

Continuing with our view of $C_{2}$ as the intersection of $S$ with a plane in $\mathbb{C}^{3}$, in our search for the weight $\unicode[STIX]{x1D706}$ we should exploit the idea used in Section 2.4 in the proof of Theorem 1.2. That is to say, we should set

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{r}(x,y,z):=\frac{1}{\unicode[STIX]{x1D70B}r^{2}}\int _{D_{r}(z)}\log |\unicode[STIX]{x1D701}-1|^{2}\,dA(\unicode[STIX]{x1D701}).\end{eqnarray}$$

Since this is the same defining data as one uses for a plane in $\mathbb{C}^{3}$, one sees that there is a constant $L_{r}>0$ such that

(19)$$\begin{eqnarray}|dT(x,y,1)|_{\unicode[STIX]{x1D714}_{o}}^{2}e^{-\unicode[STIX]{x1D706}_{r}(1)}\geqslant L_{r}\end{eqnarray}$$

for all $(x,y)\in \mathbb{C}^{2}$. (This fact can of course easily be verified directly as well.) Moreover, the curvature of $\unicode[STIX]{x1D706}_{r}$ is nonnegative and bounded above by $(1/r^{2})dd^{c}|z|^{2}$, which can be made as small as we like by taking $r$ sufficiently large but fixed.

6.1.2 Step 2: (Weight modification)

Since $dd^{c}\unicode[STIX]{x1D706}_{r}\geqslant 0$, Theorem 2.6 can be applied with the weight $\tilde{\unicode[STIX]{x1D711}}$ if there exists $\unicode[STIX]{x1D6FF}>0$ so that

(20)$$\begin{eqnarray}dd^{c}\tilde{\unicode[STIX]{x1D711}}+\text{Ricci}(\unicode[STIX]{x1D714}_{o}|_{S})\geqslant (1+\unicode[STIX]{x1D6FF})dd^{c}\unicode[STIX]{x1D706}_{r}\end{eqnarray}$$

everywhere on $S$.

Since $S$ is a complex submanifold of $\mathbb{C}^{3}$, the curvature of $\unicode[STIX]{x1D714}_{o}|_{S}$ can be (and in fact is) smaller that the restriction to $S$ of the curvature of $\unicode[STIX]{x1D714}_{o}$. The condition (20) might not be satisfied even for some weights $\unicode[STIX]{x1D711}$ satisfying the Bargmann–Fock curvature bound (1). (As it turns out, (20) is satisfied for weights $\unicode[STIX]{x1D711}$ satisfying (1) with $m$ large enough.) We therefore need to modify the weight $\tilde{\unicode[STIX]{x1D711}}$ slightly.

To see things more clearly, it is useful to parametrize the surface $S$ by the map $\unicode[STIX]{x1D6F7}$ defined by (2). On the other hand, it is also useful not to attach oneself too much to this parametrization. First, note that

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{o}|_{S}=\unicode[STIX]{x1D6F7}^{\ast }\unicode[STIX]{x1D714}_{o}=dd^{c}(|s|^{2}+|t|^{2}+|st^{2}|^{2}).\end{eqnarray}$$

An easy computation shows that

$$\begin{eqnarray}(\unicode[STIX]{x1D714}_{o}|_{S})^{2}=(1+|2st|^{2}+|t^{2}|^{2})dd^{c}|s|^{2}\wedge dd^{c}|t|^{2}.\end{eqnarray}$$

Letting $\unicode[STIX]{x1D6EF}(s,t)=(2st,t^{2})$, we see that

$$\begin{eqnarray}\text{Ricci}(\unicode[STIX]{x1D714}_{o}|_{S})=\unicode[STIX]{x1D6EF}^{\ast }(dd^{c}(-\log (1+|z^{1}|^{2}+|z^{2}|^{2})))\end{eqnarray}$$

is the pullback by $\unicode[STIX]{x1D6EF}$ of the curvature of the Euclidean metric on ${\mathcal{O}}(-1)\rightarrow \mathbb{P}_{2}$ in the affine chart $U_{o}\cong \mathbb{C}^{2}\subset \mathbb{P}_{2}$. Now,

