Local extension of boundary holomorphic forms on real hypersurfaces in \(\mathbb C^{n}\)
type: Item
instance of: scholarly article
title: Local extension of boundary holomorphic forms on real hypersurfaces in \(\mathbb C^{n}\)
zbMATH Open document ID: 1110.32014
author: Jaeseo Choi
author: Sanghyun Cho
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: Let \(M\) be real hypersurface in \(\mathbb{C}^n\), with a smooth defining function \(\rho\); for \(p \in M\), a fixed point and \(U\) a neighbourhood of \(p\) in \(\mathbb{C}^n\) set \(U^- = \{z\in U \mid p(z)\leq 0\}\), \(U^+ = \{z \in U \mid p(z) \geq 0\}\). The main result of the authors is: if \(M\) is as above, \(n \geq 3\) and \(p \in M\), and if we suppose that the Levi form of \(\rho\) at \(p\) has \(q + 1\) positive eigenvalues \((1 \leq q \leq n - 2)\) then there is a neighbourhood \(U\) of \(p\) such that for any \( \alpha \in \Lambda^{0, r}(M \cap U)\), \(1 \leq r \leq q\) satisfying \(\overline\partial_{b}\alpha = 0\) on \(M \cap U\) there exists \(\widetilde \alpha \in \Lambda^{0, r} (U^-)\) such that \(\overline\partial \widetilde\alpha = 0\) in \(U^- \backslash M\) and \((\widetilde\alpha - \alpha) \wedge\overline\partial \rho = 0\) on \(M \cap U\). Here \(\Lambda^{0, r}(U^-)\) denotes the space of smooth \((p, q)\) forms on \(U^-\). Thus the local extension problem is solved generalizing a result of \textit{J. Xu} [Ph. D thesis, Purdue (1999)] which proved the local one sided \(\overline\partial\)-closed extension problem for \((0, 1)\) forms, except some bad set \(B \subset M\). The method of proof is based on the following result which has an independent interest. Namely, let \(\{\Omega_t\}_{t \in I}\) be a smooth family of strongly pseudoconvex domain in \(\mathbb{C}^{k_1}\) and \(N_t\) be the \(\overline\partial\)-Neumann operator in \(\Omega_t\), for each \(t \in I\), \(I\) being an open subset of \(\mathbb{C}^{k_2}\). Then \(N_t\) depends smoothly on \(t \in I\), and if \(u(z, t) = N_t \alpha_t(z)\) (which is proved to be a smooth form in \({\text{di}}\cup_{t \in I} \Omega_t\) which is open in \(\mathbb{C}^{k_1 + k_2}\)) it verifies \(\| u\| _{m(U \Omega_t)} \leq C_m \| \alpha\| _{3m} (U \Omega_t)\) provided \(\alpha \in H_{3m} (U \Omega_t)\) where the \(H_m\) are the Sobolev norms. This proof of this result is a slight modification of \texti…
Mathematics Subject Classification ID: 32W10
Mathematics Subject Classification ID: 32V25
zbMATH DE Number: 5078925
zbMATH Keywords: CR extension
zbMATH Keywords: \(\overline\partial_b\) problem
reviewed by: Gheorghe Gussi
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.01.036
OpenAlex ID: W2099722833
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DOI: 10.1016/J.JMAA.2006.01.036
description: scientific article; zbMATH DE number 5078925