Bargmann-Fock extension from singular hypersurfaces (Q305160)

While the extension from an analytic subvariety to an ambient bounded domain with weighted \(L^2\)-estimates and the corresponding Bergman spaces of holomorphic functions are relatively well understood, much less is known about the unbounded situation.NEWLINENEWLINEIn this paper the authors consider the extension of holomorphic functions from a possibly singular closed complex hypersurface \(W \subset \mathbb{C}^n\) to the ambient space \(\mathbb{C}^n\) with weighted \(L^2\)-estimates.NEWLINENEWLINEGiven a smooth weight function \(\varphi : \mathbb{C}^n \to \mathbb{R}\) and the standard Kähler form \(\omega\) on \(\mathbb{C}^n\) the following Hilbert spaces can be defined: NEWLINE\[NEWLINE \mathcal{H}(\mathbb{C}^n, \varphi) := \left\{ f \in \mathcal{O}(\mathbb{C}^n) : \int_{\mathbb{C}^n} |f|^2 e^{-\varphi} \omega^n < \infty \right\}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \mathfrak{H}(W, \varphi) := \left\{ f \in \mathcal{O}(W) : \int_{W_{\mathrm{reg}}} |f|^2 e^{-\varphi} \omega^{n-1} < \infty \right\}. NEWLINE\]NEWLINE By \(\mathcal{R}_W : \mathcal{H}(\mathbb{C}^n, \varphi) \to \mathfrak{H}(W, \varphi)\) we denote restriction map.NEWLINENEWLINEA smooth closed complex hypersurface \(W \subset \mathbb{C}^n\) is called \textit{uniformly flat} if there exists an \(\varepsilon_0 > 0\) such that the \(\varepsilon_0\)-neighborhood of \(W\) is a tubular neighborhood. For a singular closed complex hypersurface \(W \subset \mathbb{C}^n\) the authors use a slightly more involved definition, see their Def.~2.4.NEWLINENEWLINEWe also need the notion of the \textit{total density tensor} \(\Upsilon^W_r\) associated to the hypersurface \(W \subset \mathbb{C}^n\) which can be defined as NEWLINE\[NEWLINE \Upsilon^W_r := [W] \ast \frac{\mathbf{1}_{B(0,r)}}{\mathrm{vol}(B(0,r))} NEWLINE\]NEWLINE where \([W]\) denotes the current of integration over \(W\). The \textit{upper density} of \(W\) can then be given as NEWLINE\[NEWLINE D_\varphi^+(W) := \inf \left\{ \alpha \geq 0 : \sqrt{-1}\partial\bar{\partial} \varphi_r - \frac{1}{\alpha} \Upsilon^W_r \geq 0 \text{ for all } r \gg 0 \right\} NEWLINE\]NEWLINE where \(\varphi_r(z)\) is the average of \(\varphi\) over \(B(z,r)\).NEWLINENEWLINEThe authors prove the following: Let \(\varphi : \mathbb{C}^n \to \mathbb{R}\) be a \(\mathcal{C}^2\)-function satisfying NEWLINE\[NEWLINE \varepsilon \omega \leq \sqrt{-1}\partial\overline{\partial}\varphi \leq C \omega NEWLINE\]NEWLINE for some constants \(\varepsilon > 0\) and \(C > 0\), and let \(W \subset \mathbb{C}^n\) be a possibly singular closed complex hypersurface that is uniformly flat and such that \(D_\varphi^+(W) < 1\). Then \(\mathcal{R}_W : \mathcal{H}(\mathbb{C}^n, \varphi) \to \mathfrak{H}(W, \varphi)\) is surjective.NEWLINENEWLINEIt is clear that the uniform flatness, even in the smooth case, imposes very strong conditions on the global geometry of the hypersurface. The authors give also examples of hypersurfaces that do not satisfy the assumptions of their theorem, in particular without uniform flatness, but still with surjective restriction map.