$$\begin{eqnarray}\displaystyle dd^{c}(-\log (1+|z|^{2})) & = & \displaystyle \frac{(1+|z|^{2})\sqrt{-1}\,dz\dot{\wedge }d\bar{z}-\sqrt{-1}\bar{z}\cdot dz\wedge z\cdot d\bar{z}}{(1+|z|^{2})^{2}}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{2\sqrt{-1}\,dz\dot{\wedge }d\bar{z}}{1+|z|^{2}}\leqslant 2\unicode[STIX]{x1D714}_{o}|_{S},\nonumber\end{eqnarray}$$

where $dz\dot{\wedge }d\bar{z}=dz^{1}\wedge d\bar{z}^{1}+dz^{2}\wedge d\bar{z}^{2}$ and $\bar{z}\cdot dz=z^{1}\,d\bar{z}^{1}+z^{2}\,d\bar{z}^{2}$. Therefore, if $\unicode[STIX]{x1D711}$ satisfies the Bargmann–Fock curvature condition with $m>2$ then one can already apply the $L^{2}$ Extension Theorem 2.6 to the weight $\unicode[STIX]{x1D713}(x,y,z)=\unicode[STIX]{x1D711}(x,y)+m|z|^{2}$.

There is, however, another modification that allows us to use the extension theorem. Namely, we let

$$\begin{eqnarray}\unicode[STIX]{x1D713}(x,y,z):=\unicode[STIX]{x1D711}(x,y)+\unicode[STIX]{x1D700}|z-1|^{2}-\unicode[STIX]{x1D707}(x,y)+\log (1+|2xy|^{2}+|y^{2}|^{2}),\end{eqnarray}$$

where

$$\begin{eqnarray}\unicode[STIX]{x1D707}(x,y)=\frac{1}{\unicode[STIX]{x1D70B}r^{2}}\int _{D_{r}(x)}\log (1+|2\unicode[STIX]{x1D709}y|^{2}+|y^{2}|^{2})\,dA(\unicode[STIX]{x1D709}).\end{eqnarray}$$

By the sub-mean value property for subharmonic functions

$$\begin{eqnarray}\log (1+|2xy|^{2}+|y^{2}|^{2})\leqslant \unicode[STIX]{x1D707}(x,y).\end{eqnarray}$$

And along the curve $C_{2}$

$$\begin{eqnarray}\displaystyle & & \displaystyle \sup _{(x,y)\in C_{2}}\unicode[STIX]{x1D707}(x,y)-\log (1+|2xy|^{2}+|y^{2}|^{2})\nonumber\\ \displaystyle & & \displaystyle \quad =\sup _{t\in \mathbb{C}^{\ast }}\frac{1}{\unicode[STIX]{x1D70B}r^{2}}\int _{D_{r}(0)}\log \frac{1+|2(\unicode[STIX]{x1D709}+t^{2})t^{-1}|^{2}+|t|^{-4}}{1+|2t|^{2}+|t|^{-4}}\,dA(\unicode[STIX]{x1D709}).\nonumber\end{eqnarray}$$

For $|t|$ large

$$\begin{eqnarray}\frac{1+|2(\unicode[STIX]{x1D709}+t^{2})t^{-1}|^{2}+|t|^{-4}}{1+|2t|^{2}+|t|^{-4}}\lesssim r^{2},\end{eqnarray}$$

while for $|t|\sim 0$

$$\begin{eqnarray}\frac{1+|2(\unicode[STIX]{x1D709}+t^{2})t^{-1}|^{2}+|t|^{-4}}{1+|2t|^{2}+|t|^{-4}}\lesssim 1,\end{eqnarray}$$

and hence there exists $A_{r}$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D707}(x,y)-\log (1+|2xy|^{2}+|y|^{4})\leqslant A_{r}\quad \text{for all}~(x,y)\in C_{2}.\end{eqnarray}$$

By taking $r$ sufficiently large we can ensure that $dd^{c}\unicode[STIX]{x1D707}$ is as small as we like.

Now, in $\mathbb{C}^{3}$ we have

$$\begin{eqnarray}\displaystyle dd^{c}\unicode[STIX]{x1D713} & = & \displaystyle dd^{c}\unicode[STIX]{x1D711}+\unicode[STIX]{x1D700}dd^{c}|z-1|^{2}-dd^{c}\unicode[STIX]{x1D707}+dd^{c}(\log (1+|\unicode[STIX]{x1D6EF}(x,y)|^{2}))\nonumber\\ \displaystyle & {\geqslant} & \displaystyle (m-\unicode[STIX]{x1D700})dd^{c}(|x|^{2}+|y|^{2})+\unicode[STIX]{x1D700}dd^{c}|z-1|^{2}+dd^{c}(\log (1+|\unicode[STIX]{x1D6EF}(x,y)|^{2})).\nonumber\end{eqnarray}$$

Therefore, on the surface $S$ we have

$$\begin{eqnarray}\displaystyle dd^{c}\unicode[STIX]{x1D713}+\text{Ricci}(\unicode[STIX]{x1D714}_{o}|_{S})-(1+\unicode[STIX]{x1D6FF})dd^{c}\unicode[STIX]{x1D706}_{r} & {\geqslant} & \displaystyle (m-\unicode[STIX]{x1D700})dd^{c}(|x|^{2}+|y|^{2})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D700}dd^{c}|z-1|^{2}-(1+\unicode[STIX]{x1D6FF})dd^{c}\unicode[STIX]{x1D706}_{r},\nonumber\end{eqnarray}$$

and the right hand side is nonnegative if one takes $r$ sufficiently large.

Proof of Theorem 6.1.

Let $f\in \mathfrak{B}_{1}(C_{2},\unicode[STIX]{x1D711})$. Then

$$\begin{eqnarray}\int _{C_{2}}\frac{|f|^{2}e^{-\unicode[STIX]{x1D713}}}{|dT|_{\unicode[STIX]{x1D714}_{o}}^{2}e^{-\unicode[STIX]{x1D706}_{r}}}\unicode[STIX]{x1D714}_{o}\leqslant \frac{e^{A_{r}}}{L_{r}}\int _{C_{2}}|f|^{2}e^{-\unicode[STIX]{x1D711}}\unicode[STIX]{x1D714}_{o}<+\infty .\end{eqnarray}$$

By Theorem 2.6 there exists $g\in \mathfrak{B}_{2}(S,\unicode[STIX]{x1D713})$ such that

$$\begin{eqnarray}g|_{C_{2}}=f.\end{eqnarray}$$

Since $\unicode[STIX]{x1D713}\leqslant \tilde{\unicode[STIX]{x1D711}}$, $g\in \mathfrak{B}_{2}(S,\tilde{\unicode[STIX]{x1D711}})$, as desired.◻

6.2 Extension from $S$ to $\mathbb{C}^{3}$

By Theorem 3 there exists $\tilde{F}\in \mathscr{B}_{3}(\tilde{\unicode[STIX]{x1D711}})$ such that

$$\begin{eqnarray}\tilde{F}|_{S}=g.\end{eqnarray}$$

6.3 Restriction from $\mathbb{C}^{3}$ to $\mathbb{C}^{2}\times \{1\}$

Let $F(x,y):=\tilde{F}(x,y,1)$. Then

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\mathbb{C}^{2}}|F(x,y)|^{2}e^{-\unicode[STIX]{x1D711}(x,y)}\,dV(x,y)\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{\mathbb{C}^{2}}|\tilde{F}(x,y,1)|^{2}e^{-\unicode[STIX]{x1D711}(x,y)}\,dV(x,y)\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \frac{\unicode[STIX]{x1D700}}{\unicode[STIX]{x1D70B}}\int _{\mathbb{C}^{3}}|\tilde{F}(x,y,z)|^{2}e^{-\unicode[STIX]{x1D711}(x,y)-\unicode[STIX]{x1D700}|z-1|^{2}}dV(x,y,z),\nonumber\end{eqnarray}$$

which shows that $F\in \mathscr{B}_{2}(\unicode[STIX]{x1D711})$. Finally,

$$\begin{eqnarray}F|_{C_{2}}=\tilde{F}|_{C_{2}\times \{1\}}=g|_{C_{2}\times \{1\}}=f,\end{eqnarray}$$

and the proof of Theorem 2 is complete. ◻

6.4 Postscript: the nonflat pairs of $C_{2}$ are too close together

Theorem 2 was utterly surprising to us. We fully expected that the curve $C_{2}$ would not be interpolating for any Bargmann–Fock weight on $\mathbb{C}^{2}$. Indeed, we noted that the points $(\unicode[STIX]{x1D6FF}^{-2},\pm \unicode[STIX]{x1D6FF})$ were very close together in $\mathbb{C}^{2}$ but quite far apart in $C_{2}$, so we expected to be able to find a function that is large at $(\unicode[STIX]{x1D6FF}^{-2},\unicode[STIX]{x1D6FF})$ and vanishes at $(\unicode[STIX]{x1D6FF}^{-2},-\unicode[STIX]{x1D6FF})$. The problem with the latter goal is that it is vague; particularly, the quantitative meaning of the word “large” is here crucial. To show that $C_{2}$ is not interpolating, one would need to find, for all $\unicode[STIX]{x1D6FF}>0$ sufficiently small, functions $f_{\unicode[STIX]{x1D6FF}}\in \mathfrak{B}_{1}(C_{2},\unicode[STIX]{x1D711})$ satisfying

(21)$$\begin{eqnarray}|f_{\unicode[STIX]{x1D6FF}}(\unicode[STIX]{x1D6FF}^{-2},\unicode[STIX]{x1D6FF})|^{2}e^{-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FF}^{-2},\unicode[STIX]{x1D6FF})}=1,\qquad f_{\unicode[STIX]{x1D6FF}}(\unicode[STIX]{x1D6FF}^{-2},-\unicode[STIX]{x1D6FF})=0\qquad \text{and}\qquad \unicode[STIX]{x1D6FF}\Vert f_{\unicode[STIX]{x1D6FF}}\Vert ^{2}\leqslant r\end{eqnarray}$$

for some $r<1$ independent of $\unicode[STIX]{x1D6FF}$. Indeed, such functions would contradict (8) of Proposition 3.3.

We were able to construct functions $f_{\unicode[STIX]{x1D6FF}}$ satisfying the first two conditions of (21), but the best estimate we could find for such functions is

$$\begin{eqnarray}\Vert f_{\unicode[STIX]{x1D6FF}}\Vert ^{2}\sim \unicode[STIX]{x1D6FF}^{-2}.\end{eqnarray}$$

Unable to find functions satisfying (21), we reluctantly had to admit to ourselves that perhaps $C_{2}$ is, after all, interpolating.

7.1 Extension from $C_{1}$ to $\unicode[STIX]{x1D6F4}$

Proof. We use the same function $T(x,y,z)=z-1$ and $\unicode[STIX]{x1D706}_{r}$ as in the proof of Theorem 6.1.

For the surface $\unicode[STIX]{x1D6F4}$ we use the parametrization $\unicode[STIX]{x1D6F9}(s,t)=(s,t,s^{2}t^{2})$. The metric induced on $\unicode[STIX]{x1D6F4}$ by the Euclidean metric is then

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{o}|_{\unicode[STIX]{x1D6F4}}=\unicode[STIX]{x1D6F9}^{\ast }\unicode[STIX]{x1D714}_{o}=dd^{c}(|s|^{2}+|t|^{2}+|s^{2}t^{2}|),\end{eqnarray}$$

and

$$\begin{eqnarray}(\unicode[STIX]{x1D714}_{o}|_{\unicode[STIX]{x1D6F4}})^{2}=(1+|2st^{2}|^{2}+|2ts^{2}|^{2})dd^{c}|s|^{2}\wedge dd^{c}|t|^{2}.\end{eqnarray}$$

Thus

$$\begin{eqnarray}\text{Ricci}(\unicode[STIX]{x1D714}_{o}|_{\unicode[STIX]{x1D6F4}})=-\unicode[STIX]{x1D6F1}^{\ast }dd^{c}\log (1+|z^{1}|^{2}+|z^{2}|^{2}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6F1}(s,t)=(2st^{2},2ts^{2})=2st(s,t)$.

To apply the $L^{2}$ extension theorem, we again need to rid ourselves of the negative curvature contributed by $\text{Ricci}(\unicode[STIX]{x1D714}_{o}|_{\unicode[STIX]{x1D6F4}})$. As with the surface $S$, we use a modified weight

$$\begin{eqnarray}\unicode[STIX]{x1D713}(x,y,z):=\unicode[STIX]{x1D711}(x,y,z)-\unicode[STIX]{x1D708}(x,y,z)+\log (1+4|zx^{2}|+4|zy^{2}|),\end{eqnarray}$$

where

$$\begin{eqnarray}\unicode[STIX]{x1D708}(x,y,z)=\frac{4!}{\unicode[STIX]{x1D70B}^{2}r^{4}}\int _{B_{r}(x,y)}\log (1+4|z\unicode[STIX]{x1D709}^{2}|+4|z\unicode[STIX]{x1D702}^{2}|)\,dV(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}).\end{eqnarray}$$

By the sub-mean value property

$$\begin{eqnarray}\log (1+4|zx^{2}|+4|zy^{2}|)\leqslant \unicode[STIX]{x1D708}(x,y,z),\end{eqnarray}$$

Along the curve $C_{1}$

(22)$$\begin{eqnarray}\displaystyle & & \displaystyle \sup _{(x,y,z)\in C_{2}\times \{1\}}\unicode[STIX]{x1D708}(x,y,z)-\log (1+4|zx^{2}|+4|zy^{2}|)\nonumber\\ \displaystyle & & \displaystyle \quad =\sup _{(x,y)\in C_{2}}\frac{4!}{\unicode[STIX]{x1D70B}^{2}r^{4}}\int _{B_{r}(0)}\log \frac{1+4|\unicode[STIX]{x1D709}+x|^{2}+4|\unicode[STIX]{x1D702}+y|^{2}}{1+4|x^{2}|+4|y^{2}|}\,dV(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})\nonumber\\ \displaystyle & & \displaystyle \quad =\sup _{t\in \mathbb{C}^{\ast }}\frac{4!}{\unicode[STIX]{x1D70B}^{2}r^{4}}\int _{B_{r}(0)}\log \frac{1+4|\unicode[STIX]{x1D709}+t|^{2}+4|\unicode[STIX]{x1D702}+t^{-1}|^{2}}{1+4|t|^{2}+4|t|^{-2}}\,dV(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}),\end{eqnarray}$$

and again the last expression is bounded, as one can check by looking near $t=0$ and $t=\infty$. Lastly, by taking $r\gg 1$ we can guarantee that $dd^{c}\unicode[STIX]{x1D708}$ is as small as we like.

Now, $\log (1+4|zx^{2}|+4|zy^{2}|)=\log (1+|2yx^{2}|^{2}+|2xy^{2}|^{2})$ on the surface $\unicode[STIX]{x1D6F4}$, and hence

$$\begin{eqnarray}dd^{c}\unicode[STIX]{x1D713}+\text{Ricci}(\unicode[STIX]{x1D714}_{o}|_{S})=dd^{c}\unicode[STIX]{x1D711}-dd^{c}\unicode[STIX]{x1D708}.\end{eqnarray}$$

It follows that

$$\begin{eqnarray}\displaystyle dd^{c}\unicode[STIX]{x1D713}+\text{Ricci}(\unicode[STIX]{x1D714}_{o}|_{S})-(1+\unicode[STIX]{x1D6FF})dd^{c}\unicode[STIX]{x1D706}_{r} & {\geqslant} & \displaystyle (m-\unicode[STIX]{x1D700})dd^{c}(|x|^{2}+|y|^{2}+|z|^{2})\nonumber\\ \displaystyle & & \displaystyle -\,(1+\unicode[STIX]{x1D6FF})dd^{c}\unicode[STIX]{x1D706}_{r},\nonumber\end{eqnarray}$$

for some small $\unicode[STIX]{x1D716}$, and the latter is positive as long as $r$ is sufficiently large, which we have assumed is the case.

Now let $f\in \mathfrak{B}_{1}(C_{1},\unicode[STIX]{x1D711})$. Then by (19) and (22)

$$\begin{eqnarray}\int _{C_{2}}\frac{|f|^{2}e^{-\unicode[STIX]{x1D713}}}{|dT|_{\unicode[STIX]{x1D714}_{o}}^{2}e^{-\unicode[STIX]{x1D706}_{r}}}\unicode[STIX]{x1D714}_{o}\lesssim \int _{C_{2}}|f|^{2}e^{-\unicode[STIX]{x1D711}}\unicode[STIX]{x1D714}_{o}.\end{eqnarray}$$

By Theorem 2.6 there exists $g\in \mathfrak{B}_{2}(\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D713})$ such that

$$\begin{eqnarray}g|_{C_{1}}=f.\end{eqnarray}$$

Therefore, $g\in \mathfrak{B}_{2}(\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D711})$, as desired.◻

7.2 End of the proof of Theorem 4

To achieve our contradiction, suppose $\unicode[STIX]{x1D6F4}$ is interpolating with respect to some Bargmann–Fock weight function $\unicode[STIX]{x1D711}\in \mathscr{C}^{2}(\mathbb{C}^{3})$.

Let $f\in \mathfrak{B}_{1}(C_{1},\unicode[STIX]{x1D711})$. By Theorem 7.1 there exists $g\in \mathfrak{B}_{2}(\unicode[STIX]{x1D6F4},\tilde{\unicode[STIX]{x1D711}})$ such that $g|_{C_{1}}=f$. Since $\tilde{\unicode[STIX]{x1D711}}\geqslant \unicode[STIX]{x1D711}$, $g\in \mathfrak{B}_{2}(\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D711})$. By our hypothesis there exists $\tilde{F}\in \mathscr{B}_{3}(\unicode[STIX]{x1D711})$ such that $F|_{\unicode[STIX]{x1D6F4}}=g$, and hence $\tilde{F}|_{C_{1}}=f$. Since $\unicode[STIX]{x1D711}$ is a Bargmann–Fock weight, for each $(x,y)\in \mathbb{C}^{2}\unicode[STIX]{x1D711}(x,y,\cdot )$ is a Bargmann–Fock weight in $\mathbb{C}$. Hence by (7) of Proposition 3.3

$$\begin{eqnarray}|\tilde{F}(x,y,1)|^{2}e^{-\unicode[STIX]{x1D711}(x,y,1)}\lesssim \int _{\mathbb{ C}}|\tilde{F}(x,y,z)|^{2}e^{-\unicode[STIX]{x1D711}(x,y,z)}\,dA(z).\end{eqnarray}$$

Integration over $(x,y)\in \mathbb{C}^{2}$ with respect to Lebesgue measure yields

$$\begin{eqnarray}\displaystyle \int _{\mathbb{C}^{2}}|\tilde{F}(x,y,1)|^{2}e^{-\unicode[STIX]{x1D711}(x,y,1)}\,dV(x,y) & {\lesssim} & \displaystyle \int _{\mathbb{C}^{3}}|\tilde{F}(x,y,z)|^{2}e^{-\unicode[STIX]{x1D711}(x,y,z)}\,dV(x,y,z)\nonumber\\ \displaystyle & {<} & \displaystyle +\infty .\nonumber\end{eqnarray}$$

Letting $F(x,y):=\tilde{F}(x,y,1)$, we find that $F|_{C_{1}}=f$ and $F\in \mathscr{B}_{2}(\unicode[STIX]{x1D711}(\cdot ,\cdot ,1))$. In other words,

$$\begin{eqnarray}\mathscr{R}_{C_{1}}:\mathscr{B}_{2}(\unicode[STIX]{x1D711}(\cdot ,\cdot ,1))\rightarrow \mathfrak{B}_{1}(C_{1},\unicode[STIX]{x1D711}(\cdot ,\cdot ,1))\end{eqnarray}$$

is surjective. Since $\unicode[STIX]{x1D711}(\cdot ,\cdot ,1)$ is also a Bargmann–Fock weight, Theorem 1 is contradicted.◻

7.3 Postscript: the crucial difference between $S$ and $\unicode[STIX]{x1D6F4}$

As mentioned in the second-to-last paragraph of Section 1.2 of the introduction, the proof of Theorem 4 by contradiction to Theorem 1 came to us relatively quickly by that point in our research. Nevertheless, we wondered why we could not just repeat the proof of Theorem 3 for the surface $\unicode[STIX]{x1D6F4}$ in place of $S$. Since our proof of Theorem 4 is indirect, it does not help one understand what precisely goes wrong in the proof of Theorem 3 when $S$ is replaced by $\unicode[STIX]{x1D6F4}$.

As we see it, the difficulty is by trying to imitate the step appearing in Section 5.3.3. In the case of $S$, the uniform nonflatness is concentrated along the line $L_{o}$, and for this line we have a nice $L^{2}$ extension theorem. But in the case of $\unicode[STIX]{x1D6F4}$ the nonflat points concentrate along the variety $\unicode[STIX]{x1D6EC}_{o}=\{xy=0,z=0\}$. The union $\{|x|<\unicode[STIX]{x1D6FF}\}\cup \{|y|<\unicode[STIX]{x1D6FF}\}\subset \mathbb{C}^{3}$ is not pseudoconvex, and it is one of the standard exercises relating to Hartogs’ phenomenon that taking the pseudoconvex hull of this set increases its size significantly. It is here that our method breaks down.

Of course, at the end of the day there is no way to remedy this problem. Indeed, Theorem 4, and not its opposite, is true.